Theorem aomclem8 40005

Description: Lemma for dfac11 40006. Perform variable substitutions. This is the most we can say without invoking regularity. (Contributed by Stefan O'Rear, 20-Jan-2015.)

Hypotheses

Ref	Expression
aomclem8.a	⊢ (𝜑 → 𝐴 ∈ On)
aomclem8.y	⊢ (𝜑 → ∀𝑎 ∈ 𝒫 (𝑅1‘𝐴)(𝑎 ≠ ∅ → (𝑦‘𝑎) ∈ ((𝒫 𝑎 ∩ Fin) ∖ {∅})))

Assertion

Ref	Expression
aomclem8	⊢ (𝜑 → ∃𝑏 𝑏 We (𝑅1‘𝐴))

Distinct variable groups:   𝜑,𝑏   𝐴,𝑎,𝑏   𝑦,𝑎,𝑏

Allowed substitution hints:   𝜑(𝑦,𝑎)   𝐴(𝑦)

Proof of Theorem aomclem8

Dummy variables 𝑐 𝑑 𝑒 𝑓 𝑔 ℎ 𝑖 𝑗 𝑙 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		elequ2 2126	. . . . . . 7 ⊢ (ℎ = 𝑏 → (𝑖 ∈ ℎ ↔ 𝑖 ∈ 𝑏))
2		elequ2 2126	. . . . . . . 8 ⊢ (𝑔 = 𝑐 → (𝑖 ∈ 𝑔 ↔ 𝑖 ∈ 𝑐))
3	2	notbid 321	. . . . . . 7 ⊢ (𝑔 = 𝑐 → (¬ 𝑖 ∈ 𝑔 ↔ ¬ 𝑖 ∈ 𝑐))
4	1, 3	bi2anan9r 639	. . . . . 6 ⊢ ((𝑔 = 𝑐 ∧ ℎ = 𝑏) → ((𝑖 ∈ ℎ ∧ ¬ 𝑖 ∈ 𝑔) ↔ (𝑖 ∈ 𝑏 ∧ ¬ 𝑖 ∈ 𝑐)))
5		elequ2 2126	. . . . . . . . 9 ⊢ (𝑔 = 𝑐 → (𝑗 ∈ 𝑔 ↔ 𝑗 ∈ 𝑐))
6		elequ2 2126	. . . . . . . . 9 ⊢ (ℎ = 𝑏 → (𝑗 ∈ ℎ ↔ 𝑗 ∈ 𝑏))
7	5, 6	bi2bian9 640	. . . . . . . 8 ⊢ ((𝑔 = 𝑐 ∧ ℎ = 𝑏) → ((𝑗 ∈ 𝑔 ↔ 𝑗 ∈ ℎ) ↔ (𝑗 ∈ 𝑐 ↔ 𝑗 ∈ 𝑏)))
8	7	imbi2d 344	. . . . . . 7 ⊢ ((𝑔 = 𝑐 ∧ ℎ = 𝑏) → ((𝑗(𝑒‘∪ dom 𝑒)𝑖 → (𝑗 ∈ 𝑔 ↔ 𝑗 ∈ ℎ)) ↔ (𝑗(𝑒‘∪ dom 𝑒)𝑖 → (𝑗 ∈ 𝑐 ↔ 𝑗 ∈ 𝑏))))
9	8	ralbidv 3162	. . . . . 6 ⊢ ((𝑔 = 𝑐 ∧ ℎ = 𝑏) → (∀𝑗 ∈ (𝑅1‘∪ dom 𝑒)(𝑗(𝑒‘∪ dom 𝑒)𝑖 → (𝑗 ∈ 𝑔 ↔ 𝑗 ∈ ℎ)) ↔ ∀𝑗 ∈ (𝑅1‘∪ dom 𝑒)(𝑗(𝑒‘∪ dom 𝑒)𝑖 → (𝑗 ∈ 𝑐 ↔ 𝑗 ∈ 𝑏))))
10	4, 9	anbi12d 633	. . . . 5 ⊢ ((𝑔 = 𝑐 ∧ ℎ = 𝑏) → (((𝑖 ∈ ℎ ∧ ¬ 𝑖 ∈ 𝑔) ∧ ∀𝑗 ∈ (𝑅1‘∪ dom 𝑒)(𝑗(𝑒‘∪ dom 𝑒)𝑖 → (𝑗 ∈ 𝑔 ↔ 𝑗 ∈ ℎ))) ↔ ((𝑖 ∈ 𝑏 ∧ ¬ 𝑖 ∈ 𝑐) ∧ ∀𝑗 ∈ (𝑅1‘∪ dom 𝑒)(𝑗(𝑒‘∪ dom 𝑒)𝑖 → (𝑗 ∈ 𝑐 ↔ 𝑗 ∈ 𝑏)))))
11	10	rexbidv 3256	. . . 4 ⊢ ((𝑔 = 𝑐 ∧ ℎ = 𝑏) → (∃𝑖 ∈ (𝑅1‘∪ dom 𝑒)((𝑖 ∈ ℎ ∧ ¬ 𝑖 ∈ 𝑔) ∧ ∀𝑗 ∈ (𝑅1‘∪ dom 𝑒)(𝑗(𝑒‘∪ dom 𝑒)𝑖 → (𝑗 ∈ 𝑔 ↔ 𝑗 ∈ ℎ))) ↔ ∃𝑖 ∈ (𝑅1‘∪ dom 𝑒)((𝑖 ∈ 𝑏 ∧ ¬ 𝑖 ∈ 𝑐) ∧ ∀𝑗 ∈ (𝑅1‘∪ dom 𝑒)(𝑗(𝑒‘∪ dom 𝑒)𝑖 → (𝑗 ∈ 𝑐 ↔ 𝑗 ∈ 𝑏)))))
12		elequ1 2118	. . . . . . 7 ⊢ (𝑖 = 𝑑 → (𝑖 ∈ 𝑏 ↔ 𝑑 ∈ 𝑏))
13		elequ1 2118	. . . . . . . 8 ⊢ (𝑖 = 𝑑 → (𝑖 ∈ 𝑐 ↔ 𝑑 ∈ 𝑐))
14	13	notbid 321	. . . . . . 7 ⊢ (𝑖 = 𝑑 → (¬ 𝑖 ∈ 𝑐 ↔ ¬ 𝑑 ∈ 𝑐))
15	12, 14	anbi12d 633	. . . . . 6 ⊢ (𝑖 = 𝑑 → ((𝑖 ∈ 𝑏 ∧ ¬ 𝑖 ∈ 𝑐) ↔ (𝑑 ∈ 𝑏 ∧ ¬ 𝑑 ∈ 𝑐)))
16		breq2 5034	. . . . . . . . 9 ⊢ (𝑖 = 𝑑 → (𝑗(𝑒‘∪ dom 𝑒)𝑖 ↔ 𝑗(𝑒‘∪ dom 𝑒)𝑑))
17	16	imbi1d 345	. . . . . . . 8 ⊢ (𝑖 = 𝑑 → ((𝑗(𝑒‘∪ dom 𝑒)𝑖 → (𝑗 ∈ 𝑐 ↔ 𝑗 ∈ 𝑏)) ↔ (𝑗(𝑒‘∪ dom 𝑒)𝑑 → (𝑗 ∈ 𝑐 ↔ 𝑗 ∈ 𝑏))))
18	17	ralbidv 3162	. . . . . . 7 ⊢ (𝑖 = 𝑑 → (∀𝑗 ∈ (𝑅1‘∪ dom 𝑒)(𝑗(𝑒‘∪ dom 𝑒)𝑖 → (𝑗 ∈ 𝑐 ↔ 𝑗 ∈ 𝑏)) ↔ ∀𝑗 ∈ (𝑅1‘∪ dom 𝑒)(𝑗(𝑒‘∪ dom 𝑒)𝑑 → (𝑗 ∈ 𝑐 ↔ 𝑗 ∈ 𝑏))))
19		breq1 5033	. . . . . . . . 9 ⊢ (𝑗 = 𝑓 → (𝑗(𝑒‘∪ dom 𝑒)𝑑 ↔ 𝑓(𝑒‘∪ dom 𝑒)𝑑))
20		elequ1 2118	. . . . . . . . . 10 ⊢ (𝑗 = 𝑓 → (𝑗 ∈ 𝑐 ↔ 𝑓 ∈ 𝑐))
21		elequ1 2118	. . . . . . . . . 10 ⊢ (𝑗 = 𝑓 → (𝑗 ∈ 𝑏 ↔ 𝑓 ∈ 𝑏))
22	20, 21	bibi12d 349	. . . . . . . . 9 ⊢ (𝑗 = 𝑓 → ((𝑗 ∈ 𝑐 ↔ 𝑗 ∈ 𝑏) ↔ (𝑓 ∈ 𝑐 ↔ 𝑓 ∈ 𝑏)))
23	19, 22	imbi12d 348	. . . . . . . 8 ⊢ (𝑗 = 𝑓 → ((𝑗(𝑒‘∪ dom 𝑒)𝑑 → (𝑗 ∈ 𝑐 ↔ 𝑗 ∈ 𝑏)) ↔ (𝑓(𝑒‘∪ dom 𝑒)𝑑 → (𝑓 ∈ 𝑐 ↔ 𝑓 ∈ 𝑏))))
24	23	cbvralvw 3396	. . . . . . 7 ⊢ (∀𝑗 ∈ (𝑅1‘∪ dom 𝑒)(𝑗(𝑒‘∪ dom 𝑒)𝑑 → (𝑗 ∈ 𝑐 ↔ 𝑗 ∈ 𝑏)) ↔ ∀𝑓 ∈ (𝑅1‘∪ dom 𝑒)(𝑓(𝑒‘∪ dom 𝑒)𝑑 → (𝑓 ∈ 𝑐 ↔ 𝑓 ∈ 𝑏)))
25	18, 24	syl6bb 290	. . . . . 6 ⊢ (𝑖 = 𝑑 → (∀𝑗 ∈ (𝑅1‘∪ dom 𝑒)(𝑗(𝑒‘∪ dom 𝑒)𝑖 → (𝑗 ∈ 𝑐 ↔ 𝑗 ∈ 𝑏)) ↔ ∀𝑓 ∈ (𝑅1‘∪ dom 𝑒)(𝑓(𝑒‘∪ dom 𝑒)𝑑 → (𝑓 ∈ 𝑐 ↔ 𝑓 ∈ 𝑏))))
26	15, 25	anbi12d 633	. . . . 5 ⊢ (𝑖 = 𝑑 → (((𝑖 ∈ 𝑏 ∧ ¬ 𝑖 ∈ 𝑐) ∧ ∀𝑗 ∈ (𝑅1‘∪ dom 𝑒)(𝑗(𝑒‘∪ dom 𝑒)𝑖 → (𝑗 ∈ 𝑐 ↔ 𝑗 ∈ 𝑏))) ↔ ((𝑑 ∈ 𝑏 ∧ ¬ 𝑑 ∈ 𝑐) ∧ ∀𝑓 ∈ (𝑅1‘∪ dom 𝑒)(𝑓(𝑒‘∪ dom 𝑒)𝑑 → (𝑓 ∈ 𝑐 ↔ 𝑓 ∈ 𝑏)))))
27	26	cbvrexvw 3397	. . . 4 ⊢ (∃𝑖 ∈ (𝑅1‘∪ dom 𝑒)((𝑖 ∈ 𝑏 ∧ ¬ 𝑖 ∈ 𝑐) ∧ ∀𝑗 ∈ (𝑅1‘∪ dom 𝑒)(𝑗(𝑒‘∪ dom 𝑒)𝑖 → (𝑗 ∈ 𝑐 ↔ 𝑗 ∈ 𝑏))) ↔ ∃𝑑 ∈ (𝑅1‘∪ dom 𝑒)((𝑑 ∈ 𝑏 ∧ ¬ 𝑑 ∈ 𝑐) ∧ ∀𝑓 ∈ (𝑅1‘∪ dom 𝑒)(𝑓(𝑒‘∪ dom 𝑒)𝑑 → (𝑓 ∈ 𝑐 ↔ 𝑓 ∈ 𝑏))))
28	11, 27	syl6bb 290	. . 3 ⊢ ((𝑔 = 𝑐 ∧ ℎ = 𝑏) → (∃𝑖 ∈ (𝑅1‘∪ dom 𝑒)((𝑖 ∈ ℎ ∧ ¬ 𝑖 ∈ 𝑔) ∧ ∀𝑗 ∈ (𝑅1‘∪ dom 𝑒)(𝑗(𝑒‘∪ dom 𝑒)𝑖 → (𝑗 ∈ 𝑔 ↔ 𝑗 ∈ ℎ))) ↔ ∃𝑑 ∈ (𝑅1‘∪ dom 𝑒)((𝑑 ∈ 𝑏 ∧ ¬ 𝑑 ∈ 𝑐) ∧ ∀𝑓 ∈ (𝑅1‘∪ dom 𝑒)(𝑓(𝑒‘∪ dom 𝑒)𝑑 → (𝑓 ∈ 𝑐 ↔ 𝑓 ∈ 𝑏)))))
29	28	cbvopabv 5102	. 2 ⊢ {⟨𝑔, ℎ⟩ ∣ ∃𝑖 ∈ (𝑅1‘∪ dom 𝑒)((𝑖 ∈ ℎ ∧ ¬ 𝑖 ∈ 𝑔) ∧ ∀𝑗 ∈ (𝑅1‘∪ dom 𝑒)(𝑗(𝑒‘∪ dom 𝑒)𝑖 → (𝑗 ∈ 𝑔 ↔ 𝑗 ∈ ℎ)))} = {⟨𝑐, 𝑏⟩ ∣ ∃𝑑 ∈ (𝑅1‘∪ dom 𝑒)((𝑑 ∈ 𝑏 ∧ ¬ 𝑑 ∈ 𝑐) ∧ ∀𝑓 ∈ (𝑅1‘∪ dom 𝑒)(𝑓(𝑒‘∪ dom 𝑒)𝑑 → (𝑓 ∈ 𝑐 ↔ 𝑓 ∈ 𝑏)))}
30		nfcv 2955	. . 3 ⊢ Ⅎ𝑐sup((𝑦‘𝑔), (𝑅1‘dom 𝑒), {⟨𝑔, ℎ⟩ ∣ ∃𝑖 ∈ (𝑅1‘∪ dom 𝑒)((𝑖 ∈ ℎ ∧ ¬ 𝑖 ∈ 𝑔) ∧ ∀𝑗 ∈ (𝑅1‘∪ dom 𝑒)(𝑗(𝑒‘∪ dom 𝑒)𝑖 → (𝑗 ∈ 𝑔 ↔ 𝑗 ∈ ℎ)))})
31		nfcv 2955	. . . 4 ⊢ Ⅎ𝑔(𝑦‘𝑐)
32		nfcv 2955	. . . 4 ⊢ Ⅎ𝑔(𝑅1‘dom 𝑒)
33		nfopab1 5099	. . . 4 ⊢ Ⅎ𝑔{⟨𝑔, ℎ⟩ ∣ ∃𝑖 ∈ (𝑅1‘∪ dom 𝑒)((𝑖 ∈ ℎ ∧ ¬ 𝑖 ∈ 𝑔) ∧ ∀𝑗 ∈ (𝑅1‘∪ dom 𝑒)(𝑗(𝑒‘∪ dom 𝑒)𝑖 → (𝑗 ∈ 𝑔 ↔ 𝑗 ∈ ℎ)))}
34	31, 32, 33	nfsup 8899	. . 3 ⊢ Ⅎ𝑔sup((𝑦‘𝑐), (𝑅1‘dom 𝑒), {⟨𝑔, ℎ⟩ ∣ ∃𝑖 ∈ (𝑅1‘∪ dom 𝑒)((𝑖 ∈ ℎ ∧ ¬ 𝑖 ∈ 𝑔) ∧ ∀𝑗 ∈ (𝑅1‘∪ dom 𝑒)(𝑗(𝑒‘∪ dom 𝑒)𝑖 → (𝑗 ∈ 𝑔 ↔ 𝑗 ∈ ℎ)))})
35		fveq2 6645	. . . 4 ⊢ (𝑔 = 𝑐 → (𝑦‘𝑔) = (𝑦‘𝑐))
36	35	supeq1d 8894	. . 3 ⊢ (𝑔 = 𝑐 → sup((𝑦‘𝑔), (𝑅1‘dom 𝑒), {⟨𝑔, ℎ⟩ ∣ ∃𝑖 ∈ (𝑅1‘∪ dom 𝑒)((𝑖 ∈ ℎ ∧ ¬ 𝑖 ∈ 𝑔) ∧ ∀𝑗 ∈ (𝑅1‘∪ dom 𝑒)(𝑗(𝑒‘∪ dom 𝑒)𝑖 → (𝑗 ∈ 𝑔 ↔ 𝑗 ∈ ℎ)))}) = sup((𝑦‘𝑐), (𝑅1‘dom 𝑒), {⟨𝑔, ℎ⟩ ∣ ∃𝑖 ∈ (𝑅1‘∪ dom 𝑒)((𝑖 ∈ ℎ ∧ ¬ 𝑖 ∈ 𝑔) ∧ ∀𝑗 ∈ (𝑅1‘∪ dom 𝑒)(𝑗(𝑒‘∪ dom 𝑒)𝑖 → (𝑗 ∈ 𝑔 ↔ 𝑗 ∈ ℎ)))}))
37	30, 34, 36	cbvmpt 5131	. 2 ⊢ (𝑔 ∈ V ↦ sup((𝑦‘𝑔), (𝑅1‘dom 𝑒), {⟨𝑔, ℎ⟩ ∣ ∃𝑖 ∈ (𝑅1‘∪ dom 𝑒)((𝑖 ∈ ℎ ∧ ¬ 𝑖 ∈ 𝑔) ∧ ∀𝑗 ∈ (𝑅1‘∪ dom 𝑒)(𝑗(𝑒‘∪ dom 𝑒)𝑖 → (𝑗 ∈ 𝑔 ↔ 𝑗 ∈ ℎ)))})) = (𝑐 ∈ V ↦ sup((𝑦‘𝑐), (𝑅1‘dom 𝑒), {⟨𝑔, ℎ⟩ ∣ ∃𝑖 ∈ (𝑅1‘∪ dom 𝑒)((𝑖 ∈ ℎ ∧ ¬ 𝑖 ∈ 𝑔) ∧ ∀𝑗 ∈ (𝑅1‘∪ dom 𝑒)(𝑗(𝑒‘∪ dom 𝑒)𝑖 → (𝑗 ∈ 𝑔 ↔ 𝑗 ∈ ℎ)))}))
38		nfcv 2955	. . . 4 ⊢ Ⅎ𝑐((𝑔 ∈ V ↦ sup((𝑦‘𝑔), (𝑅1‘dom 𝑒), {⟨𝑔, ℎ⟩ ∣ ∃𝑖 ∈ (𝑅1‘∪ dom 𝑒)((𝑖 ∈ ℎ ∧ ¬ 𝑖 ∈ 𝑔) ∧ ∀𝑗 ∈ (𝑅1‘∪ dom 𝑒)(𝑗(𝑒‘∪ dom 𝑒)𝑖 → (𝑗 ∈ 𝑔 ↔ 𝑗 ∈ ℎ)))}))‘((𝑅1‘dom 𝑒) ∖ ran 𝑔))
39		nffvmpt1 6656	. . . 4 ⊢ Ⅎ𝑔((𝑔 ∈ V ↦ sup((𝑦‘𝑔), (𝑅1‘dom 𝑒), {⟨𝑔, ℎ⟩ ∣ ∃𝑖 ∈ (𝑅1‘∪ dom 𝑒)((𝑖 ∈ ℎ ∧ ¬ 𝑖 ∈ 𝑔) ∧ ∀𝑗 ∈ (𝑅1‘∪ dom 𝑒)(𝑗(𝑒‘∪ dom 𝑒)𝑖 → (𝑗 ∈ 𝑔 ↔ 𝑗 ∈ ℎ)))}))‘((𝑅1‘dom 𝑒) ∖ ran 𝑐))
40		rneq 5770	. . . . . 6 ⊢ (𝑔 = 𝑐 → ran 𝑔 = ran 𝑐)
41	40	difeq2d 4050	. . . . 5 ⊢ (𝑔 = 𝑐 → ((𝑅1‘dom 𝑒) ∖ ran 𝑔) = ((𝑅1‘dom 𝑒) ∖ ran 𝑐))
42	41	fveq2d 6649	. . . 4 ⊢ (𝑔 = 𝑐 → ((𝑔 ∈ V ↦ sup((𝑦‘𝑔), (𝑅1‘dom 𝑒), {⟨𝑔, ℎ⟩ ∣ ∃𝑖 ∈ (𝑅1‘∪ dom 𝑒)((𝑖 ∈ ℎ ∧ ¬ 𝑖 ∈ 𝑔) ∧ ∀𝑗 ∈ (𝑅1‘∪ dom 𝑒)(𝑗(𝑒‘∪ dom 𝑒)𝑖 → (𝑗 ∈ 𝑔 ↔ 𝑗 ∈ ℎ)))}))‘((𝑅1‘dom 𝑒) ∖ ran 𝑔)) = ((𝑔 ∈ V ↦ sup((𝑦‘𝑔), (𝑅1‘dom 𝑒), {⟨𝑔, ℎ⟩ ∣ ∃𝑖 ∈ (𝑅1‘∪ dom 𝑒)((𝑖 ∈ ℎ ∧ ¬ 𝑖 ∈ 𝑔) ∧ ∀𝑗 ∈ (𝑅1‘∪ dom 𝑒)(𝑗(𝑒‘∪ dom 𝑒)𝑖 → (𝑗 ∈ 𝑔 ↔ 𝑗 ∈ ℎ)))}))‘((𝑅1‘dom 𝑒) ∖ ran 𝑐)))
43	38, 39, 42	cbvmpt 5131	. . 3 ⊢ (𝑔 ∈ V ↦ ((𝑔 ∈ V ↦ sup((𝑦‘𝑔), (𝑅1‘dom 𝑒), {⟨𝑔, ℎ⟩ ∣ ∃𝑖 ∈ (𝑅1‘∪ dom 𝑒)((𝑖 ∈ ℎ ∧ ¬ 𝑖 ∈ 𝑔) ∧ ∀𝑗 ∈ (𝑅1‘∪ dom 𝑒)(𝑗(𝑒‘∪ dom 𝑒)𝑖 → (𝑗 ∈ 𝑔 ↔ 𝑗 ∈ ℎ)))}))‘((𝑅1‘dom 𝑒) ∖ ran 𝑔))) = (𝑐 ∈ V ↦ ((𝑔 ∈ V ↦ sup((𝑦‘𝑔), (𝑅1‘dom 𝑒), {⟨𝑔, ℎ⟩ ∣ ∃𝑖 ∈ (𝑅1‘∪ dom 𝑒)((𝑖 ∈ ℎ ∧ ¬ 𝑖 ∈ 𝑔) ∧ ∀𝑗 ∈ (𝑅1‘∪ dom 𝑒)(𝑗(𝑒‘∪ dom 𝑒)𝑖 → (𝑗 ∈ 𝑔 ↔ 𝑗 ∈ ℎ)))}))‘((𝑅1‘dom 𝑒) ∖ ran 𝑐)))
44		recseq 7993	. . 3 ⊢ ((𝑔 ∈ V ↦ ((𝑔 ∈ V ↦ sup((𝑦‘𝑔), (𝑅1‘dom 𝑒), {⟨𝑔, ℎ⟩ ∣ ∃𝑖 ∈ (𝑅1‘∪ dom 𝑒)((𝑖 ∈ ℎ ∧ ¬ 𝑖 ∈ 𝑔) ∧ ∀𝑗 ∈ (𝑅1‘∪ dom 𝑒)(𝑗(𝑒‘∪ dom 𝑒)𝑖 → (𝑗 ∈ 𝑔 ↔ 𝑗 ∈ ℎ)))}))‘((𝑅1‘dom 𝑒) ∖ ran 𝑔))) = (𝑐 ∈ V ↦ ((𝑔 ∈ V ↦ sup((𝑦‘𝑔), (𝑅1‘dom 𝑒), {⟨𝑔, ℎ⟩ ∣ ∃𝑖 ∈ (𝑅1‘∪ dom 𝑒)((𝑖 ∈ ℎ ∧ ¬ 𝑖 ∈ 𝑔) ∧ ∀𝑗 ∈ (𝑅1‘∪ dom 𝑒)(𝑗(𝑒‘∪ dom 𝑒)𝑖 → (𝑗 ∈ 𝑔 ↔ 𝑗 ∈ ℎ)))}))‘((𝑅1‘dom 𝑒) ∖ ran 𝑐))) → recs((𝑔 ∈ V ↦ ((𝑔 ∈ V ↦ sup((𝑦‘𝑔), (𝑅1‘dom 𝑒), {⟨𝑔, ℎ⟩ ∣ ∃𝑖 ∈ (𝑅1‘∪ dom 𝑒)((𝑖 ∈ ℎ ∧ ¬ 𝑖 ∈ 𝑔) ∧ ∀𝑗 ∈ (𝑅1‘∪ dom 𝑒)(𝑗(𝑒‘∪ dom 𝑒)𝑖 → (𝑗 ∈ 𝑔 ↔ 𝑗 ∈ ℎ)))}))‘((𝑅1‘dom 𝑒) ∖ ran 𝑔)))) = recs((𝑐 ∈ V ↦ ((𝑔 ∈ V ↦ sup((𝑦‘𝑔), (𝑅1‘dom 𝑒), {⟨𝑔, ℎ⟩ ∣ ∃𝑖 ∈ (𝑅1‘∪ dom 𝑒)((𝑖 ∈ ℎ ∧ ¬ 𝑖 ∈ 𝑔) ∧ ∀𝑗 ∈ (𝑅1‘∪ dom 𝑒)(𝑗(𝑒‘∪ dom 𝑒)𝑖 → (𝑗 ∈ 𝑔 ↔ 𝑗 ∈ ℎ)))}))‘((𝑅1‘dom 𝑒) ∖ ran 𝑐)))))
45	43, 44	ax-mp 5	. 2 ⊢ recs((𝑔 ∈ V ↦ ((𝑔 ∈ V ↦ sup((𝑦‘𝑔), (𝑅1‘dom 𝑒), {⟨𝑔, ℎ⟩ ∣ ∃𝑖 ∈ (𝑅1‘∪ dom 𝑒)((𝑖 ∈ ℎ ∧ ¬ 𝑖 ∈ 𝑔) ∧ ∀𝑗 ∈ (𝑅1‘∪ dom 𝑒)(𝑗(𝑒‘∪ dom 𝑒)𝑖 → (𝑗 ∈ 𝑔 ↔ 𝑗 ∈ ℎ)))}))‘((𝑅1‘dom 𝑒) ∖ ran 𝑔)))) = recs((𝑐 ∈ V ↦ ((𝑔 ∈ V ↦ sup((𝑦‘𝑔), (𝑅1‘dom 𝑒), {⟨𝑔, ℎ⟩ ∣ ∃𝑖 ∈ (𝑅1‘∪ dom 𝑒)((𝑖 ∈ ℎ ∧ ¬ 𝑖 ∈ 𝑔) ∧ ∀𝑗 ∈ (𝑅1‘∪ dom 𝑒)(𝑗(𝑒‘∪ dom 𝑒)𝑖 → (𝑗 ∈ 𝑔 ↔ 𝑗 ∈ ℎ)))}))‘((𝑅1‘dom 𝑒) ∖ ran 𝑐))))
46		nfv 1915	. . 3 ⊢ Ⅎ𝑐∩ (◡recs((𝑔 ∈ V ↦ ((𝑔 ∈ V ↦ sup((𝑦‘𝑔), (𝑅1‘dom 𝑒), {⟨𝑔, ℎ⟩ ∣ ∃𝑖 ∈ (𝑅1‘∪ dom 𝑒)((𝑖 ∈ ℎ ∧ ¬ 𝑖 ∈ 𝑔) ∧ ∀𝑗 ∈ (𝑅1‘∪ dom 𝑒)(𝑗(𝑒‘∪ dom 𝑒)𝑖 → (𝑗 ∈ 𝑔 ↔ 𝑗 ∈ ℎ)))}))‘((𝑅1‘dom 𝑒) ∖ ran 𝑔)))) “ {𝑔}) ∈ ∩ (◡recs((𝑔 ∈ V ↦ ((𝑔 ∈ V ↦ sup((𝑦‘𝑔), (𝑅1‘dom 𝑒), {⟨𝑔, ℎ⟩ ∣ ∃𝑖 ∈ (𝑅1‘∪ dom 𝑒)((𝑖 ∈ ℎ ∧ ¬ 𝑖 ∈ 𝑔) ∧ ∀𝑗 ∈ (𝑅1‘∪ dom 𝑒)(𝑗(𝑒‘∪ dom 𝑒)𝑖 → (𝑗 ∈ 𝑔 ↔ 𝑗 ∈ ℎ)))}))‘((𝑅1‘dom 𝑒) ∖ ran 𝑔)))) “ {ℎ})
47		nfv 1915	. . 3 ⊢ Ⅎ𝑏∩ (◡recs((𝑔 ∈ V ↦ ((𝑔 ∈ V ↦ sup((𝑦‘𝑔), (𝑅1‘dom 𝑒), {⟨𝑔, ℎ⟩ ∣ ∃𝑖 ∈ (𝑅1‘∪ dom 𝑒)((𝑖 ∈ ℎ ∧ ¬ 𝑖 ∈ 𝑔) ∧ ∀𝑗 ∈ (𝑅1‘∪ dom 𝑒)(𝑗(𝑒‘∪ dom 𝑒)𝑖 → (𝑗 ∈ 𝑔 ↔ 𝑗 ∈ ℎ)))}))‘((𝑅1‘dom 𝑒) ∖ ran 𝑔)))) “ {𝑔}) ∈ ∩ (◡recs((𝑔 ∈ V ↦ ((𝑔 ∈ V ↦ sup((𝑦‘𝑔), (𝑅1‘dom 𝑒), {⟨𝑔, ℎ⟩ ∣ ∃𝑖 ∈ (𝑅1‘∪ dom 𝑒)((𝑖 ∈ ℎ ∧ ¬ 𝑖 ∈ 𝑔) ∧ ∀𝑗 ∈ (𝑅1‘∪ dom 𝑒)(𝑗(𝑒‘∪ dom 𝑒)𝑖 → (𝑗 ∈ 𝑔 ↔ 𝑗 ∈ ℎ)))}))‘((𝑅1‘dom 𝑒) ∖ ran 𝑔)))) “ {ℎ})
48		nfmpt1 5128	. . . . . . . 8 ⊢ Ⅎ𝑔(𝑔 ∈ V ↦ ((𝑔 ∈ V ↦ sup((𝑦‘𝑔), (𝑅1‘dom 𝑒), {⟨𝑔, ℎ⟩ ∣ ∃𝑖 ∈ (𝑅1‘∪ dom 𝑒)((𝑖 ∈ ℎ ∧ ¬ 𝑖 ∈ 𝑔) ∧ ∀𝑗 ∈ (𝑅1‘∪ dom 𝑒)(𝑗(𝑒‘∪ dom 𝑒)𝑖 → (𝑗 ∈ 𝑔 ↔ 𝑗 ∈ ℎ)))}))‘((𝑅1‘dom 𝑒) ∖ ran 𝑔)))
49	48	nfrecs 7994	. . . . . . 7 ⊢ Ⅎ𝑔recs((𝑔 ∈ V ↦ ((𝑔 ∈ V ↦ sup((𝑦‘𝑔), (𝑅1‘dom 𝑒), {⟨𝑔, ℎ⟩ ∣ ∃𝑖 ∈ (𝑅1‘∪ dom 𝑒)((𝑖 ∈ ℎ ∧ ¬ 𝑖 ∈ 𝑔) ∧ ∀𝑗 ∈ (𝑅1‘∪ dom 𝑒)(𝑗(𝑒‘∪ dom 𝑒)𝑖 → (𝑗 ∈ 𝑔 ↔ 𝑗 ∈ ℎ)))}))‘((𝑅1‘dom 𝑒) ∖ ran 𝑔))))
50	49	nfcnv 5713	. . . . . 6 ⊢ Ⅎ𝑔◡recs((𝑔 ∈ V ↦ ((𝑔 ∈ V ↦ sup((𝑦‘𝑔), (𝑅1‘dom 𝑒), {⟨𝑔, ℎ⟩ ∣ ∃𝑖 ∈ (𝑅1‘∪ dom 𝑒)((𝑖 ∈ ℎ ∧ ¬ 𝑖 ∈ 𝑔) ∧ ∀𝑗 ∈ (𝑅1‘∪ dom 𝑒)(𝑗(𝑒‘∪ dom 𝑒)𝑖 → (𝑗 ∈ 𝑔 ↔ 𝑗 ∈ ℎ)))}))‘((𝑅1‘dom 𝑒) ∖ ran 𝑔))))
51		nfcv 2955	. . . . . 6 ⊢ Ⅎ𝑔{𝑐}
52	50, 51	nfima 5904	. . . . 5 ⊢ Ⅎ𝑔(◡recs((𝑔 ∈ V ↦ ((𝑔 ∈ V ↦ sup((𝑦‘𝑔), (𝑅1‘dom 𝑒), {⟨𝑔, ℎ⟩ ∣ ∃𝑖 ∈ (𝑅1‘∪ dom 𝑒)((𝑖 ∈ ℎ ∧ ¬ 𝑖 ∈ 𝑔) ∧ ∀𝑗 ∈ (𝑅1‘∪ dom 𝑒)(𝑗(𝑒‘∪ dom 𝑒)𝑖 → (𝑗 ∈ 𝑔 ↔ 𝑗 ∈ ℎ)))}))‘((𝑅1‘dom 𝑒) ∖ ran 𝑔)))) “ {𝑐})
53	52	nfint 4848	. . . 4 ⊢ Ⅎ𝑔∩ (◡recs((𝑔 ∈ V ↦ ((𝑔 ∈ V ↦ sup((𝑦‘𝑔), (𝑅1‘dom 𝑒), {⟨𝑔, ℎ⟩ ∣ ∃𝑖 ∈ (𝑅1‘∪ dom 𝑒)((𝑖 ∈ ℎ ∧ ¬ 𝑖 ∈ 𝑔) ∧ ∀𝑗 ∈ (𝑅1‘∪ dom 𝑒)(𝑗(𝑒‘∪ dom 𝑒)𝑖 → (𝑗 ∈ 𝑔 ↔ 𝑗 ∈ ℎ)))}))‘((𝑅1‘dom 𝑒) ∖ ran 𝑔)))) “ {𝑐})
54		nfcv 2955	. . . . . 6 ⊢ Ⅎ𝑔{𝑏}
55	50, 54	nfima 5904	. . . . 5 ⊢ Ⅎ𝑔(◡recs((𝑔 ∈ V ↦ ((𝑔 ∈ V ↦ sup((𝑦‘𝑔), (𝑅1‘dom 𝑒), {⟨𝑔, ℎ⟩ ∣ ∃𝑖 ∈ (𝑅1‘∪ dom 𝑒)((𝑖 ∈ ℎ ∧ ¬ 𝑖 ∈ 𝑔) ∧ ∀𝑗 ∈ (𝑅1‘∪ dom 𝑒)(𝑗(𝑒‘∪ dom 𝑒)𝑖 → (𝑗 ∈ 𝑔 ↔ 𝑗 ∈ ℎ)))}))‘((𝑅1‘dom 𝑒) ∖ ran 𝑔)))) “ {𝑏})
56	55	nfint 4848	. . . 4 ⊢ Ⅎ𝑔∩ (◡recs((𝑔 ∈ V ↦ ((𝑔 ∈ V ↦ sup((𝑦‘𝑔), (𝑅1‘dom 𝑒), {⟨𝑔, ℎ⟩ ∣ ∃𝑖 ∈ (𝑅1‘∪ dom 𝑒)((𝑖 ∈ ℎ ∧ ¬ 𝑖 ∈ 𝑔) ∧ ∀𝑗 ∈ (𝑅1‘∪ dom 𝑒)(𝑗(𝑒‘∪ dom 𝑒)𝑖 → (𝑗 ∈ 𝑔 ↔ 𝑗 ∈ ℎ)))}))‘((𝑅1‘dom 𝑒) ∖ ran 𝑔)))) “ {𝑏})
57	53, 56	nfel 2969	. . 3 ⊢ Ⅎ𝑔∩ (◡recs((𝑔 ∈ V ↦ ((𝑔 ∈ V ↦ sup((𝑦‘𝑔), (𝑅1‘dom 𝑒), {⟨𝑔, ℎ⟩ ∣ ∃𝑖 ∈ (𝑅1‘∪ dom 𝑒)((𝑖 ∈ ℎ ∧ ¬ 𝑖 ∈ 𝑔) ∧ ∀𝑗 ∈ (𝑅1‘∪ dom 𝑒)(𝑗(𝑒‘∪ dom 𝑒)𝑖 → (𝑗 ∈ 𝑔 ↔ 𝑗 ∈ ℎ)))}))‘((𝑅1‘dom 𝑒) ∖ ran 𝑔)))) “ {𝑐}) ∈ ∩ (◡recs((𝑔 ∈ V ↦ ((𝑔 ∈ V ↦ sup((𝑦‘𝑔), (𝑅1‘dom 𝑒), {⟨𝑔, ℎ⟩ ∣ ∃𝑖 ∈ (𝑅1‘∪ dom 𝑒)((𝑖 ∈ ℎ ∧ ¬ 𝑖 ∈ 𝑔) ∧ ∀𝑗 ∈ (𝑅1‘∪ dom 𝑒)(𝑗(𝑒‘∪ dom 𝑒)𝑖 → (𝑗 ∈ 𝑔 ↔ 𝑗 ∈ ℎ)))}))‘((𝑅1‘dom 𝑒) ∖ ran 𝑔)))) “ {𝑏})
58		nfcv 2955	. . . . . . . . 9 ⊢ ℲℎV
59		nfcv 2955	. . . . . . . . . . . 12 ⊢ Ⅎℎ(𝑦‘𝑔)
60		nfcv 2955	. . . . . . . . . . . 12 ⊢ Ⅎℎ(𝑅1‘dom 𝑒)
61		nfopab2 5100	. . . . . . . . . . . 12 ⊢ Ⅎℎ{⟨𝑔, ℎ⟩ ∣ ∃𝑖 ∈ (𝑅1‘∪ dom 𝑒)((𝑖 ∈ ℎ ∧ ¬ 𝑖 ∈ 𝑔) ∧ ∀𝑗 ∈ (𝑅1‘∪ dom 𝑒)(𝑗(𝑒‘∪ dom 𝑒)𝑖 → (𝑗 ∈ 𝑔 ↔ 𝑗 ∈ ℎ)))}
62	59, 60, 61	nfsup 8899	. . . . . . . . . . 11 ⊢ Ⅎℎsup((𝑦‘𝑔), (𝑅1‘dom 𝑒), {⟨𝑔, ℎ⟩ ∣ ∃𝑖 ∈ (𝑅1‘∪ dom 𝑒)((𝑖 ∈ ℎ ∧ ¬ 𝑖 ∈ 𝑔) ∧ ∀𝑗 ∈ (𝑅1‘∪ dom 𝑒)(𝑗(𝑒‘∪ dom 𝑒)𝑖 → (𝑗 ∈ 𝑔 ↔ 𝑗 ∈ ℎ)))})
63	58, 62	nfmpt 5127	. . . . . . . . . 10 ⊢ Ⅎℎ(𝑔 ∈ V ↦ sup((𝑦‘𝑔), (𝑅1‘dom 𝑒), {⟨𝑔, ℎ⟩ ∣ ∃𝑖 ∈ (𝑅1‘∪ dom 𝑒)((𝑖 ∈ ℎ ∧ ¬ 𝑖 ∈ 𝑔) ∧ ∀𝑗 ∈ (𝑅1‘∪ dom 𝑒)(𝑗(𝑒‘∪ dom 𝑒)𝑖 → (𝑗 ∈ 𝑔 ↔ 𝑗 ∈ ℎ)))}))
64		nfcv 2955	. . . . . . . . . 10 ⊢ Ⅎℎ((𝑅1‘dom 𝑒) ∖ ran 𝑔)
65	63, 64	nffv 6655	. . . . . . . . 9 ⊢ Ⅎℎ((𝑔 ∈ V ↦ sup((𝑦‘𝑔), (𝑅1‘dom 𝑒), {⟨𝑔, ℎ⟩ ∣ ∃𝑖 ∈ (𝑅1‘∪ dom 𝑒)((𝑖 ∈ ℎ ∧ ¬ 𝑖 ∈ 𝑔) ∧ ∀𝑗 ∈ (𝑅1‘∪ dom 𝑒)(𝑗(𝑒‘∪ dom 𝑒)𝑖 → (𝑗 ∈ 𝑔 ↔ 𝑗 ∈ ℎ)))}))‘((𝑅1‘dom 𝑒) ∖ ran 𝑔))
66	58, 65	nfmpt 5127	. . . . . . . 8 ⊢ Ⅎℎ(𝑔 ∈ V ↦ ((𝑔 ∈ V ↦ sup((𝑦‘𝑔), (𝑅1‘dom 𝑒), {⟨𝑔, ℎ⟩ ∣ ∃𝑖 ∈ (𝑅1‘∪ dom 𝑒)((𝑖 ∈ ℎ ∧ ¬ 𝑖 ∈ 𝑔) ∧ ∀𝑗 ∈ (𝑅1‘∪ dom 𝑒)(𝑗(𝑒‘∪ dom 𝑒)𝑖 → (𝑗 ∈ 𝑔 ↔ 𝑗 ∈ ℎ)))}))‘((𝑅1‘dom 𝑒) ∖ ran 𝑔)))
67	66	nfrecs 7994	. . . . . . 7 ⊢ Ⅎℎrecs((𝑔 ∈ V ↦ ((𝑔 ∈ V ↦ sup((𝑦‘𝑔), (𝑅1‘dom 𝑒), {⟨𝑔, ℎ⟩ ∣ ∃𝑖 ∈ (𝑅1‘∪ dom 𝑒)((𝑖 ∈ ℎ ∧ ¬ 𝑖 ∈ 𝑔) ∧ ∀𝑗 ∈ (𝑅1‘∪ dom 𝑒)(𝑗(𝑒‘∪ dom 𝑒)𝑖 → (𝑗 ∈ 𝑔 ↔ 𝑗 ∈ ℎ)))}))‘((𝑅1‘dom 𝑒) ∖ ran 𝑔))))
68	67	nfcnv 5713	. . . . . 6 ⊢ Ⅎℎ◡recs((𝑔 ∈ V ↦ ((𝑔 ∈ V ↦ sup((𝑦‘𝑔), (𝑅1‘dom 𝑒), {⟨𝑔, ℎ⟩ ∣ ∃𝑖 ∈ (𝑅1‘∪ dom 𝑒)((𝑖 ∈ ℎ ∧ ¬ 𝑖 ∈ 𝑔) ∧ ∀𝑗 ∈ (𝑅1‘∪ dom 𝑒)(𝑗(𝑒‘∪ dom 𝑒)𝑖 → (𝑗 ∈ 𝑔 ↔ 𝑗 ∈ ℎ)))}))‘((𝑅1‘dom 𝑒) ∖ ran 𝑔))))
69		nfcv 2955	. . . . . 6 ⊢ Ⅎℎ{𝑐}
70	68, 69	nfima 5904	. . . . 5 ⊢ Ⅎℎ(◡recs((𝑔 ∈ V ↦ ((𝑔 ∈ V ↦ sup((𝑦‘𝑔), (𝑅1‘dom 𝑒), {⟨𝑔, ℎ⟩ ∣ ∃𝑖 ∈ (𝑅1‘∪ dom 𝑒)((𝑖 ∈ ℎ ∧ ¬ 𝑖 ∈ 𝑔) ∧ ∀𝑗 ∈ (𝑅1‘∪ dom 𝑒)(𝑗(𝑒‘∪ dom 𝑒)𝑖 → (𝑗 ∈ 𝑔 ↔ 𝑗 ∈ ℎ)))}))‘((𝑅1‘dom 𝑒) ∖ ran 𝑔)))) “ {𝑐})
71	70	nfint 4848	. . . 4 ⊢ Ⅎℎ∩ (◡recs((𝑔 ∈ V ↦ ((𝑔 ∈ V ↦ sup((𝑦‘𝑔), (𝑅1‘dom 𝑒), {⟨𝑔, ℎ⟩ ∣ ∃𝑖 ∈ (𝑅1‘∪ dom 𝑒)((𝑖 ∈ ℎ ∧ ¬ 𝑖 ∈ 𝑔) ∧ ∀𝑗 ∈ (𝑅1‘∪ dom 𝑒)(𝑗(𝑒‘∪ dom 𝑒)𝑖 → (𝑗 ∈ 𝑔 ↔ 𝑗 ∈ ℎ)))}))‘((𝑅1‘dom 𝑒) ∖ ran 𝑔)))) “ {𝑐})
72		nfcv 2955	. . . . . 6 ⊢ Ⅎℎ{𝑏}
73	68, 72	nfima 5904	. . . . 5 ⊢ Ⅎℎ(◡recs((𝑔 ∈ V ↦ ((𝑔 ∈ V ↦ sup((𝑦‘𝑔), (𝑅1‘dom 𝑒), {⟨𝑔, ℎ⟩ ∣ ∃𝑖 ∈ (𝑅1‘∪ dom 𝑒)((𝑖 ∈ ℎ ∧ ¬ 𝑖 ∈ 𝑔) ∧ ∀𝑗 ∈ (𝑅1‘∪ dom 𝑒)(𝑗(𝑒‘∪ dom 𝑒)𝑖 → (𝑗 ∈ 𝑔 ↔ 𝑗 ∈ ℎ)))}))‘((𝑅1‘dom 𝑒) ∖ ran 𝑔)))) “ {𝑏})
74	73	nfint 4848	. . . 4 ⊢ Ⅎℎ∩ (◡recs((𝑔 ∈ V ↦ ((𝑔 ∈ V ↦ sup((𝑦‘𝑔), (𝑅1‘dom 𝑒), {⟨𝑔, ℎ⟩ ∣ ∃𝑖 ∈ (𝑅1‘∪ dom 𝑒)((𝑖 ∈ ℎ ∧ ¬ 𝑖 ∈ 𝑔) ∧ ∀𝑗 ∈ (𝑅1‘∪ dom 𝑒)(𝑗(𝑒‘∪ dom 𝑒)𝑖 → (𝑗 ∈ 𝑔 ↔ 𝑗 ∈ ℎ)))}))‘((𝑅1‘dom 𝑒) ∖ ran 𝑔)))) “ {𝑏})
75	71, 74	nfel 2969	. . 3 ⊢ Ⅎℎ∩ (◡recs((𝑔 ∈ V ↦ ((𝑔 ∈ V ↦ sup((𝑦‘𝑔), (𝑅1‘dom 𝑒), {⟨𝑔, ℎ⟩ ∣ ∃𝑖 ∈ (𝑅1‘∪ dom 𝑒)((𝑖 ∈ ℎ ∧ ¬ 𝑖 ∈ 𝑔) ∧ ∀𝑗 ∈ (𝑅1‘∪ dom 𝑒)(𝑗(𝑒‘∪ dom 𝑒)𝑖 → (𝑗 ∈ 𝑔 ↔ 𝑗 ∈ ℎ)))}))‘((𝑅1‘dom 𝑒) ∖ ran 𝑔)))) “ {𝑐}) ∈ ∩ (◡recs((𝑔 ∈ V ↦ ((𝑔 ∈ V ↦ sup((𝑦‘𝑔), (𝑅1‘dom 𝑒), {⟨𝑔, ℎ⟩ ∣ ∃𝑖 ∈ (𝑅1‘∪ dom 𝑒)((𝑖 ∈ ℎ ∧ ¬ 𝑖 ∈ 𝑔) ∧ ∀𝑗 ∈ (𝑅1‘∪ dom 𝑒)(𝑗(𝑒‘∪ dom 𝑒)𝑖 → (𝑗 ∈ 𝑔 ↔ 𝑗 ∈ ℎ)))}))‘((𝑅1‘dom 𝑒) ∖ ran 𝑔)))) “ {𝑏})
76		sneq 4535	. . . . . 6 ⊢ (𝑔 = 𝑐 → {𝑔} = {𝑐})
77	76	imaeq2d 5896	. . . . 5 ⊢ (𝑔 = 𝑐 → (◡recs((𝑔 ∈ V ↦ ((𝑔 ∈ V ↦ sup((𝑦‘𝑔), (𝑅1‘dom 𝑒), {⟨𝑔, ℎ⟩ ∣ ∃𝑖 ∈ (𝑅1‘∪ dom 𝑒)((𝑖 ∈ ℎ ∧ ¬ 𝑖 ∈ 𝑔) ∧ ∀𝑗 ∈ (𝑅1‘∪ dom 𝑒)(𝑗(𝑒‘∪ dom 𝑒)𝑖 → (𝑗 ∈ 𝑔 ↔ 𝑗 ∈ ℎ)))}))‘((𝑅1‘dom 𝑒) ∖ ran 𝑔)))) “ {𝑔}) = (◡recs((𝑔 ∈ V ↦ ((𝑔 ∈ V ↦ sup((𝑦‘𝑔), (𝑅1‘dom 𝑒), {⟨𝑔, ℎ⟩ ∣ ∃𝑖 ∈ (𝑅1‘∪ dom 𝑒)((𝑖 ∈ ℎ ∧ ¬ 𝑖 ∈ 𝑔) ∧ ∀𝑗 ∈ (𝑅1‘∪ dom 𝑒)(𝑗(𝑒‘∪ dom 𝑒)𝑖 → (𝑗 ∈ 𝑔 ↔ 𝑗 ∈ ℎ)))}))‘((𝑅1‘dom 𝑒) ∖ ran 𝑔)))) “ {𝑐}))
78	77	inteqd 4843	. . . 4 ⊢ (𝑔 = 𝑐 → ∩ (◡recs((𝑔 ∈ V ↦ ((𝑔 ∈ V ↦ sup((𝑦‘𝑔), (𝑅1‘dom 𝑒), {⟨𝑔, ℎ⟩ ∣ ∃𝑖 ∈ (𝑅1‘∪ dom 𝑒)((𝑖 ∈ ℎ ∧ ¬ 𝑖 ∈ 𝑔) ∧ ∀𝑗 ∈ (𝑅1‘∪ dom 𝑒)(𝑗(𝑒‘∪ dom 𝑒)𝑖 → (𝑗 ∈ 𝑔 ↔ 𝑗 ∈ ℎ)))}))‘((𝑅1‘dom 𝑒) ∖ ran 𝑔)))) “ {𝑔}) = ∩ (◡recs((𝑔 ∈ V ↦ ((𝑔 ∈ V ↦ sup((𝑦‘𝑔), (𝑅1‘dom 𝑒), {⟨𝑔, ℎ⟩ ∣ ∃𝑖 ∈ (𝑅1‘∪ dom 𝑒)((𝑖 ∈ ℎ ∧ ¬ 𝑖 ∈ 𝑔) ∧ ∀𝑗 ∈ (𝑅1‘∪ dom 𝑒)(𝑗(𝑒‘∪ dom 𝑒)𝑖 → (𝑗 ∈ 𝑔 ↔ 𝑗 ∈ ℎ)))}))‘((𝑅1‘dom 𝑒) ∖ ran 𝑔)))) “ {𝑐}))
79		sneq 4535	. . . . . 6 ⊢ (ℎ = 𝑏 → {ℎ} = {𝑏})
80	79	imaeq2d 5896	. . . . 5 ⊢ (ℎ = 𝑏 → (◡recs((𝑔 ∈ V ↦ ((𝑔 ∈ V ↦ sup((𝑦‘𝑔), (𝑅1‘dom 𝑒), {⟨𝑔, ℎ⟩ ∣ ∃𝑖 ∈ (𝑅1‘∪ dom 𝑒)((𝑖 ∈ ℎ ∧ ¬ 𝑖 ∈ 𝑔) ∧ ∀𝑗 ∈ (𝑅1‘∪ dom 𝑒)(𝑗(𝑒‘∪ dom 𝑒)𝑖 → (𝑗 ∈ 𝑔 ↔ 𝑗 ∈ ℎ)))}))‘((𝑅1‘dom 𝑒) ∖ ran 𝑔)))) “ {ℎ}) = (◡recs((𝑔 ∈ V ↦ ((𝑔 ∈ V ↦ sup((𝑦‘𝑔), (𝑅1‘dom 𝑒), {⟨𝑔, ℎ⟩ ∣ ∃𝑖 ∈ (𝑅1‘∪ dom 𝑒)((𝑖 ∈ ℎ ∧ ¬ 𝑖 ∈ 𝑔) ∧ ∀𝑗 ∈ (𝑅1‘∪ dom 𝑒)(𝑗(𝑒‘∪ dom 𝑒)𝑖 → (𝑗 ∈ 𝑔 ↔ 𝑗 ∈ ℎ)))}))‘((𝑅1‘dom 𝑒) ∖ ran 𝑔)))) “ {𝑏}))
81	80	inteqd 4843	. . . 4 ⊢ (ℎ = 𝑏 → ∩ (◡recs((𝑔 ∈ V ↦ ((𝑔 ∈ V ↦ sup((𝑦‘𝑔), (𝑅1‘dom 𝑒), {⟨𝑔, ℎ⟩ ∣ ∃𝑖 ∈ (𝑅1‘∪ dom 𝑒)((𝑖 ∈ ℎ ∧ ¬ 𝑖 ∈ 𝑔) ∧ ∀𝑗 ∈ (𝑅1‘∪ dom 𝑒)(𝑗(𝑒‘∪ dom 𝑒)𝑖 → (𝑗 ∈ 𝑔 ↔ 𝑗 ∈ ℎ)))}))‘((𝑅1‘dom 𝑒) ∖ ran 𝑔)))) “ {ℎ}) = ∩ (◡recs((𝑔 ∈ V ↦ ((𝑔 ∈ V ↦ sup((𝑦‘𝑔), (𝑅1‘dom 𝑒), {⟨𝑔, ℎ⟩ ∣ ∃𝑖 ∈ (𝑅1‘∪ dom 𝑒)((𝑖 ∈ ℎ ∧ ¬ 𝑖 ∈ 𝑔) ∧ ∀𝑗 ∈ (𝑅1‘∪ dom 𝑒)(𝑗(𝑒‘∪ dom 𝑒)𝑖 → (𝑗 ∈ 𝑔 ↔ 𝑗 ∈ ℎ)))}))‘((𝑅1‘dom 𝑒) ∖ ran 𝑔)))) “ {𝑏}))
82		eleq12 2879	. . . 4 ⊢ ((∩ (◡recs((𝑔 ∈ V ↦ ((𝑔 ∈ V ↦ sup((𝑦‘𝑔), (𝑅1‘dom 𝑒), {⟨𝑔, ℎ⟩ ∣ ∃𝑖 ∈ (𝑅1‘∪ dom 𝑒)((𝑖 ∈ ℎ ∧ ¬ 𝑖 ∈ 𝑔) ∧ ∀𝑗 ∈ (𝑅1‘∪ dom 𝑒)(𝑗(𝑒‘∪ dom 𝑒)𝑖 → (𝑗 ∈ 𝑔 ↔ 𝑗 ∈ ℎ)))}))‘((𝑅1‘dom 𝑒) ∖ ran 𝑔)))) “ {𝑔}) = ∩ (◡recs((𝑔 ∈ V ↦ ((𝑔 ∈ V ↦ sup((𝑦‘𝑔), (𝑅1‘dom 𝑒), {⟨𝑔, ℎ⟩ ∣ ∃𝑖 ∈ (𝑅1‘∪ dom 𝑒)((𝑖 ∈ ℎ ∧ ¬ 𝑖 ∈ 𝑔) ∧ ∀𝑗 ∈ (𝑅1‘∪ dom 𝑒)(𝑗(𝑒‘∪ dom 𝑒)𝑖 → (𝑗 ∈ 𝑔 ↔ 𝑗 ∈ ℎ)))}))‘((𝑅1‘dom 𝑒) ∖ ran 𝑔)))) “ {𝑐}) ∧ ∩ (◡recs((𝑔 ∈ V ↦ ((𝑔 ∈ V ↦ sup((𝑦‘𝑔), (𝑅1‘dom 𝑒), {⟨𝑔, ℎ⟩ ∣ ∃𝑖 ∈ (𝑅1‘∪ dom 𝑒)((𝑖 ∈ ℎ ∧ ¬ 𝑖 ∈ 𝑔) ∧ ∀𝑗 ∈ (𝑅1‘∪ dom 𝑒)(𝑗(𝑒‘∪ dom 𝑒)𝑖 → (𝑗 ∈ 𝑔 ↔ 𝑗 ∈ ℎ)))}))‘((𝑅1‘dom 𝑒) ∖ ran 𝑔)))) “ {ℎ}) = ∩ (◡recs((𝑔 ∈ V ↦ ((𝑔 ∈ V ↦ sup((𝑦‘𝑔), (𝑅1‘dom 𝑒), {⟨𝑔, ℎ⟩ ∣ ∃𝑖 ∈ (𝑅1‘∪ dom 𝑒)((𝑖 ∈ ℎ ∧ ¬ 𝑖 ∈ 𝑔) ∧ ∀𝑗 ∈ (𝑅1‘∪ dom 𝑒)(𝑗(𝑒‘∪ dom 𝑒)𝑖 → (𝑗 ∈ 𝑔 ↔ 𝑗 ∈ ℎ)))}))‘((𝑅1‘dom 𝑒) ∖ ran 𝑔)))) “ {𝑏})) → (∩ (◡recs((𝑔 ∈ V ↦ ((𝑔 ∈ V ↦ sup((𝑦‘𝑔), (𝑅1‘dom 𝑒), {⟨𝑔, ℎ⟩ ∣ ∃𝑖 ∈ (𝑅1‘∪ dom 𝑒)((𝑖 ∈ ℎ ∧ ¬ 𝑖 ∈ 𝑔) ∧ ∀𝑗 ∈ (𝑅1‘∪ dom 𝑒)(𝑗(𝑒‘∪ dom 𝑒)𝑖 → (𝑗 ∈ 𝑔 ↔ 𝑗 ∈ ℎ)))}))‘((𝑅1‘dom 𝑒) ∖ ran 𝑔)))) “ {𝑔}) ∈ ∩ (◡recs((𝑔 ∈ V ↦ ((𝑔 ∈ V ↦ sup((𝑦‘𝑔), (𝑅1‘dom 𝑒), {⟨𝑔, ℎ⟩ ∣ ∃𝑖 ∈ (𝑅1‘∪ dom 𝑒)((𝑖 ∈ ℎ ∧ ¬ 𝑖 ∈ 𝑔) ∧ ∀𝑗 ∈ (𝑅1‘∪ dom 𝑒)(𝑗(𝑒‘∪ dom 𝑒)𝑖 → (𝑗 ∈ 𝑔 ↔ 𝑗 ∈ ℎ)))}))‘((𝑅1‘dom 𝑒) ∖ ran 𝑔)))) “ {ℎ}) ↔ ∩ (◡recs((𝑔 ∈ V ↦ ((𝑔 ∈ V ↦ sup((𝑦‘𝑔), (𝑅1‘dom 𝑒), {⟨𝑔, ℎ⟩ ∣ ∃𝑖 ∈ (𝑅1‘∪ dom 𝑒)((𝑖 ∈ ℎ ∧ ¬ 𝑖 ∈ 𝑔) ∧ ∀𝑗 ∈ (𝑅1‘∪ dom 𝑒)(𝑗(𝑒‘∪ dom 𝑒)𝑖 → (𝑗 ∈ 𝑔 ↔ 𝑗 ∈ ℎ)))}))‘((𝑅1‘dom 𝑒) ∖ ran 𝑔)))) “ {𝑐}) ∈ ∩ (◡recs((𝑔 ∈ V ↦ ((𝑔 ∈ V ↦ sup((𝑦‘𝑔), (𝑅1‘dom 𝑒), {⟨𝑔, ℎ⟩ ∣ ∃𝑖 ∈ (𝑅1‘∪ dom 𝑒)((𝑖 ∈ ℎ ∧ ¬ 𝑖 ∈ 𝑔) ∧ ∀𝑗 ∈ (𝑅1‘∪ dom 𝑒)(𝑗(𝑒‘∪ dom 𝑒)𝑖 → (𝑗 ∈ 𝑔 ↔ 𝑗 ∈ ℎ)))}))‘((𝑅1‘dom 𝑒) ∖ ran 𝑔)))) “ {𝑏})))
83	78, 81, 82	syl2an 598	. . 3 ⊢ ((𝑔 = 𝑐 ∧ ℎ = 𝑏) → (∩ (◡recs((𝑔 ∈ V ↦ ((𝑔 ∈ V ↦ sup((𝑦‘𝑔), (𝑅1‘dom 𝑒), {⟨𝑔, ℎ⟩ ∣ ∃𝑖 ∈ (𝑅1‘∪ dom 𝑒)((𝑖 ∈ ℎ ∧ ¬ 𝑖 ∈ 𝑔) ∧ ∀𝑗 ∈ (𝑅1‘∪ dom 𝑒)(𝑗(𝑒‘∪ dom 𝑒)𝑖 → (𝑗 ∈ 𝑔 ↔ 𝑗 ∈ ℎ)))}))‘((𝑅1‘dom 𝑒) ∖ ran 𝑔)))) “ {𝑔}) ∈ ∩ (◡recs((𝑔 ∈ V ↦ ((𝑔 ∈ V ↦ sup((𝑦‘𝑔), (𝑅1‘dom 𝑒), {⟨𝑔, ℎ⟩ ∣ ∃𝑖 ∈ (𝑅1‘∪ dom 𝑒)((𝑖 ∈ ℎ ∧ ¬ 𝑖 ∈ 𝑔) ∧ ∀𝑗 ∈ (𝑅1‘∪ dom 𝑒)(𝑗(𝑒‘∪ dom 𝑒)𝑖 → (𝑗 ∈ 𝑔 ↔ 𝑗 ∈ ℎ)))}))‘((𝑅1‘dom 𝑒) ∖ ran 𝑔)))) “ {ℎ}) ↔ ∩ (◡recs((𝑔 ∈ V ↦ ((𝑔 ∈ V ↦ sup((𝑦‘𝑔), (𝑅1‘dom 𝑒), {⟨𝑔, ℎ⟩ ∣ ∃𝑖 ∈ (𝑅1‘∪ dom 𝑒)((𝑖 ∈ ℎ ∧ ¬ 𝑖 ∈ 𝑔) ∧ ∀𝑗 ∈ (𝑅1‘∪ dom 𝑒)(𝑗(𝑒‘∪ dom 𝑒)𝑖 → (𝑗 ∈ 𝑔 ↔ 𝑗 ∈ ℎ)))}))‘((𝑅1‘dom 𝑒) ∖ ran 𝑔)))) “ {𝑐}) ∈ ∩ (◡recs((𝑔 ∈ V ↦ ((𝑔 ∈ V ↦ sup((𝑦‘𝑔), (𝑅1‘dom 𝑒), {⟨𝑔, ℎ⟩ ∣ ∃𝑖 ∈ (𝑅1‘∪ dom 𝑒)((𝑖 ∈ ℎ ∧ ¬ 𝑖 ∈ 𝑔) ∧ ∀𝑗 ∈ (𝑅1‘∪ dom 𝑒)(𝑗(𝑒‘∪ dom 𝑒)𝑖 → (𝑗 ∈ 𝑔 ↔ 𝑗 ∈ ℎ)))}))‘((𝑅1‘dom 𝑒) ∖ ran 𝑔)))) “ {𝑏})))
84	46, 47, 57, 75, 83	cbvopab 5101	. 2 ⊢ {⟨𝑔, ℎ⟩ ∣ ∩ (◡recs((𝑔 ∈ V ↦ ((𝑔 ∈ V ↦ sup((𝑦‘𝑔), (𝑅1‘dom 𝑒), {⟨𝑔, ℎ⟩ ∣ ∃𝑖 ∈ (𝑅1‘∪ dom 𝑒)((𝑖 ∈ ℎ ∧ ¬ 𝑖 ∈ 𝑔) ∧ ∀𝑗 ∈ (𝑅1‘∪ dom 𝑒)(𝑗(𝑒‘∪ dom 𝑒)𝑖 → (𝑗 ∈ 𝑔 ↔ 𝑗 ∈ ℎ)))}))‘((𝑅1‘dom 𝑒) ∖ ran 𝑔)))) “ {𝑔}) ∈ ∩ (◡recs((𝑔 ∈ V ↦ ((𝑔 ∈ V ↦ sup((𝑦‘𝑔), (𝑅1‘dom 𝑒), {⟨𝑔, ℎ⟩ ∣ ∃𝑖 ∈ (𝑅1‘∪ dom 𝑒)((𝑖 ∈ ℎ ∧ ¬ 𝑖 ∈ 𝑔) ∧ ∀𝑗 ∈ (𝑅1‘∪ dom 𝑒)(𝑗(𝑒‘∪ dom 𝑒)𝑖 → (𝑗 ∈ 𝑔 ↔ 𝑗 ∈ ℎ)))}))‘((𝑅1‘dom 𝑒) ∖ ran 𝑔)))) “ {ℎ})} = {⟨𝑐, 𝑏⟩ ∣ ∩ (◡recs((𝑔 ∈ V ↦ ((𝑔 ∈ V ↦ sup((𝑦‘𝑔), (𝑅1‘dom 𝑒), {⟨𝑔, ℎ⟩ ∣ ∃𝑖 ∈ (𝑅1‘∪ dom 𝑒)((𝑖 ∈ ℎ ∧ ¬ 𝑖 ∈ 𝑔) ∧ ∀𝑗 ∈ (𝑅1‘∪ dom 𝑒)(𝑗(𝑒‘∪ dom 𝑒)𝑖 → (𝑗 ∈ 𝑔 ↔ 𝑗 ∈ ℎ)))}))‘((𝑅1‘dom 𝑒) ∖ ran 𝑔)))) “ {𝑐}) ∈ ∩ (◡recs((𝑔 ∈ V ↦ ((𝑔 ∈ V ↦ sup((𝑦‘𝑔), (𝑅1‘dom 𝑒), {⟨𝑔, ℎ⟩ ∣ ∃𝑖 ∈ (𝑅1‘∪ dom 𝑒)((𝑖 ∈ ℎ ∧ ¬ 𝑖 ∈ 𝑔) ∧ ∀𝑗 ∈ (𝑅1‘∪ dom 𝑒)(𝑗(𝑒‘∪ dom 𝑒)𝑖 → (𝑗 ∈ 𝑔 ↔ 𝑗 ∈ ℎ)))}))‘((𝑅1‘dom 𝑒) ∖ ran 𝑔)))) “ {𝑏})}
85		fveq2 6645	. . . . 5 ⊢ (𝑔 = 𝑐 → (rank‘𝑔) = (rank‘𝑐))
86		fveq2 6645	. . . . 5 ⊢ (ℎ = 𝑏 → (rank‘ℎ) = (rank‘𝑏))
87	85, 86	breqan12d 5046	. . . 4 ⊢ ((𝑔 = 𝑐 ∧ ℎ = 𝑏) → ((rank‘𝑔) E (rank‘ℎ) ↔ (rank‘𝑐) E (rank‘𝑏)))
88	85, 86	eqeqan12d 2815	. . . . 5 ⊢ ((𝑔 = 𝑐 ∧ ℎ = 𝑏) → ((rank‘𝑔) = (rank‘ℎ) ↔ (rank‘𝑐) = (rank‘𝑏)))
89		simpl 486	. . . . . 6 ⊢ ((𝑔 = 𝑐 ∧ ℎ = 𝑏) → 𝑔 = 𝑐)
90		suceq 6224	. . . . . . . . 9 ⊢ ((rank‘𝑔) = (rank‘𝑐) → suc (rank‘𝑔) = suc (rank‘𝑐))
91	85, 90	syl 17	. . . . . . . 8 ⊢ (𝑔 = 𝑐 → suc (rank‘𝑔) = suc (rank‘𝑐))
92	91	adantr 484	. . . . . . 7 ⊢ ((𝑔 = 𝑐 ∧ ℎ = 𝑏) → suc (rank‘𝑔) = suc (rank‘𝑐))
93	92	fveq2d 6649	. . . . . 6 ⊢ ((𝑔 = 𝑐 ∧ ℎ = 𝑏) → (𝑒‘suc (rank‘𝑔)) = (𝑒‘suc (rank‘𝑐)))
94		simpr 488	. . . . . 6 ⊢ ((𝑔 = 𝑐 ∧ ℎ = 𝑏) → ℎ = 𝑏)
95	89, 93, 94	breq123d 5044	. . . . 5 ⊢ ((𝑔 = 𝑐 ∧ ℎ = 𝑏) → (𝑔(𝑒‘suc (rank‘𝑔))ℎ ↔ 𝑐(𝑒‘suc (rank‘𝑐))𝑏))
96	88, 95	anbi12d 633	. . . 4 ⊢ ((𝑔 = 𝑐 ∧ ℎ = 𝑏) → (((rank‘𝑔) = (rank‘ℎ) ∧ 𝑔(𝑒‘suc (rank‘𝑔))ℎ) ↔ ((rank‘𝑐) = (rank‘𝑏) ∧ 𝑐(𝑒‘suc (rank‘𝑐))𝑏)))
97	87, 96	orbi12d 916	. . 3 ⊢ ((𝑔 = 𝑐 ∧ ℎ = 𝑏) → (((rank‘𝑔) E (rank‘ℎ) ∨ ((rank‘𝑔) = (rank‘ℎ) ∧ 𝑔(𝑒‘suc (rank‘𝑔))ℎ)) ↔ ((rank‘𝑐) E (rank‘𝑏) ∨ ((rank‘𝑐) = (rank‘𝑏) ∧ 𝑐(𝑒‘suc (rank‘𝑐))𝑏))))
98	97	cbvopabv 5102	. 2 ⊢ {⟨𝑔, ℎ⟩ ∣ ((rank‘𝑔) E (rank‘ℎ) ∨ ((rank‘𝑔) = (rank‘ℎ) ∧ 𝑔(𝑒‘suc (rank‘𝑔))ℎ))} = {⟨𝑐, 𝑏⟩ ∣ ((rank‘𝑐) E (rank‘𝑏) ∨ ((rank‘𝑐) = (rank‘𝑏) ∧ 𝑐(𝑒‘suc (rank‘𝑐))𝑏))}
99		eqid 2798	. 2 ⊢ (if(dom 𝑒 = ∪ dom 𝑒, {⟨𝑔, ℎ⟩ ∣ ((rank‘𝑔) E (rank‘ℎ) ∨ ((rank‘𝑔) = (rank‘ℎ) ∧ 𝑔(𝑒‘suc (rank‘𝑔))ℎ))}, {⟨𝑔, ℎ⟩ ∣ ∩ (◡recs((𝑔 ∈ V ↦ ((𝑔 ∈ V ↦ sup((𝑦‘𝑔), (𝑅1‘dom 𝑒), {⟨𝑔, ℎ⟩ ∣ ∃𝑖 ∈ (𝑅1‘∪ dom 𝑒)((𝑖 ∈ ℎ ∧ ¬ 𝑖 ∈ 𝑔) ∧ ∀𝑗 ∈ (𝑅1‘∪ dom 𝑒)(𝑗(𝑒‘∪ dom 𝑒)𝑖 → (𝑗 ∈ 𝑔 ↔ 𝑗 ∈ ℎ)))}))‘((𝑅1‘dom 𝑒) ∖ ran 𝑔)))) “ {𝑔}) ∈ ∩ (◡recs((𝑔 ∈ V ↦ ((𝑔 ∈ V ↦ sup((𝑦‘𝑔), (𝑅1‘dom 𝑒), {⟨𝑔, ℎ⟩ ∣ ∃𝑖 ∈ (𝑅1‘∪ dom 𝑒)((𝑖 ∈ ℎ ∧ ¬ 𝑖 ∈ 𝑔) ∧ ∀𝑗 ∈ (𝑅1‘∪ dom 𝑒)(𝑗(𝑒‘∪ dom 𝑒)𝑖 → (𝑗 ∈ 𝑔 ↔ 𝑗 ∈ ℎ)))}))‘((𝑅1‘dom 𝑒) ∖ ran 𝑔)))) “ {ℎ})}) ∩ ((𝑅1‘dom 𝑒) × (𝑅1‘dom 𝑒))) = (if(dom 𝑒 = ∪ dom 𝑒, {⟨𝑔, ℎ⟩ ∣ ((rank‘𝑔) E (rank‘ℎ) ∨ ((rank‘𝑔) = (rank‘ℎ) ∧ 𝑔(𝑒‘suc (rank‘𝑔))ℎ))}, {⟨𝑔, ℎ⟩ ∣ ∩ (◡recs((𝑔 ∈ V ↦ ((𝑔 ∈ V ↦ sup((𝑦‘𝑔), (𝑅1‘dom 𝑒), {⟨𝑔, ℎ⟩ ∣ ∃𝑖 ∈ (𝑅1‘∪ dom 𝑒)((𝑖 ∈ ℎ ∧ ¬ 𝑖 ∈ 𝑔) ∧ ∀𝑗 ∈ (𝑅1‘∪ dom 𝑒)(𝑗(𝑒‘∪ dom 𝑒)𝑖 → (𝑗 ∈ 𝑔 ↔ 𝑗 ∈ ℎ)))}))‘((𝑅1‘dom 𝑒) ∖ ran 𝑔)))) “ {𝑔}) ∈ ∩ (◡recs((𝑔 ∈ V ↦ ((𝑔 ∈ V ↦ sup((𝑦‘𝑔), (𝑅1‘dom 𝑒), {⟨𝑔, ℎ⟩ ∣ ∃𝑖 ∈ (𝑅1‘∪ dom 𝑒)((𝑖 ∈ ℎ ∧ ¬ 𝑖 ∈ 𝑔) ∧ ∀𝑗 ∈ (𝑅1‘∪ dom 𝑒)(𝑗(𝑒‘∪ dom 𝑒)𝑖 → (𝑗 ∈ 𝑔 ↔ 𝑗 ∈ ℎ)))}))‘((𝑅1‘dom 𝑒) ∖ ran 𝑔)))) “ {ℎ})}) ∩ ((𝑅1‘dom 𝑒) × (𝑅1‘dom 𝑒)))
100		dmeq 5736	. . . . . . 7 ⊢ (𝑙 = 𝑒 → dom 𝑙 = dom 𝑒)
101	100	unieqd 4814	. . . . . . 7 ⊢ (𝑙 = 𝑒 → ∪ dom 𝑙 = ∪ dom 𝑒)
102	100, 101	eqeq12d 2814	. . . . . 6 ⊢ (𝑙 = 𝑒 → (dom 𝑙 = ∪ dom 𝑙 ↔ dom 𝑒 = ∪ dom 𝑒))
103		fveq1 6644	. . . . . . . . . 10 ⊢ (𝑙 = 𝑒 → (𝑙‘suc (rank‘𝑔)) = (𝑒‘suc (rank‘𝑔)))
104	103	breqd 5041	. . . . . . . . 9 ⊢ (𝑙 = 𝑒 → (𝑔(𝑙‘suc (rank‘𝑔))ℎ ↔ 𝑔(𝑒‘suc (rank‘𝑔))ℎ))
105	104	anbi2d 631	. . . . . . . 8 ⊢ (𝑙 = 𝑒 → (((rank‘𝑔) = (rank‘ℎ) ∧ 𝑔(𝑙‘suc (rank‘𝑔))ℎ) ↔ ((rank‘𝑔) = (rank‘ℎ) ∧ 𝑔(𝑒‘suc (rank‘𝑔))ℎ)))
106	105	orbi2d 913	. . . . . . 7 ⊢ (𝑙 = 𝑒 → (((rank‘𝑔) E (rank‘ℎ) ∨ ((rank‘𝑔) = (rank‘ℎ) ∧ 𝑔(𝑙‘suc (rank‘𝑔))ℎ)) ↔ ((rank‘𝑔) E (rank‘ℎ) ∨ ((rank‘𝑔) = (rank‘ℎ) ∧ 𝑔(𝑒‘suc (rank‘𝑔))ℎ))))
107	106	opabbidv 5096	. . . . . 6 ⊢ (𝑙 = 𝑒 → {⟨𝑔, ℎ⟩ ∣ ((rank‘𝑔) E (rank‘ℎ) ∨ ((rank‘𝑔) = (rank‘ℎ) ∧ 𝑔(𝑙‘suc (rank‘𝑔))ℎ))} = {⟨𝑔, ℎ⟩ ∣ ((rank‘𝑔) E (rank‘ℎ) ∨ ((rank‘𝑔) = (rank‘ℎ) ∧ 𝑔(𝑒‘suc (rank‘𝑔))ℎ))})
108		eqidd 2799	. . . . . . . . . . . . . . . 16 ⊢ (𝑙 = 𝑒 → (𝑦‘𝑔) = (𝑦‘𝑔))
109	100	fveq2d 6649	. . . . . . . . . . . . . . . 16 ⊢ (𝑙 = 𝑒 → (𝑅1‘dom 𝑙) = (𝑅1‘dom 𝑒))
110	101	fveq2d 6649	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑙 = 𝑒 → (𝑅1‘∪ dom 𝑙) = (𝑅1‘∪ dom 𝑒))
111		id 22	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑙 = 𝑒 → 𝑙 = 𝑒)
112	111, 101	fveq12d 6652	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑙 = 𝑒 → (𝑙‘∪ dom 𝑙) = (𝑒‘∪ dom 𝑒))
113	112	breqd 5041	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑙 = 𝑒 → (𝑗(𝑙‘∪ dom 𝑙)𝑖 ↔ 𝑗(𝑒‘∪ dom 𝑒)𝑖))
114	113	imbi1d 345	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑙 = 𝑒 → ((𝑗(𝑙‘∪ dom 𝑙)𝑖 → (𝑗 ∈ 𝑔 ↔ 𝑗 ∈ ℎ)) ↔ (𝑗(𝑒‘∪ dom 𝑒)𝑖 → (𝑗 ∈ 𝑔 ↔ 𝑗 ∈ ℎ))))
115	110, 114	raleqbidv 3354	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑙 = 𝑒 → (∀𝑗 ∈ (𝑅1‘∪ dom 𝑙)(𝑗(𝑙‘∪ dom 𝑙)𝑖 → (𝑗 ∈ 𝑔 ↔ 𝑗 ∈ ℎ)) ↔ ∀𝑗 ∈ (𝑅1‘∪ dom 𝑒)(𝑗(𝑒‘∪ dom 𝑒)𝑖 → (𝑗 ∈ 𝑔 ↔ 𝑗 ∈ ℎ))))
116	115	anbi2d 631	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑙 = 𝑒 → (((𝑖 ∈ ℎ ∧ ¬ 𝑖 ∈ 𝑔) ∧ ∀𝑗 ∈ (𝑅1‘∪ dom 𝑙)(𝑗(𝑙‘∪ dom 𝑙)𝑖 → (𝑗 ∈ 𝑔 ↔ 𝑗 ∈ ℎ))) ↔ ((𝑖 ∈ ℎ ∧ ¬ 𝑖 ∈ 𝑔) ∧ ∀𝑗 ∈ (𝑅1‘∪ dom 𝑒)(𝑗(𝑒‘∪ dom 𝑒)𝑖 → (𝑗 ∈ 𝑔 ↔ 𝑗 ∈ ℎ)))))
117	110, 116	rexeqbidv 3355	. . . . . . . . . . . . . . . . 17 ⊢ (𝑙 = 𝑒 → (∃𝑖 ∈ (𝑅1‘∪ dom 𝑙)((𝑖 ∈ ℎ ∧ ¬ 𝑖 ∈ 𝑔) ∧ ∀𝑗 ∈ (𝑅1‘∪ dom 𝑙)(𝑗(𝑙‘∪ dom 𝑙)𝑖 → (𝑗 ∈ 𝑔 ↔ 𝑗 ∈ ℎ))) ↔ ∃𝑖 ∈ (𝑅1‘∪ dom 𝑒)((𝑖 ∈ ℎ ∧ ¬ 𝑖 ∈ 𝑔) ∧ ∀𝑗 ∈ (𝑅1‘∪ dom 𝑒)(𝑗(𝑒‘∪ dom 𝑒)𝑖 → (𝑗 ∈ 𝑔 ↔ 𝑗 ∈ ℎ)))))
118	117	opabbidv 5096	. . . . . . . . . . . . . . . 16 ⊢ (𝑙 = 𝑒 → {⟨𝑔, ℎ⟩ ∣ ∃𝑖 ∈ (𝑅1‘∪ dom 𝑙)((𝑖 ∈ ℎ ∧ ¬ 𝑖 ∈ 𝑔) ∧ ∀𝑗 ∈ (𝑅1‘∪ dom 𝑙)(𝑗(𝑙‘∪ dom 𝑙)𝑖 → (𝑗 ∈ 𝑔 ↔ 𝑗 ∈ ℎ)))} = {⟨𝑔, ℎ⟩ ∣ ∃𝑖 ∈ (𝑅1‘∪ dom 𝑒)((𝑖 ∈ ℎ ∧ ¬ 𝑖 ∈ 𝑔) ∧ ∀𝑗 ∈ (𝑅1‘∪ dom 𝑒)(𝑗(𝑒‘∪ dom 𝑒)𝑖 → (𝑗 ∈ 𝑔 ↔ 𝑗 ∈ ℎ)))})
119	108, 109, 118	supeq123d 8898	. . . . . . . . . . . . . . 15 ⊢ (𝑙 = 𝑒 → sup((𝑦‘𝑔), (𝑅1‘dom 𝑙), {⟨𝑔, ℎ⟩ ∣ ∃𝑖 ∈ (𝑅1‘∪ dom 𝑙)((𝑖 ∈ ℎ ∧ ¬ 𝑖 ∈ 𝑔) ∧ ∀𝑗 ∈ (𝑅1‘∪ dom 𝑙)(𝑗(𝑙‘∪ dom 𝑙)𝑖 → (𝑗 ∈ 𝑔 ↔ 𝑗 ∈ ℎ)))}) = sup((𝑦‘𝑔), (𝑅1‘dom 𝑒), {⟨𝑔, ℎ⟩ ∣ ∃𝑖 ∈ (𝑅1‘∪ dom 𝑒)((𝑖 ∈ ℎ ∧ ¬ 𝑖 ∈ 𝑔) ∧ ∀𝑗 ∈ (𝑅1‘∪ dom 𝑒)(𝑗(𝑒‘∪ dom 𝑒)𝑖 → (𝑗 ∈ 𝑔 ↔ 𝑗 ∈ ℎ)))}))
120	119	mpteq2dv 5126	. . . . . . . . . . . . . 14 ⊢ (𝑙 = 𝑒 → (𝑔 ∈ V ↦ sup((𝑦‘𝑔), (𝑅1‘dom 𝑙), {⟨𝑔, ℎ⟩ ∣ ∃𝑖 ∈ (𝑅1‘∪ dom 𝑙)((𝑖 ∈ ℎ ∧ ¬ 𝑖 ∈ 𝑔) ∧ ∀𝑗 ∈ (𝑅1‘∪ dom 𝑙)(𝑗(𝑙‘∪ dom 𝑙)𝑖 → (𝑗 ∈ 𝑔 ↔ 𝑗 ∈ ℎ)))})) = (𝑔 ∈ V ↦ sup((𝑦‘𝑔), (𝑅1‘dom 𝑒), {⟨𝑔, ℎ⟩ ∣ ∃𝑖 ∈ (𝑅1‘∪ dom 𝑒)((𝑖 ∈ ℎ ∧ ¬ 𝑖 ∈ 𝑔) ∧ ∀𝑗 ∈ (𝑅1‘∪ dom 𝑒)(𝑗(𝑒‘∪ dom 𝑒)𝑖 → (𝑗 ∈ 𝑔 ↔ 𝑗 ∈ ℎ)))})))
121	109	difeq1d 4049	. . . . . . . . . . . . . 14 ⊢ (𝑙 = 𝑒 → ((𝑅1‘dom 𝑙) ∖ ran 𝑔) = ((𝑅1‘dom 𝑒) ∖ ran 𝑔))
122	120, 121	fveq12d 6652	. . . . . . . . . . . . 13 ⊢ (𝑙 = 𝑒 → ((𝑔 ∈ V ↦ sup((𝑦‘𝑔), (𝑅1‘dom 𝑙), {⟨𝑔, ℎ⟩ ∣ ∃𝑖 ∈ (𝑅1‘∪ dom 𝑙)((𝑖 ∈ ℎ ∧ ¬ 𝑖 ∈ 𝑔) ∧ ∀𝑗 ∈ (𝑅1‘∪ dom 𝑙)(𝑗(𝑙‘∪ dom 𝑙)𝑖 → (𝑗 ∈ 𝑔 ↔ 𝑗 ∈ ℎ)))}))‘((𝑅1‘dom 𝑙) ∖ ran 𝑔)) = ((𝑔 ∈ V ↦ sup((𝑦‘𝑔), (𝑅1‘dom 𝑒), {⟨𝑔, ℎ⟩ ∣ ∃𝑖 ∈ (𝑅1‘∪ dom 𝑒)((𝑖 ∈ ℎ ∧ ¬ 𝑖 ∈ 𝑔) ∧ ∀𝑗 ∈ (𝑅1‘∪ dom 𝑒)(𝑗(𝑒‘∪ dom 𝑒)𝑖 → (𝑗 ∈ 𝑔 ↔ 𝑗 ∈ ℎ)))}))‘((𝑅1‘dom 𝑒) ∖ ran 𝑔)))
123	122	mpteq2dv 5126	. . . . . . . . . . . 12 ⊢ (𝑙 = 𝑒 → (𝑔 ∈ V ↦ ((𝑔 ∈ V ↦ sup((𝑦‘𝑔), (𝑅1‘dom 𝑙), {⟨𝑔, ℎ⟩ ∣ ∃𝑖 ∈ (𝑅1‘∪ dom 𝑙)((𝑖 ∈ ℎ ∧ ¬ 𝑖 ∈ 𝑔) ∧ ∀𝑗 ∈ (𝑅1‘∪ dom 𝑙)(𝑗(𝑙‘∪ dom 𝑙)𝑖 → (𝑗 ∈ 𝑔 ↔ 𝑗 ∈ ℎ)))}))‘((𝑅1‘dom 𝑙) ∖ ran 𝑔))) = (𝑔 ∈ V ↦ ((𝑔 ∈ V ↦ sup((𝑦‘𝑔), (𝑅1‘dom 𝑒), {⟨𝑔, ℎ⟩ ∣ ∃𝑖 ∈ (𝑅1‘∪ dom 𝑒)((𝑖 ∈ ℎ ∧ ¬ 𝑖 ∈ 𝑔) ∧ ∀𝑗 ∈ (𝑅1‘∪ dom 𝑒)(𝑗(𝑒‘∪ dom 𝑒)𝑖 → (𝑗 ∈ 𝑔 ↔ 𝑗 ∈ ℎ)))}))‘((𝑅1‘dom 𝑒) ∖ ran 𝑔))))
124		recseq 7993	. . . . . . . . . . . 12 ⊢ ((𝑔 ∈ V ↦ ((𝑔 ∈ V ↦ sup((𝑦‘𝑔), (𝑅1‘dom 𝑙), {⟨𝑔, ℎ⟩ ∣ ∃𝑖 ∈ (𝑅1‘∪ dom 𝑙)((𝑖 ∈ ℎ ∧ ¬ 𝑖 ∈ 𝑔) ∧ ∀𝑗 ∈ (𝑅1‘∪ dom 𝑙)(𝑗(𝑙‘∪ dom 𝑙)𝑖 → (𝑗 ∈ 𝑔 ↔ 𝑗 ∈ ℎ)))}))‘((𝑅1‘dom 𝑙) ∖ ran 𝑔))) = (𝑔 ∈ V ↦ ((𝑔 ∈ V ↦ sup((𝑦‘𝑔), (𝑅1‘dom 𝑒), {⟨𝑔, ℎ⟩ ∣ ∃𝑖 ∈ (𝑅1‘∪ dom 𝑒)((𝑖 ∈ ℎ ∧ ¬ 𝑖 ∈ 𝑔) ∧ ∀𝑗 ∈ (𝑅1‘∪ dom 𝑒)(𝑗(𝑒‘∪ dom 𝑒)𝑖 → (𝑗 ∈ 𝑔 ↔ 𝑗 ∈ ℎ)))}))‘((𝑅1‘dom 𝑒) ∖ ran 𝑔))) → recs((𝑔 ∈ V ↦ ((𝑔 ∈ V ↦ sup((𝑦‘𝑔), (𝑅1‘dom 𝑙), {⟨𝑔, ℎ⟩ ∣ ∃𝑖 ∈ (𝑅1‘∪ dom 𝑙)((𝑖 ∈ ℎ ∧ ¬ 𝑖 ∈ 𝑔) ∧ ∀𝑗 ∈ (𝑅1‘∪ dom 𝑙)(𝑗(𝑙‘∪ dom 𝑙)𝑖 → (𝑗 ∈ 𝑔 ↔ 𝑗 ∈ ℎ)))}))‘((𝑅1‘dom 𝑙) ∖ ran 𝑔)))) = recs((𝑔 ∈ V ↦ ((𝑔 ∈ V ↦ sup((𝑦‘𝑔), (𝑅1‘dom 𝑒), {⟨𝑔, ℎ⟩ ∣ ∃𝑖 ∈ (𝑅1‘∪ dom 𝑒)((𝑖 ∈ ℎ ∧ ¬ 𝑖 ∈ 𝑔) ∧ ∀𝑗 ∈ (𝑅1‘∪ dom 𝑒)(𝑗(𝑒‘∪ dom 𝑒)𝑖 → (𝑗 ∈ 𝑔 ↔ 𝑗 ∈ ℎ)))}))‘((𝑅1‘dom 𝑒) ∖ ran 𝑔)))))
125	123, 124	syl 17	. . . . . . . . . . 11 ⊢ (𝑙 = 𝑒 → recs((𝑔 ∈ V ↦ ((𝑔 ∈ V ↦ sup((𝑦‘𝑔), (𝑅1‘dom 𝑙), {⟨𝑔, ℎ⟩ ∣ ∃𝑖 ∈ (𝑅1‘∪ dom 𝑙)((𝑖 ∈ ℎ ∧ ¬ 𝑖 ∈ 𝑔) ∧ ∀𝑗 ∈ (𝑅1‘∪ dom 𝑙)(𝑗(𝑙‘∪ dom 𝑙)𝑖 → (𝑗 ∈ 𝑔 ↔ 𝑗 ∈ ℎ)))}))‘((𝑅1‘dom 𝑙) ∖ ran 𝑔)))) = recs((𝑔 ∈ V ↦ ((𝑔 ∈ V ↦ sup((𝑦‘𝑔), (𝑅1‘dom 𝑒), {⟨𝑔, ℎ⟩ ∣ ∃𝑖 ∈ (𝑅1‘∪ dom 𝑒)((𝑖 ∈ ℎ ∧ ¬ 𝑖 ∈ 𝑔) ∧ ∀𝑗 ∈ (𝑅1‘∪ dom 𝑒)(𝑗(𝑒‘∪ dom 𝑒)𝑖 → (𝑗 ∈ 𝑔 ↔ 𝑗 ∈ ℎ)))}))‘((𝑅1‘dom 𝑒) ∖ ran 𝑔)))))
126	125	cnveqd 5710	. . . . . . . . . 10 ⊢ (𝑙 = 𝑒 → ◡recs((𝑔 ∈ V ↦ ((𝑔 ∈ V ↦ sup((𝑦‘𝑔), (𝑅1‘dom 𝑙), {⟨𝑔, ℎ⟩ ∣ ∃𝑖 ∈ (𝑅1‘∪ dom 𝑙)((𝑖 ∈ ℎ ∧ ¬ 𝑖 ∈ 𝑔) ∧ ∀𝑗 ∈ (𝑅1‘∪ dom 𝑙)(𝑗(𝑙‘∪ dom 𝑙)𝑖 → (𝑗 ∈ 𝑔 ↔ 𝑗 ∈ ℎ)))}))‘((𝑅1‘dom 𝑙) ∖ ran 𝑔)))) = ◡recs((𝑔 ∈ V ↦ ((𝑔 ∈ V ↦ sup((𝑦‘𝑔), (𝑅1‘dom 𝑒), {⟨𝑔, ℎ⟩ ∣ ∃𝑖 ∈ (𝑅1‘∪ dom 𝑒)((𝑖 ∈ ℎ ∧ ¬ 𝑖 ∈ 𝑔) ∧ ∀𝑗 ∈ (𝑅1‘∪ dom 𝑒)(𝑗(𝑒‘∪ dom 𝑒)𝑖 → (𝑗 ∈ 𝑔 ↔ 𝑗 ∈ ℎ)))}))‘((𝑅1‘dom 𝑒) ∖ ran 𝑔)))))
127	126	imaeq1d 5895	. . . . . . . . 9 ⊢ (𝑙 = 𝑒 → (◡recs((𝑔 ∈ V ↦ ((𝑔 ∈ V ↦ sup((𝑦‘𝑔), (𝑅1‘dom 𝑙), {⟨𝑔, ℎ⟩ ∣ ∃𝑖 ∈ (𝑅1‘∪ dom 𝑙)((𝑖 ∈ ℎ ∧ ¬ 𝑖 ∈ 𝑔) ∧ ∀𝑗 ∈ (𝑅1‘∪ dom 𝑙)(𝑗(𝑙‘∪ dom 𝑙)𝑖 → (𝑗 ∈ 𝑔 ↔ 𝑗 ∈ ℎ)))}))‘((𝑅1‘dom 𝑙) ∖ ran 𝑔)))) “ {𝑔}) = (◡recs((𝑔 ∈ V ↦ ((𝑔 ∈ V ↦ sup((𝑦‘𝑔), (𝑅1‘dom 𝑒), {⟨𝑔, ℎ⟩ ∣ ∃𝑖 ∈ (𝑅1‘∪ dom 𝑒)((𝑖 ∈ ℎ ∧ ¬ 𝑖 ∈ 𝑔) ∧ ∀𝑗 ∈ (𝑅1‘∪ dom 𝑒)(𝑗(𝑒‘∪ dom 𝑒)𝑖 → (𝑗 ∈ 𝑔 ↔ 𝑗 ∈ ℎ)))}))‘((𝑅1‘dom 𝑒) ∖ ran 𝑔)))) “ {𝑔}))
128	127	inteqd 4843	. . . . . . . 8 ⊢ (𝑙 = 𝑒 → ∩ (◡recs((𝑔 ∈ V ↦ ((𝑔 ∈ V ↦ sup((𝑦‘𝑔), (𝑅1‘dom 𝑙), {⟨𝑔, ℎ⟩ ∣ ∃𝑖 ∈ (𝑅1‘∪ dom 𝑙)((𝑖 ∈ ℎ ∧ ¬ 𝑖 ∈ 𝑔) ∧ ∀𝑗 ∈ (𝑅1‘∪ dom 𝑙)(𝑗(𝑙‘∪ dom 𝑙)𝑖 → (𝑗 ∈ 𝑔 ↔ 𝑗 ∈ ℎ)))}))‘((𝑅1‘dom 𝑙) ∖ ran 𝑔)))) “ {𝑔}) = ∩ (◡recs((𝑔 ∈ V ↦ ((𝑔 ∈ V ↦ sup((𝑦‘𝑔), (𝑅1‘dom 𝑒), {⟨𝑔, ℎ⟩ ∣ ∃𝑖 ∈ (𝑅1‘∪ dom 𝑒)((𝑖 ∈ ℎ ∧ ¬ 𝑖 ∈ 𝑔) ∧ ∀𝑗 ∈ (𝑅1‘∪ dom 𝑒)(𝑗(𝑒‘∪ dom 𝑒)𝑖 → (𝑗 ∈ 𝑔 ↔ 𝑗 ∈ ℎ)))}))‘((𝑅1‘dom 𝑒) ∖ ran 𝑔)))) “ {𝑔}))
129	126	imaeq1d 5895	. . . . . . . . 9 ⊢ (𝑙 = 𝑒 → (◡recs((𝑔 ∈ V ↦ ((𝑔 ∈ V ↦ sup((𝑦‘𝑔), (𝑅1‘dom 𝑙), {⟨𝑔, ℎ⟩ ∣ ∃𝑖 ∈ (𝑅1‘∪ dom 𝑙)((𝑖 ∈ ℎ ∧ ¬ 𝑖 ∈ 𝑔) ∧ ∀𝑗 ∈ (𝑅1‘∪ dom 𝑙)(𝑗(𝑙‘∪ dom 𝑙)𝑖 → (𝑗 ∈ 𝑔 ↔ 𝑗 ∈ ℎ)))}))‘((𝑅1‘dom 𝑙) ∖ ran 𝑔)))) “ {ℎ}) = (◡recs((𝑔 ∈ V ↦ ((𝑔 ∈ V ↦ sup((𝑦‘𝑔), (𝑅1‘dom 𝑒), {⟨𝑔, ℎ⟩ ∣ ∃𝑖 ∈ (𝑅1‘∪ dom 𝑒)((𝑖 ∈ ℎ ∧ ¬ 𝑖 ∈ 𝑔) ∧ ∀𝑗 ∈ (𝑅1‘∪ dom 𝑒)(𝑗(𝑒‘∪ dom 𝑒)𝑖 → (𝑗 ∈ 𝑔 ↔ 𝑗 ∈ ℎ)))}))‘((𝑅1‘dom 𝑒) ∖ ran 𝑔)))) “ {ℎ}))
130	129	inteqd 4843	. . . . . . . 8 ⊢ (𝑙 = 𝑒 → ∩ (◡recs((𝑔 ∈ V ↦ ((𝑔 ∈ V ↦ sup((𝑦‘𝑔), (𝑅1‘dom 𝑙), {⟨𝑔, ℎ⟩ ∣ ∃𝑖 ∈ (𝑅1‘∪ dom 𝑙)((𝑖 ∈ ℎ ∧ ¬ 𝑖 ∈ 𝑔) ∧ ∀𝑗 ∈ (𝑅1‘∪ dom 𝑙)(𝑗(𝑙‘∪ dom 𝑙)𝑖 → (𝑗 ∈ 𝑔 ↔ 𝑗 ∈ ℎ)))}))‘((𝑅1‘dom 𝑙) ∖ ran 𝑔)))) “ {ℎ}) = ∩ (◡recs((𝑔 ∈ V ↦ ((𝑔 ∈ V ↦ sup((𝑦‘𝑔), (𝑅1‘dom 𝑒), {⟨𝑔, ℎ⟩ ∣ ∃𝑖 ∈ (𝑅1‘∪ dom 𝑒)((𝑖 ∈ ℎ ∧ ¬ 𝑖 ∈ 𝑔) ∧ ∀𝑗 ∈ (𝑅1‘∪ dom 𝑒)(𝑗(𝑒‘∪ dom 𝑒)𝑖 → (𝑗 ∈ 𝑔 ↔ 𝑗 ∈ ℎ)))}))‘((𝑅1‘dom 𝑒) ∖ ran 𝑔)))) “ {ℎ}))
131	128, 130	eleq12d 2884	. . . . . . 7 ⊢ (𝑙 = 𝑒 → (∩ (◡recs((𝑔 ∈ V ↦ ((𝑔 ∈ V ↦ sup((𝑦‘𝑔), (𝑅1‘dom 𝑙), {⟨𝑔, ℎ⟩ ∣ ∃𝑖 ∈ (𝑅1‘∪ dom 𝑙)((𝑖 ∈ ℎ ∧ ¬ 𝑖 ∈ 𝑔) ∧ ∀𝑗 ∈ (𝑅1‘∪ dom 𝑙)(𝑗(𝑙‘∪ dom 𝑙)𝑖 → (𝑗 ∈ 𝑔 ↔ 𝑗 ∈ ℎ)))}))‘((𝑅1‘dom 𝑙) ∖ ran 𝑔)))) “ {𝑔}) ∈ ∩ (◡recs((𝑔 ∈ V ↦ ((𝑔 ∈ V ↦ sup((𝑦‘𝑔), (𝑅1‘dom 𝑙), {⟨𝑔, ℎ⟩ ∣ ∃𝑖 ∈ (𝑅1‘∪ dom 𝑙)((𝑖 ∈ ℎ ∧ ¬ 𝑖 ∈ 𝑔) ∧ ∀𝑗 ∈ (𝑅1‘∪ dom 𝑙)(𝑗(𝑙‘∪ dom 𝑙)𝑖 → (𝑗 ∈ 𝑔 ↔ 𝑗 ∈ ℎ)))}))‘((𝑅1‘dom 𝑙) ∖ ran 𝑔)))) “ {ℎ}) ↔ ∩ (◡recs((𝑔 ∈ V ↦ ((𝑔 ∈ V ↦ sup((𝑦‘𝑔), (𝑅1‘dom 𝑒), {⟨𝑔, ℎ⟩ ∣ ∃𝑖 ∈ (𝑅1‘∪ dom 𝑒)((𝑖 ∈ ℎ ∧ ¬ 𝑖 ∈ 𝑔) ∧ ∀𝑗 ∈ (𝑅1‘∪ dom 𝑒)(𝑗(𝑒‘∪ dom 𝑒)𝑖 → (𝑗 ∈ 𝑔 ↔ 𝑗 ∈ ℎ)))}))‘((𝑅1‘dom 𝑒) ∖ ran 𝑔)))) “ {𝑔}) ∈ ∩ (◡recs((𝑔 ∈ V ↦ ((𝑔 ∈ V ↦ sup((𝑦‘𝑔), (𝑅1‘dom 𝑒), {⟨𝑔, ℎ⟩ ∣ ∃𝑖 ∈ (𝑅1‘∪ dom 𝑒)((𝑖 ∈ ℎ ∧ ¬ 𝑖 ∈ 𝑔) ∧ ∀𝑗 ∈ (𝑅1‘∪ dom 𝑒)(𝑗(𝑒‘∪ dom 𝑒)𝑖 → (𝑗 ∈ 𝑔 ↔ 𝑗 ∈ ℎ)))}))‘((𝑅1‘dom 𝑒) ∖ ran 𝑔)))) “ {ℎ})))
132	131	opabbidv 5096	. . . . . 6 ⊢ (𝑙 = 𝑒 → {⟨𝑔, ℎ⟩ ∣ ∩ (◡recs((𝑔 ∈ V ↦ ((𝑔 ∈ V ↦ sup((𝑦‘𝑔), (𝑅1‘dom 𝑙), {⟨𝑔, ℎ⟩ ∣ ∃𝑖 ∈ (𝑅1‘∪ dom 𝑙)((𝑖 ∈ ℎ ∧ ¬ 𝑖 ∈ 𝑔) ∧ ∀𝑗 ∈ (𝑅1‘∪ dom 𝑙)(𝑗(𝑙‘∪ dom 𝑙)𝑖 → (𝑗 ∈ 𝑔 ↔ 𝑗 ∈ ℎ)))}))‘((𝑅1‘dom 𝑙) ∖ ran 𝑔)))) “ {𝑔}) ∈ ∩ (◡recs((𝑔 ∈ V ↦ ((𝑔 ∈ V ↦ sup((𝑦‘𝑔), (𝑅1‘dom 𝑙), {⟨𝑔, ℎ⟩ ∣ ∃𝑖 ∈ (𝑅1‘∪ dom 𝑙)((𝑖 ∈ ℎ ∧ ¬ 𝑖 ∈ 𝑔) ∧ ∀𝑗 ∈ (𝑅1‘∪ dom 𝑙)(𝑗(𝑙‘∪ dom 𝑙)𝑖 → (𝑗 ∈ 𝑔 ↔ 𝑗 ∈ ℎ)))}))‘((𝑅1‘dom 𝑙) ∖ ran 𝑔)))) “ {ℎ})} = {⟨𝑔, ℎ⟩ ∣ ∩ (◡recs((𝑔 ∈ V ↦ ((𝑔 ∈ V ↦ sup((𝑦‘𝑔), (𝑅1‘dom 𝑒), {⟨𝑔, ℎ⟩ ∣ ∃𝑖 ∈ (𝑅1‘∪ dom 𝑒)((𝑖 ∈ ℎ ∧ ¬ 𝑖 ∈ 𝑔) ∧ ∀𝑗 ∈ (𝑅1‘∪ dom 𝑒)(𝑗(𝑒‘∪ dom 𝑒)𝑖 → (𝑗 ∈ 𝑔 ↔ 𝑗 ∈ ℎ)))}))‘((𝑅1‘dom 𝑒) ∖ ran 𝑔)))) “ {𝑔}) ∈ ∩ (◡recs((𝑔 ∈ V ↦ ((𝑔 ∈ V ↦ sup((𝑦‘𝑔), (𝑅1‘dom 𝑒), {⟨𝑔, ℎ⟩ ∣ ∃𝑖 ∈ (𝑅1‘∪ dom 𝑒)((𝑖 ∈ ℎ ∧ ¬ 𝑖 ∈ 𝑔) ∧ ∀𝑗 ∈ (𝑅1‘∪ dom 𝑒)(𝑗(𝑒‘∪ dom 𝑒)𝑖 → (𝑗 ∈ 𝑔 ↔ 𝑗 ∈ ℎ)))}))‘((𝑅1‘dom 𝑒) ∖ ran 𝑔)))) “ {ℎ})})
133	102, 107, 132	ifbieq12d 4452	. . . . 5 ⊢ (𝑙 = 𝑒 → if(dom 𝑙 = ∪ dom 𝑙, {⟨𝑔, ℎ⟩ ∣ ((rank‘𝑔) E (rank‘ℎ) ∨ ((rank‘𝑔) = (rank‘ℎ) ∧ 𝑔(𝑙‘suc (rank‘𝑔))ℎ))}, {⟨𝑔, ℎ⟩ ∣ ∩ (◡recs((𝑔 ∈ V ↦ ((𝑔 ∈ V ↦ sup((𝑦‘𝑔), (𝑅1‘dom 𝑙), {⟨𝑔, ℎ⟩ ∣ ∃𝑖 ∈ (𝑅1‘∪ dom 𝑙)((𝑖 ∈ ℎ ∧ ¬ 𝑖 ∈ 𝑔) ∧ ∀𝑗 ∈ (𝑅1‘∪ dom 𝑙)(𝑗(𝑙‘∪ dom 𝑙)𝑖 → (𝑗 ∈ 𝑔 ↔ 𝑗 ∈ ℎ)))}))‘((𝑅1‘dom 𝑙) ∖ ran 𝑔)))) “ {𝑔}) ∈ ∩ (◡recs((𝑔 ∈ V ↦ ((𝑔 ∈ V ↦ sup((𝑦‘𝑔), (𝑅1‘dom 𝑙), {⟨𝑔, ℎ⟩ ∣ ∃𝑖 ∈ (𝑅1‘∪ dom 𝑙)((𝑖 ∈ ℎ ∧ ¬ 𝑖 ∈ 𝑔) ∧ ∀𝑗 ∈ (𝑅1‘∪ dom 𝑙)(𝑗(𝑙‘∪ dom 𝑙)𝑖 → (𝑗 ∈ 𝑔 ↔ 𝑗 ∈ ℎ)))}))‘((𝑅1‘dom 𝑙) ∖ ran 𝑔)))) “ {ℎ})}) = if(dom 𝑒 = ∪ dom 𝑒, {⟨𝑔, ℎ⟩ ∣ ((rank‘𝑔) E (rank‘ℎ) ∨ ((rank‘𝑔) = (rank‘ℎ) ∧ 𝑔(𝑒‘suc (rank‘𝑔))ℎ))}, {⟨𝑔, ℎ⟩ ∣ ∩ (◡recs((𝑔 ∈ V ↦ ((𝑔 ∈ V ↦ sup((𝑦‘𝑔), (𝑅1‘dom 𝑒), {⟨𝑔, ℎ⟩ ∣ ∃𝑖 ∈ (𝑅1‘∪ dom 𝑒)((𝑖 ∈ ℎ ∧ ¬ 𝑖 ∈ 𝑔) ∧ ∀𝑗 ∈ (𝑅1‘∪ dom 𝑒)(𝑗(𝑒‘∪ dom 𝑒)𝑖 → (𝑗 ∈ 𝑔 ↔ 𝑗 ∈ ℎ)))}))‘((𝑅1‘dom 𝑒) ∖ ran 𝑔)))) “ {𝑔}) ∈ ∩ (◡recs((𝑔 ∈ V ↦ ((𝑔 ∈ V ↦ sup((𝑦‘𝑔), (𝑅1‘dom 𝑒), {⟨𝑔, ℎ⟩ ∣ ∃𝑖 ∈ (𝑅1‘∪ dom 𝑒)((𝑖 ∈ ℎ ∧ ¬ 𝑖 ∈ 𝑔) ∧ ∀𝑗 ∈ (𝑅1‘∪ dom 𝑒)(𝑗(𝑒‘∪ dom 𝑒)𝑖 → (𝑗 ∈ 𝑔 ↔ 𝑗 ∈ ℎ)))}))‘((𝑅1‘dom 𝑒) ∖ ran 𝑔)))) “ {ℎ})}))
134	109	sqxpeqd 5551	. . . . 5 ⊢ (𝑙 = 𝑒 → ((𝑅1‘dom 𝑙) × (𝑅1‘dom 𝑙)) = ((𝑅1‘dom 𝑒) × (𝑅1‘dom 𝑒)))
135	133, 134	ineq12d 4140	. . . 4 ⊢ (𝑙 = 𝑒 → (if(dom 𝑙 = ∪ dom 𝑙, {⟨𝑔, ℎ⟩ ∣ ((rank‘𝑔) E (rank‘ℎ) ∨ ((rank‘𝑔) = (rank‘ℎ) ∧ 𝑔(𝑙‘suc (rank‘𝑔))ℎ))}, {⟨𝑔, ℎ⟩ ∣ ∩ (◡recs((𝑔 ∈ V ↦ ((𝑔 ∈ V ↦ sup((𝑦‘𝑔), (𝑅1‘dom 𝑙), {⟨𝑔, ℎ⟩ ∣ ∃𝑖 ∈ (𝑅1‘∪ dom 𝑙)((𝑖 ∈ ℎ ∧ ¬ 𝑖 ∈ 𝑔) ∧ ∀𝑗 ∈ (𝑅1‘∪ dom 𝑙)(𝑗(𝑙‘∪ dom 𝑙)𝑖 → (𝑗 ∈ 𝑔 ↔ 𝑗 ∈ ℎ)))}))‘((𝑅1‘dom 𝑙) ∖ ran 𝑔)))) “ {𝑔}) ∈ ∩ (◡recs((𝑔 ∈ V ↦ ((𝑔 ∈ V ↦ sup((𝑦‘𝑔), (𝑅1‘dom 𝑙), {⟨𝑔, ℎ⟩ ∣ ∃𝑖 ∈ (𝑅1‘∪ dom 𝑙)((𝑖 ∈ ℎ ∧ ¬ 𝑖 ∈ 𝑔) ∧ ∀𝑗 ∈ (𝑅1‘∪ dom 𝑙)(𝑗(𝑙‘∪ dom 𝑙)𝑖 → (𝑗 ∈ 𝑔 ↔ 𝑗 ∈ ℎ)))}))‘((𝑅1‘dom 𝑙) ∖ ran 𝑔)))) “ {ℎ})}) ∩ ((𝑅1‘dom 𝑙) × (𝑅1‘dom 𝑙))) = (if(dom 𝑒 = ∪ dom 𝑒, {⟨𝑔, ℎ⟩ ∣ ((rank‘𝑔) E (rank‘ℎ) ∨ ((rank‘𝑔) = (rank‘ℎ) ∧ 𝑔(𝑒‘suc (rank‘𝑔))ℎ))}, {⟨𝑔, ℎ⟩ ∣ ∩ (◡recs((𝑔 ∈ V ↦ ((𝑔 ∈ V ↦ sup((𝑦‘𝑔), (𝑅1‘dom 𝑒), {⟨𝑔, ℎ⟩ ∣ ∃𝑖 ∈ (𝑅1‘∪ dom 𝑒)((𝑖 ∈ ℎ ∧ ¬ 𝑖 ∈ 𝑔) ∧ ∀𝑗 ∈ (𝑅1‘∪ dom 𝑒)(𝑗(𝑒‘∪ dom 𝑒)𝑖 → (𝑗 ∈ 𝑔 ↔ 𝑗 ∈ ℎ)))}))‘((𝑅1‘dom 𝑒) ∖ ran 𝑔)))) “ {𝑔}) ∈ ∩ (◡recs((𝑔 ∈ V ↦ ((𝑔 ∈ V ↦ sup((𝑦‘𝑔), (𝑅1‘dom 𝑒), {⟨𝑔, ℎ⟩ ∣ ∃𝑖 ∈ (𝑅1‘∪ dom 𝑒)((𝑖 ∈ ℎ ∧ ¬ 𝑖 ∈ 𝑔) ∧ ∀𝑗 ∈ (𝑅1‘∪ dom 𝑒)(𝑗(𝑒‘∪ dom 𝑒)𝑖 → (𝑗 ∈ 𝑔 ↔ 𝑗 ∈ ℎ)))}))‘((𝑅1‘dom 𝑒) ∖ ran 𝑔)))) “ {ℎ})}) ∩ ((𝑅1‘dom 𝑒) × (𝑅1‘dom 𝑒))))
136	135	cbvmptv 5133	. . 3 ⊢ (𝑙 ∈ V ↦ (if(dom 𝑙 = ∪ dom 𝑙, {⟨𝑔, ℎ⟩ ∣ ((rank‘𝑔) E (rank‘ℎ) ∨ ((rank‘𝑔) = (rank‘ℎ) ∧ 𝑔(𝑙‘suc (rank‘𝑔))ℎ))}, {⟨𝑔, ℎ⟩ ∣ ∩ (◡recs((𝑔 ∈ V ↦ ((𝑔 ∈ V ↦ sup((𝑦‘𝑔), (𝑅1‘dom 𝑙), {⟨𝑔, ℎ⟩ ∣ ∃𝑖 ∈ (𝑅1‘∪ dom 𝑙)((𝑖 ∈ ℎ ∧ ¬ 𝑖 ∈ 𝑔) ∧ ∀𝑗 ∈ (𝑅1‘∪ dom 𝑙)(𝑗(𝑙‘∪ dom 𝑙)𝑖 → (𝑗 ∈ 𝑔 ↔ 𝑗 ∈ ℎ)))}))‘((𝑅1‘dom 𝑙) ∖ ran 𝑔)))) “ {𝑔}) ∈ ∩ (◡recs((𝑔 ∈ V ↦ ((𝑔 ∈ V ↦ sup((𝑦‘𝑔), (𝑅1‘dom 𝑙), {⟨𝑔, ℎ⟩ ∣ ∃𝑖 ∈ (𝑅1‘∪ dom 𝑙)((𝑖 ∈ ℎ ∧ ¬ 𝑖 ∈ 𝑔) ∧ ∀𝑗 ∈ (𝑅1‘∪ dom 𝑙)(𝑗(𝑙‘∪ dom 𝑙)𝑖 → (𝑗 ∈ 𝑔 ↔ 𝑗 ∈ ℎ)))}))‘((𝑅1‘dom 𝑙) ∖ ran 𝑔)))) “ {ℎ})}) ∩ ((𝑅1‘dom 𝑙) × (𝑅1‘dom 𝑙)))) = (𝑒 ∈ V ↦ (if(dom 𝑒 = ∪ dom 𝑒, {⟨𝑔, ℎ⟩ ∣ ((rank‘𝑔) E (rank‘ℎ) ∨ ((rank‘𝑔) = (rank‘ℎ) ∧ 𝑔(𝑒‘suc (rank‘𝑔))ℎ))}, {⟨𝑔, ℎ⟩ ∣ ∩ (◡recs((𝑔 ∈ V ↦ ((𝑔 ∈ V ↦ sup((𝑦‘𝑔), (𝑅1‘dom 𝑒), {⟨𝑔, ℎ⟩ ∣ ∃𝑖 ∈ (𝑅1‘∪ dom 𝑒)((𝑖 ∈ ℎ ∧ ¬ 𝑖 ∈ 𝑔) ∧ ∀𝑗 ∈ (𝑅1‘∪ dom 𝑒)(𝑗(𝑒‘∪ dom 𝑒)𝑖 → (𝑗 ∈ 𝑔 ↔ 𝑗 ∈ ℎ)))}))‘((𝑅1‘dom 𝑒) ∖ ran 𝑔)))) “ {𝑔}) ∈ ∩ (◡recs((𝑔 ∈ V ↦ ((𝑔 ∈ V ↦ sup((𝑦‘𝑔), (𝑅1‘dom 𝑒), {⟨𝑔, ℎ⟩ ∣ ∃𝑖 ∈ (𝑅1‘∪ dom 𝑒)((𝑖 ∈ ℎ ∧ ¬ 𝑖 ∈ 𝑔) ∧ ∀𝑗 ∈ (𝑅1‘∪ dom 𝑒)(𝑗(𝑒‘∪ dom 𝑒)𝑖 → (𝑗 ∈ 𝑔 ↔ 𝑗 ∈ ℎ)))}))‘((𝑅1‘dom 𝑒) ∖ ran 𝑔)))) “ {ℎ})}) ∩ ((𝑅1‘dom 𝑒) × (𝑅1‘dom 𝑒))))
137		recseq 7993	. . 3 ⊢ ((𝑙 ∈ V ↦ (if(dom 𝑙 = ∪ dom 𝑙, {⟨𝑔, ℎ⟩ ∣ ((rank‘𝑔) E (rank‘ℎ) ∨ ((rank‘𝑔) = (rank‘ℎ) ∧ 𝑔(𝑙‘suc (rank‘𝑔))ℎ))}, {⟨𝑔, ℎ⟩ ∣ ∩ (◡recs((𝑔 ∈ V ↦ ((𝑔 ∈ V ↦ sup((𝑦‘𝑔), (𝑅1‘dom 𝑙), {⟨𝑔, ℎ⟩ ∣ ∃𝑖 ∈ (𝑅1‘∪ dom 𝑙)((𝑖 ∈ ℎ ∧ ¬ 𝑖 ∈ 𝑔) ∧ ∀𝑗 ∈ (𝑅1‘∪ dom 𝑙)(𝑗(𝑙‘∪ dom 𝑙)𝑖 → (𝑗 ∈ 𝑔 ↔ 𝑗 ∈ ℎ)))}))‘((𝑅1‘dom 𝑙) ∖ ran 𝑔)))) “ {𝑔}) ∈ ∩ (◡recs((𝑔 ∈ V ↦ ((𝑔 ∈ V ↦ sup((𝑦‘𝑔), (𝑅1‘dom 𝑙), {⟨𝑔, ℎ⟩ ∣ ∃𝑖 ∈ (𝑅1‘∪ dom 𝑙)((𝑖 ∈ ℎ ∧ ¬ 𝑖 ∈ 𝑔) ∧ ∀𝑗 ∈ (𝑅1‘∪ dom 𝑙)(𝑗(𝑙‘∪ dom 𝑙)𝑖 → (𝑗 ∈ 𝑔 ↔ 𝑗 ∈ ℎ)))}))‘((𝑅1‘dom 𝑙) ∖ ran 𝑔)))) “ {ℎ})}) ∩ ((𝑅1‘dom 𝑙) × (𝑅1‘dom 𝑙)))) = (𝑒 ∈ V ↦ (if(dom 𝑒 = ∪ dom 𝑒, {⟨𝑔, ℎ⟩ ∣ ((rank‘𝑔) E (rank‘ℎ) ∨ ((rank‘𝑔) = (rank‘ℎ) ∧ 𝑔(𝑒‘suc (rank‘𝑔))ℎ))}, {⟨𝑔, ℎ⟩ ∣ ∩ (◡recs((𝑔 ∈ V ↦ ((𝑔 ∈ V ↦ sup((𝑦‘𝑔), (𝑅1‘dom 𝑒), {⟨𝑔, ℎ⟩ ∣ ∃𝑖 ∈ (𝑅1‘∪ dom 𝑒)((𝑖 ∈ ℎ ∧ ¬ 𝑖 ∈ 𝑔) ∧ ∀𝑗 ∈ (𝑅1‘∪ dom 𝑒)(𝑗(𝑒‘∪ dom 𝑒)𝑖 → (𝑗 ∈ 𝑔 ↔ 𝑗 ∈ ℎ)))}))‘((𝑅1‘dom 𝑒) ∖ ran 𝑔)))) “ {𝑔}) ∈ ∩ (◡recs((𝑔 ∈ V ↦ ((𝑔 ∈ V ↦ sup((𝑦‘𝑔), (𝑅1‘dom 𝑒), {⟨𝑔, ℎ⟩ ∣ ∃𝑖 ∈ (𝑅1‘∪ dom 𝑒)((𝑖 ∈ ℎ ∧ ¬ 𝑖 ∈ 𝑔) ∧ ∀𝑗 ∈ (𝑅1‘∪ dom 𝑒)(𝑗(𝑒‘∪ dom 𝑒)𝑖 → (𝑗 ∈ 𝑔 ↔ 𝑗 ∈ ℎ)))}))‘((𝑅1‘dom 𝑒) ∖ ran 𝑔)))) “ {ℎ})}) ∩ ((𝑅1‘dom 𝑒) × (𝑅1‘dom 𝑒)))) → recs((𝑙 ∈ V ↦ (if(dom 𝑙 = ∪ dom 𝑙, {⟨𝑔, ℎ⟩ ∣ ((rank‘𝑔) E (rank‘ℎ) ∨ ((rank‘𝑔) = (rank‘ℎ) ∧ 𝑔(𝑙‘suc (rank‘𝑔))ℎ))}, {⟨𝑔, ℎ⟩ ∣ ∩ (◡recs((𝑔 ∈ V ↦ ((𝑔 ∈ V ↦ sup((𝑦‘𝑔), (𝑅1‘dom 𝑙), {⟨𝑔, ℎ⟩ ∣ ∃𝑖 ∈ (𝑅1‘∪ dom 𝑙)((𝑖 ∈ ℎ ∧ ¬ 𝑖 ∈ 𝑔) ∧ ∀𝑗 ∈ (𝑅1‘∪ dom 𝑙)(𝑗(𝑙‘∪ dom 𝑙)𝑖 → (𝑗 ∈ 𝑔 ↔ 𝑗 ∈ ℎ)))}))‘((𝑅1‘dom 𝑙) ∖ ran 𝑔)))) “ {𝑔}) ∈ ∩ (◡recs((𝑔 ∈ V ↦ ((𝑔 ∈ V ↦ sup((𝑦‘𝑔), (𝑅1‘dom 𝑙), {⟨𝑔, ℎ⟩ ∣ ∃𝑖 ∈ (𝑅1‘∪ dom 𝑙)((𝑖 ∈ ℎ ∧ ¬ 𝑖 ∈ 𝑔) ∧ ∀𝑗 ∈ (𝑅1‘∪ dom 𝑙)(𝑗(𝑙‘∪ dom 𝑙)𝑖 → (𝑗 ∈ 𝑔 ↔ 𝑗 ∈ ℎ)))}))‘((𝑅1‘dom 𝑙) ∖ ran 𝑔)))) “ {ℎ})}) ∩ ((𝑅1‘dom 𝑙) × (𝑅1‘dom 𝑙))))) = recs((𝑒 ∈ V ↦ (if(dom 𝑒 = ∪ dom 𝑒, {⟨𝑔, ℎ⟩ ∣ ((rank‘𝑔) E (rank‘ℎ) ∨ ((rank‘𝑔) = (rank‘ℎ) ∧ 𝑔(𝑒‘suc (rank‘𝑔))ℎ))}, {⟨𝑔, ℎ⟩ ∣ ∩ (◡recs((𝑔 ∈ V ↦ ((𝑔 ∈ V ↦ sup((𝑦‘𝑔), (𝑅1‘dom 𝑒), {⟨𝑔, ℎ⟩ ∣ ∃𝑖 ∈ (𝑅1‘∪ dom 𝑒)((𝑖 ∈ ℎ ∧ ¬ 𝑖 ∈ 𝑔) ∧ ∀𝑗 ∈ (𝑅1‘∪ dom 𝑒)(𝑗(𝑒‘∪ dom 𝑒)𝑖 → (𝑗 ∈ 𝑔 ↔ 𝑗 ∈ ℎ)))}))‘((𝑅1‘dom 𝑒) ∖ ran 𝑔)))) “ {𝑔}) ∈ ∩ (◡recs((𝑔 ∈ V ↦ ((𝑔 ∈ V ↦ sup((𝑦‘𝑔), (𝑅1‘dom 𝑒), {⟨𝑔, ℎ⟩ ∣ ∃𝑖 ∈ (𝑅1‘∪ dom 𝑒)((𝑖 ∈ ℎ ∧ ¬ 𝑖 ∈ 𝑔) ∧ ∀𝑗 ∈ (𝑅1‘∪ dom 𝑒)(𝑗(𝑒‘∪ dom 𝑒)𝑖 → (𝑗 ∈ 𝑔 ↔ 𝑗 ∈ ℎ)))}))‘((𝑅1‘dom 𝑒) ∖ ran 𝑔)))) “ {ℎ})}) ∩ ((𝑅1‘dom 𝑒) × (𝑅1‘dom 𝑒))))))
138	136, 137	ax-mp 5	. 2 ⊢ recs((𝑙 ∈ V ↦ (if(dom 𝑙 = ∪ dom 𝑙, {⟨𝑔, ℎ⟩ ∣ ((rank‘𝑔) E (rank‘ℎ) ∨ ((rank‘𝑔) = (rank‘ℎ) ∧ 𝑔(𝑙‘suc (rank‘𝑔))ℎ))}, {⟨𝑔, ℎ⟩ ∣ ∩ (◡recs((𝑔 ∈ V ↦ ((𝑔 ∈ V ↦ sup((𝑦‘𝑔), (𝑅1‘dom 𝑙), {⟨𝑔, ℎ⟩ ∣ ∃𝑖 ∈ (𝑅1‘∪ dom 𝑙)((𝑖 ∈ ℎ ∧ ¬ 𝑖 ∈ 𝑔) ∧ ∀𝑗 ∈ (𝑅1‘∪ dom 𝑙)(𝑗(𝑙‘∪ dom 𝑙)𝑖 → (𝑗 ∈ 𝑔 ↔ 𝑗 ∈ ℎ)))}))‘((𝑅1‘dom 𝑙) ∖ ran 𝑔)))) “ {𝑔}) ∈ ∩ (◡recs((𝑔 ∈ V ↦ ((𝑔 ∈ V ↦ sup((𝑦‘𝑔), (𝑅1‘dom 𝑙), {⟨𝑔, ℎ⟩ ∣ ∃𝑖 ∈ (𝑅1‘∪ dom 𝑙)((𝑖 ∈ ℎ ∧ ¬ 𝑖 ∈ 𝑔) ∧ ∀𝑗 ∈ (𝑅1‘∪ dom 𝑙)(𝑗(𝑙‘∪ dom 𝑙)𝑖 → (𝑗 ∈ 𝑔 ↔ 𝑗 ∈ ℎ)))}))‘((𝑅1‘dom 𝑙) ∖ ran 𝑔)))) “ {ℎ})}) ∩ ((𝑅1‘dom 𝑙) × (𝑅1‘dom 𝑙))))) = recs((𝑒 ∈ V ↦ (if(dom 𝑒 = ∪ dom 𝑒, {⟨𝑔, ℎ⟩ ∣ ((rank‘𝑔) E (rank‘ℎ) ∨ ((rank‘𝑔) = (rank‘ℎ) ∧ 𝑔(𝑒‘suc (rank‘𝑔))ℎ))}, {⟨𝑔, ℎ⟩ ∣ ∩ (◡recs((𝑔 ∈ V ↦ ((𝑔 ∈ V ↦ sup((𝑦‘𝑔), (𝑅1‘dom 𝑒), {⟨𝑔, ℎ⟩ ∣ ∃𝑖 ∈ (𝑅1‘∪ dom 𝑒)((𝑖 ∈ ℎ ∧ ¬ 𝑖 ∈ 𝑔) ∧ ∀𝑗 ∈ (𝑅1‘∪ dom 𝑒)(𝑗(𝑒‘∪ dom 𝑒)𝑖 → (𝑗 ∈ 𝑔 ↔ 𝑗 ∈ ℎ)))}))‘((𝑅1‘dom 𝑒) ∖ ran 𝑔)))) “ {𝑔}) ∈ ∩ (◡recs((𝑔 ∈ V ↦ ((𝑔 ∈ V ↦ sup((𝑦‘𝑔), (𝑅1‘dom 𝑒), {⟨𝑔, ℎ⟩ ∣ ∃𝑖 ∈ (𝑅1‘∪ dom 𝑒)((𝑖 ∈ ℎ ∧ ¬ 𝑖 ∈ 𝑔) ∧ ∀𝑗 ∈ (𝑅1‘∪ dom 𝑒)(𝑗(𝑒‘∪ dom 𝑒)𝑖 → (𝑗 ∈ 𝑔 ↔ 𝑗 ∈ ℎ)))}))‘((𝑅1‘dom 𝑒) ∖ ran 𝑔)))) “ {ℎ})}) ∩ ((𝑅1‘dom 𝑒) × (𝑅1‘dom 𝑒)))))
139		aomclem8.a	. 2 ⊢ (𝜑 → 𝐴 ∈ On)
140		aomclem8.y	. . 3 ⊢ (𝜑 → ∀𝑎 ∈ 𝒫 (𝑅1‘𝐴)(𝑎 ≠ ∅ → (𝑦‘𝑎) ∈ ((𝒫 𝑎 ∩ Fin) ∖ {∅})))
141		neeq1 3049	. . . . 5 ⊢ (𝑎 = 𝑐 → (𝑎 ≠ ∅ ↔ 𝑐 ≠ ∅))
142		fveq2 6645	. . . . . 6 ⊢ (𝑎 = 𝑐 → (𝑦‘𝑎) = (𝑦‘𝑐))
143		pweq 4513	. . . . . . . 8 ⊢ (𝑎 = 𝑐 → 𝒫 𝑎 = 𝒫 𝑐)
144	143	ineq1d 4138	. . . . . . 7 ⊢ (𝑎 = 𝑐 → (𝒫 𝑎 ∩ Fin) = (𝒫 𝑐 ∩ Fin))
145	144	difeq1d 4049	. . . . . 6 ⊢ (𝑎 = 𝑐 → ((𝒫 𝑎 ∩ Fin) ∖ {∅}) = ((𝒫 𝑐 ∩ Fin) ∖ {∅}))
146	142, 145	eleq12d 2884	. . . . 5 ⊢ (𝑎 = 𝑐 → ((𝑦‘𝑎) ∈ ((𝒫 𝑎 ∩ Fin) ∖ {∅}) ↔ (𝑦‘𝑐) ∈ ((𝒫 𝑐 ∩ Fin) ∖ {∅})))
147	141, 146	imbi12d 348	. . . 4 ⊢ (𝑎 = 𝑐 → ((𝑎 ≠ ∅ → (𝑦‘𝑎) ∈ ((𝒫 𝑎 ∩ Fin) ∖ {∅})) ↔ (𝑐 ≠ ∅ → (𝑦‘𝑐) ∈ ((𝒫 𝑐 ∩ Fin) ∖ {∅}))))
148	147	cbvralvw 3396	. . 3 ⊢ (∀𝑎 ∈ 𝒫 (𝑅1‘𝐴)(𝑎 ≠ ∅ → (𝑦‘𝑎) ∈ ((𝒫 𝑎 ∩ Fin) ∖ {∅})) ↔ ∀𝑐 ∈ 𝒫 (𝑅1‘𝐴)(𝑐 ≠ ∅ → (𝑦‘𝑐) ∈ ((𝒫 𝑐 ∩ Fin) ∖ {∅})))
149	140, 148	sylib 221	. 2 ⊢ (𝜑 → ∀𝑐 ∈ 𝒫 (𝑅1‘𝐴)(𝑐 ≠ ∅ → (𝑦‘𝑐) ∈ ((𝒫 𝑐 ∩ Fin) ∖ {∅})))
150	29, 37, 45, 84, 98, 99, 138, 139, 149	aomclem7 40004	1 ⊢ (𝜑 → ∃𝑏 𝑏 We (𝑅1‘𝐴))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 399   ∨ wo 844   = wceq 1538  ∃wex 1781   ∈ wcel 2111   ≠ wne 2987  ∀wral 3106  ∃wrex 3107  Vcvv 3441   ∖ cdif 3878   ∩ cin 3880  ∅c0 4243  ifcif 4425  𝒫 cpw 4497  {csn 4525  ∪ cuni 4800  ∩ cint 4838   class class class wbr 5030  {copab 5092   ↦ cmpt 5110   E cep 5429   We wwe 5477   × cxp 5517  ^◡ccnv 5518  dom cdm 5519  ran crn 5520   “ cima 5522  Oncon0 6159  suc csuc 6161  ‘cfv 6324  recscrecs 7990  Fincfn 8492  supcsup 8888  𝑅₁cr1 9175  rankcrnk 9176

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-rep 5154  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7441

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-reu 3113  df-rmo 3114  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-pss 3900  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-tp 4530  df-op 4532  df-uni 4801  df-int 4839  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-tr 5137  df-id 5425  df-eprel 5430  df-po 5438  df-so 5439  df-fr 5478  df-we 5480  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-pred 6116  df-ord 6162  df-on 6163  df-lim 6164  df-suc 6165  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-f1 6329  df-fo 6330  df-f1o 6331  df-fv 6332  df-isom 6333  df-riota 7093  df-ov 7138  df-oprab 7139  df-mpo 7140  df-om 7561  df-1st 7671  df-2nd 7672  df-wrecs 7930  df-recs 7991  df-rdg 8029  df-1o 8085  df-2o 8086  df-er 8272  df-map 8391  df-en 8493  df-fin 8496  df-sup 8890  df-r1 9177  df-rank 9178

This theorem is referenced by:  dfac11  40006