Theorem satfv0 32605

Description: The value of the satisfaction predicate as function over wff codes at ∅. (Contributed by AV, 8-Oct-2023.)

Hypothesis

Ref	Expression
satfv0.s	⊢ 𝑆 = (𝑀 Sat 𝐸)

Assertion

Ref	Expression
satfv0	⊢ ((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) → (𝑆‘∅) = {⟨𝑥, 𝑦⟩ ∣ ∃𝑖 ∈ ω ∃𝑗 ∈ ω (𝑥 = (𝑖∈𝑔𝑗) ∧ 𝑦 = {𝑎 ∈ (𝑀 ↑m ω) ∣ (𝑎‘𝑖)𝐸(𝑎‘𝑗)})})

Distinct variable groups:   𝐸,𝑎,𝑖,𝑗,𝑥,𝑦   𝑀,𝑎,𝑖,𝑗,𝑥,𝑦

Allowed substitution hints:   𝑆(𝑥,𝑦,𝑖,𝑗,𝑎)   𝑉(𝑥,𝑦,𝑖,𝑗,𝑎)   𝑊(𝑥,𝑦,𝑖,𝑗,𝑎)

Proof of Theorem satfv0

Dummy variables 𝑓 𝑚 𝑛 𝑢 𝑣 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		peano1 7601	. . . 4 ⊢ ∅ ∈ ω
2		elelsuc 6263	. . . 4 ⊢ (∅ ∈ ω → ∅ ∈ suc ω)
3	1, 2	mp1i 13	. . 3 ⊢ ((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) → ∅ ∈ suc ω)
4		satfv0.s	. . . 4 ⊢ 𝑆 = (𝑀 Sat 𝐸)
5	4	satfvsucom 32604	. . 3 ⊢ ((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ∧ ∅ ∈ suc ω) → (𝑆‘∅) = (rec((𝑓 ∈ V ↦ (𝑓 ∪ {⟨𝑥, 𝑦⟩ ∣ ∃𝑢 ∈ 𝑓 (∃𝑣 ∈ 𝑓 (𝑥 = ((1st ‘𝑢)⊼𝑔(1st ‘𝑣)) ∧ 𝑦 = ((𝑀 ↑m ω) ∖ ((2nd ‘𝑢) ∩ (2nd ‘𝑣)))) ∨ ∃𝑖 ∈ ω (𝑥 = ∀𝑔𝑖(1st ‘𝑢) ∧ 𝑦 = {𝑎 ∈ (𝑀 ↑m ω) ∣ ∀𝑧 ∈ 𝑀 ({⟨𝑖, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑖}))) ∈ (2nd ‘𝑢)}))})), {⟨𝑥, 𝑦⟩ ∣ ∃𝑖 ∈ ω ∃𝑗 ∈ ω (𝑥 = (𝑖∈𝑔𝑗) ∧ 𝑦 = {𝑎 ∈ (𝑀 ↑m ω) ∣ (𝑎‘𝑖)𝐸(𝑎‘𝑗)})})‘∅))
6	3, 5	mpd3an3 1458	. 2 ⊢ ((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) → (𝑆‘∅) = (rec((𝑓 ∈ V ↦ (𝑓 ∪ {⟨𝑥, 𝑦⟩ ∣ ∃𝑢 ∈ 𝑓 (∃𝑣 ∈ 𝑓 (𝑥 = ((1st ‘𝑢)⊼𝑔(1st ‘𝑣)) ∧ 𝑦 = ((𝑀 ↑m ω) ∖ ((2nd ‘𝑢) ∩ (2nd ‘𝑣)))) ∨ ∃𝑖 ∈ ω (𝑥 = ∀𝑔𝑖(1st ‘𝑢) ∧ 𝑦 = {𝑎 ∈ (𝑀 ↑m ω) ∣ ∀𝑧 ∈ 𝑀 ({⟨𝑖, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑖}))) ∈ (2nd ‘𝑢)}))})), {⟨𝑥, 𝑦⟩ ∣ ∃𝑖 ∈ ω ∃𝑗 ∈ ω (𝑥 = (𝑖∈𝑔𝑗) ∧ 𝑦 = {𝑎 ∈ (𝑀 ↑m ω) ∣ (𝑎‘𝑖)𝐸(𝑎‘𝑗)})})‘∅))
7		goelel3xp 32595	. . . . . . . . . 10 ⊢ ((𝑖 ∈ ω ∧ 𝑗 ∈ ω) → (𝑖∈𝑔𝑗) ∈ (ω × (ω × ω)))
8		eleq1 2900	. . . . . . . . . 10 ⊢ (𝑥 = (𝑖∈𝑔𝑗) → (𝑥 ∈ (ω × (ω × ω)) ↔ (𝑖∈𝑔𝑗) ∈ (ω × (ω × ω))))
9	7, 8	syl5ibrcom 249	. . . . . . . . 9 ⊢ ((𝑖 ∈ ω ∧ 𝑗 ∈ ω) → (𝑥 = (𝑖∈𝑔𝑗) → 𝑥 ∈ (ω × (ω × ω))))
10	9	adantrd 494	. . . . . . . 8 ⊢ ((𝑖 ∈ ω ∧ 𝑗 ∈ ω) → ((𝑥 = (𝑖∈𝑔𝑗) ∧ 𝑦 = {𝑎 ∈ (𝑀 ↑m ω) ∣ (𝑎‘𝑖)𝐸(𝑎‘𝑗)}) → 𝑥 ∈ (ω × (ω × ω))))
11	10	pm4.71d 564	. . . . . . 7 ⊢ ((𝑖 ∈ ω ∧ 𝑗 ∈ ω) → ((𝑥 = (𝑖∈𝑔𝑗) ∧ 𝑦 = {𝑎 ∈ (𝑀 ↑m ω) ∣ (𝑎‘𝑖)𝐸(𝑎‘𝑗)}) ↔ ((𝑥 = (𝑖∈𝑔𝑗) ∧ 𝑦 = {𝑎 ∈ (𝑀 ↑m ω) ∣ (𝑎‘𝑖)𝐸(𝑎‘𝑗)}) ∧ 𝑥 ∈ (ω × (ω × ω)))))
12	11	2rexbiia 3298	. . . . . 6 ⊢ (∃𝑖 ∈ ω ∃𝑗 ∈ ω (𝑥 = (𝑖∈𝑔𝑗) ∧ 𝑦 = {𝑎 ∈ (𝑀 ↑m ω) ∣ (𝑎‘𝑖)𝐸(𝑎‘𝑗)}) ↔ ∃𝑖 ∈ ω ∃𝑗 ∈ ω ((𝑥 = (𝑖∈𝑔𝑗) ∧ 𝑦 = {𝑎 ∈ (𝑀 ↑m ω) ∣ (𝑎‘𝑖)𝐸(𝑎‘𝑗)}) ∧ 𝑥 ∈ (ω × (ω × ω))))
13		r19.41vv 3349	. . . . . 6 ⊢ (∃𝑖 ∈ ω ∃𝑗 ∈ ω ((𝑥 = (𝑖∈𝑔𝑗) ∧ 𝑦 = {𝑎 ∈ (𝑀 ↑m ω) ∣ (𝑎‘𝑖)𝐸(𝑎‘𝑗)}) ∧ 𝑥 ∈ (ω × (ω × ω))) ↔ (∃𝑖 ∈ ω ∃𝑗 ∈ ω (𝑥 = (𝑖∈𝑔𝑗) ∧ 𝑦 = {𝑎 ∈ (𝑀 ↑m ω) ∣ (𝑎‘𝑖)𝐸(𝑎‘𝑗)}) ∧ 𝑥 ∈ (ω × (ω × ω))))
14		ancom 463	. . . . . 6 ⊢ ((∃𝑖 ∈ ω ∃𝑗 ∈ ω (𝑥 = (𝑖∈𝑔𝑗) ∧ 𝑦 = {𝑎 ∈ (𝑀 ↑m ω) ∣ (𝑎‘𝑖)𝐸(𝑎‘𝑗)}) ∧ 𝑥 ∈ (ω × (ω × ω))) ↔ (𝑥 ∈ (ω × (ω × ω)) ∧ ∃𝑖 ∈ ω ∃𝑗 ∈ ω (𝑥 = (𝑖∈𝑔𝑗) ∧ 𝑦 = {𝑎 ∈ (𝑀 ↑m ω) ∣ (𝑎‘𝑖)𝐸(𝑎‘𝑗)})))
15	12, 13, 14	3bitri 299	. . . . 5 ⊢ (∃𝑖 ∈ ω ∃𝑗 ∈ ω (𝑥 = (𝑖∈𝑔𝑗) ∧ 𝑦 = {𝑎 ∈ (𝑀 ↑m ω) ∣ (𝑎‘𝑖)𝐸(𝑎‘𝑗)}) ↔ (𝑥 ∈ (ω × (ω × ω)) ∧ ∃𝑖 ∈ ω ∃𝑗 ∈ ω (𝑥 = (𝑖∈𝑔𝑗) ∧ 𝑦 = {𝑎 ∈ (𝑀 ↑m ω) ∣ (𝑎‘𝑖)𝐸(𝑎‘𝑗)})))
16	15	opabbii 5133	. . . 4 ⊢ {⟨𝑥, 𝑦⟩ ∣ ∃𝑖 ∈ ω ∃𝑗 ∈ ω (𝑥 = (𝑖∈𝑔𝑗) ∧ 𝑦 = {𝑎 ∈ (𝑀 ↑m ω) ∣ (𝑎‘𝑖)𝐸(𝑎‘𝑗)})} = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ (ω × (ω × ω)) ∧ ∃𝑖 ∈ ω ∃𝑗 ∈ ω (𝑥 = (𝑖∈𝑔𝑗) ∧ 𝑦 = {𝑎 ∈ (𝑀 ↑m ω) ∣ (𝑎‘𝑖)𝐸(𝑎‘𝑗)}))}
17		omex 9106	. . . . 5 ⊢ ω ∈ V
18	17, 17	xpex 7476	. . . . 5 ⊢ (ω × ω) ∈ V
19		xpexg 7473	. . . . . 6 ⊢ ((ω ∈ V ∧ (ω × ω) ∈ V) → (ω × (ω × ω)) ∈ V)
20		oveq1 7163	. . . . . . . . . . . . . 14 ⊢ (𝑖 = 𝑚 → (𝑖∈𝑔𝑗) = (𝑚∈𝑔𝑗))
21	20	eqeq2d 2832	. . . . . . . . . . . . 13 ⊢ (𝑖 = 𝑚 → (𝑥 = (𝑖∈𝑔𝑗) ↔ 𝑥 = (𝑚∈𝑔𝑗)))
22		fveq2 6670	. . . . . . . . . . . . . . . 16 ⊢ (𝑖 = 𝑚 → (𝑎‘𝑖) = (𝑎‘𝑚))
23	22	breq1d 5076	. . . . . . . . . . . . . . 15 ⊢ (𝑖 = 𝑚 → ((𝑎‘𝑖)𝐸(𝑎‘𝑗) ↔ (𝑎‘𝑚)𝐸(𝑎‘𝑗)))
24	23	rabbidv 3480	. . . . . . . . . . . . . 14 ⊢ (𝑖 = 𝑚 → {𝑎 ∈ (𝑀 ↑m ω) ∣ (𝑎‘𝑖)𝐸(𝑎‘𝑗)} = {𝑎 ∈ (𝑀 ↑m ω) ∣ (𝑎‘𝑚)𝐸(𝑎‘𝑗)})
25	24	eqeq2d 2832	. . . . . . . . . . . . 13 ⊢ (𝑖 = 𝑚 → (𝑦 = {𝑎 ∈ (𝑀 ↑m ω) ∣ (𝑎‘𝑖)𝐸(𝑎‘𝑗)} ↔ 𝑦 = {𝑎 ∈ (𝑀 ↑m ω) ∣ (𝑎‘𝑚)𝐸(𝑎‘𝑗)}))
26	21, 25	anbi12d 632	. . . . . . . . . . . 12 ⊢ (𝑖 = 𝑚 → ((𝑥 = (𝑖∈𝑔𝑗) ∧ 𝑦 = {𝑎 ∈ (𝑀 ↑m ω) ∣ (𝑎‘𝑖)𝐸(𝑎‘𝑗)}) ↔ (𝑥 = (𝑚∈𝑔𝑗) ∧ 𝑦 = {𝑎 ∈ (𝑀 ↑m ω) ∣ (𝑎‘𝑚)𝐸(𝑎‘𝑗)})))
27		oveq2 7164	. . . . . . . . . . . . . 14 ⊢ (𝑗 = 𝑛 → (𝑚∈𝑔𝑗) = (𝑚∈𝑔𝑛))
28	27	eqeq2d 2832	. . . . . . . . . . . . 13 ⊢ (𝑗 = 𝑛 → (𝑥 = (𝑚∈𝑔𝑗) ↔ 𝑥 = (𝑚∈𝑔𝑛)))
29		fveq2 6670	. . . . . . . . . . . . . . . 16 ⊢ (𝑗 = 𝑛 → (𝑎‘𝑗) = (𝑎‘𝑛))
30	29	breq2d 5078	. . . . . . . . . . . . . . 15 ⊢ (𝑗 = 𝑛 → ((𝑎‘𝑚)𝐸(𝑎‘𝑗) ↔ (𝑎‘𝑚)𝐸(𝑎‘𝑛)))
31	30	rabbidv 3480	. . . . . . . . . . . . . 14 ⊢ (𝑗 = 𝑛 → {𝑎 ∈ (𝑀 ↑m ω) ∣ (𝑎‘𝑚)𝐸(𝑎‘𝑗)} = {𝑎 ∈ (𝑀 ↑m ω) ∣ (𝑎‘𝑚)𝐸(𝑎‘𝑛)})
32	31	eqeq2d 2832	. . . . . . . . . . . . 13 ⊢ (𝑗 = 𝑛 → (𝑦 = {𝑎 ∈ (𝑀 ↑m ω) ∣ (𝑎‘𝑚)𝐸(𝑎‘𝑗)} ↔ 𝑦 = {𝑎 ∈ (𝑀 ↑m ω) ∣ (𝑎‘𝑚)𝐸(𝑎‘𝑛)}))
33	28, 32	anbi12d 632	. . . . . . . . . . . 12 ⊢ (𝑗 = 𝑛 → ((𝑥 = (𝑚∈𝑔𝑗) ∧ 𝑦 = {𝑎 ∈ (𝑀 ↑m ω) ∣ (𝑎‘𝑚)𝐸(𝑎‘𝑗)}) ↔ (𝑥 = (𝑚∈𝑔𝑛) ∧ 𝑦 = {𝑎 ∈ (𝑀 ↑m ω) ∣ (𝑎‘𝑚)𝐸(𝑎‘𝑛)})))
34	26, 33	cbvrex2vw 3462	. . . . . . . . . . 11 ⊢ (∃𝑖 ∈ ω ∃𝑗 ∈ ω (𝑥 = (𝑖∈𝑔𝑗) ∧ 𝑦 = {𝑎 ∈ (𝑀 ↑m ω) ∣ (𝑎‘𝑖)𝐸(𝑎‘𝑗)}) ↔ ∃𝑚 ∈ ω ∃𝑛 ∈ ω (𝑥 = (𝑚∈𝑔𝑛) ∧ 𝑦 = {𝑎 ∈ (𝑀 ↑m ω) ∣ (𝑎‘𝑚)𝐸(𝑎‘𝑛)}))
35		eqeq1 2825	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 = (𝑖∈𝑔𝑗) → (𝑥 = (𝑚∈𝑔𝑛) ↔ (𝑖∈𝑔𝑗) = (𝑚∈𝑔𝑛)))
36	35	adantl 484	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝑚 ∈ ω ∧ 𝑛 ∈ ω) ∧ (𝑖 ∈ ω ∧ 𝑗 ∈ ω)) ∧ 𝑥 = (𝑖∈𝑔𝑗)) → (𝑥 = (𝑚∈𝑔𝑛) ↔ (𝑖∈𝑔𝑗) = (𝑚∈𝑔𝑛)))
37		goeleq12bg 32596	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝑚 ∈ ω ∧ 𝑛 ∈ ω) ∧ (𝑖 ∈ ω ∧ 𝑗 ∈ ω)) → ((𝑖∈𝑔𝑗) = (𝑚∈𝑔𝑛) ↔ (𝑖 = 𝑚 ∧ 𝑗 = 𝑛)))
38	22	eqcomd 2827	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑖 = 𝑚 → (𝑎‘𝑚) = (𝑎‘𝑖))
39	29	eqcomd 2827	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑗 = 𝑛 → (𝑎‘𝑛) = (𝑎‘𝑗))
40	38, 39	breqan12d 5082	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑖 = 𝑚 ∧ 𝑗 = 𝑛) → ((𝑎‘𝑚)𝐸(𝑎‘𝑛) ↔ (𝑎‘𝑖)𝐸(𝑎‘𝑗)))
41	40	rabbidv 3480	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑖 = 𝑚 ∧ 𝑗 = 𝑛) → {𝑎 ∈ (𝑀 ↑m ω) ∣ (𝑎‘𝑚)𝐸(𝑎‘𝑛)} = {𝑎 ∈ (𝑀 ↑m ω) ∣ (𝑎‘𝑖)𝐸(𝑎‘𝑗)})
42	37, 41	syl6bi 255	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝑚 ∈ ω ∧ 𝑛 ∈ ω) ∧ (𝑖 ∈ ω ∧ 𝑗 ∈ ω)) → ((𝑖∈𝑔𝑗) = (𝑚∈𝑔𝑛) → {𝑎 ∈ (𝑀 ↑m ω) ∣ (𝑎‘𝑚)𝐸(𝑎‘𝑛)} = {𝑎 ∈ (𝑀 ↑m ω) ∣ (𝑎‘𝑖)𝐸(𝑎‘𝑗)}))
43	42	imp 409	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝑚 ∈ ω ∧ 𝑛 ∈ ω) ∧ (𝑖 ∈ ω ∧ 𝑗 ∈ ω)) ∧ (𝑖∈𝑔𝑗) = (𝑚∈𝑔𝑛)) → {𝑎 ∈ (𝑀 ↑m ω) ∣ (𝑎‘𝑚)𝐸(𝑎‘𝑛)} = {𝑎 ∈ (𝑀 ↑m ω) ∣ (𝑎‘𝑖)𝐸(𝑎‘𝑗)})
44		eqeq12 2835	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑦 = {𝑎 ∈ (𝑀 ↑m ω) ∣ (𝑎‘𝑚)𝐸(𝑎‘𝑛)} ∧ 𝑧 = {𝑎 ∈ (𝑀 ↑m ω) ∣ (𝑎‘𝑖)𝐸(𝑎‘𝑗)}) → (𝑦 = 𝑧 ↔ {𝑎 ∈ (𝑀 ↑m ω) ∣ (𝑎‘𝑚)𝐸(𝑎‘𝑛)} = {𝑎 ∈ (𝑀 ↑m ω) ∣ (𝑎‘𝑖)𝐸(𝑎‘𝑗)}))
45	43, 44	syl5ibrcom 249	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝑚 ∈ ω ∧ 𝑛 ∈ ω) ∧ (𝑖 ∈ ω ∧ 𝑗 ∈ ω)) ∧ (𝑖∈𝑔𝑗) = (𝑚∈𝑔𝑛)) → ((𝑦 = {𝑎 ∈ (𝑀 ↑m ω) ∣ (𝑎‘𝑚)𝐸(𝑎‘𝑛)} ∧ 𝑧 = {𝑎 ∈ (𝑀 ↑m ω) ∣ (𝑎‘𝑖)𝐸(𝑎‘𝑗)}) → 𝑦 = 𝑧))
46	45	exp4b 433	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑚 ∈ ω ∧ 𝑛 ∈ ω) ∧ (𝑖 ∈ ω ∧ 𝑗 ∈ ω)) → ((𝑖∈𝑔𝑗) = (𝑚∈𝑔𝑛) → (𝑦 = {𝑎 ∈ (𝑀 ↑m ω) ∣ (𝑎‘𝑚)𝐸(𝑎‘𝑛)} → (𝑧 = {𝑎 ∈ (𝑀 ↑m ω) ∣ (𝑎‘𝑖)𝐸(𝑎‘𝑗)} → 𝑦 = 𝑧))))
47	46	adantr 483	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝑚 ∈ ω ∧ 𝑛 ∈ ω) ∧ (𝑖 ∈ ω ∧ 𝑗 ∈ ω)) ∧ 𝑥 = (𝑖∈𝑔𝑗)) → ((𝑖∈𝑔𝑗) = (𝑚∈𝑔𝑛) → (𝑦 = {𝑎 ∈ (𝑀 ↑m ω) ∣ (𝑎‘𝑚)𝐸(𝑎‘𝑛)} → (𝑧 = {𝑎 ∈ (𝑀 ↑m ω) ∣ (𝑎‘𝑖)𝐸(𝑎‘𝑗)} → 𝑦 = 𝑧))))
48	36, 47	sylbid 242	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝑚 ∈ ω ∧ 𝑛 ∈ ω) ∧ (𝑖 ∈ ω ∧ 𝑗 ∈ ω)) ∧ 𝑥 = (𝑖∈𝑔𝑗)) → (𝑥 = (𝑚∈𝑔𝑛) → (𝑦 = {𝑎 ∈ (𝑀 ↑m ω) ∣ (𝑎‘𝑚)𝐸(𝑎‘𝑛)} → (𝑧 = {𝑎 ∈ (𝑀 ↑m ω) ∣ (𝑎‘𝑖)𝐸(𝑎‘𝑗)} → 𝑦 = 𝑧))))
49	48	impd 413	. . . . . . . . . . . . . . . 16 ⊢ ((((𝑚 ∈ ω ∧ 𝑛 ∈ ω) ∧ (𝑖 ∈ ω ∧ 𝑗 ∈ ω)) ∧ 𝑥 = (𝑖∈𝑔𝑗)) → ((𝑥 = (𝑚∈𝑔𝑛) ∧ 𝑦 = {𝑎 ∈ (𝑀 ↑m ω) ∣ (𝑎‘𝑚)𝐸(𝑎‘𝑛)}) → (𝑧 = {𝑎 ∈ (𝑀 ↑m ω) ∣ (𝑎‘𝑖)𝐸(𝑎‘𝑗)} → 𝑦 = 𝑧)))
50	49	com23 86	. . . . . . . . . . . . . . 15 ⊢ ((((𝑚 ∈ ω ∧ 𝑛 ∈ ω) ∧ (𝑖 ∈ ω ∧ 𝑗 ∈ ω)) ∧ 𝑥 = (𝑖∈𝑔𝑗)) → (𝑧 = {𝑎 ∈ (𝑀 ↑m ω) ∣ (𝑎‘𝑖)𝐸(𝑎‘𝑗)} → ((𝑥 = (𝑚∈𝑔𝑛) ∧ 𝑦 = {𝑎 ∈ (𝑀 ↑m ω) ∣ (𝑎‘𝑚)𝐸(𝑎‘𝑛)}) → 𝑦 = 𝑧)))
51	50	expimpd 456	. . . . . . . . . . . . . 14 ⊢ (((𝑚 ∈ ω ∧ 𝑛 ∈ ω) ∧ (𝑖 ∈ ω ∧ 𝑗 ∈ ω)) → ((𝑥 = (𝑖∈𝑔𝑗) ∧ 𝑧 = {𝑎 ∈ (𝑀 ↑m ω) ∣ (𝑎‘𝑖)𝐸(𝑎‘𝑗)}) → ((𝑥 = (𝑚∈𝑔𝑛) ∧ 𝑦 = {𝑎 ∈ (𝑀 ↑m ω) ∣ (𝑎‘𝑚)𝐸(𝑎‘𝑛)}) → 𝑦 = 𝑧)))
52	51	rexlimdvva 3294	. . . . . . . . . . . . 13 ⊢ ((𝑚 ∈ ω ∧ 𝑛 ∈ ω) → (∃𝑖 ∈ ω ∃𝑗 ∈ ω (𝑥 = (𝑖∈𝑔𝑗) ∧ 𝑧 = {𝑎 ∈ (𝑀 ↑m ω) ∣ (𝑎‘𝑖)𝐸(𝑎‘𝑗)}) → ((𝑥 = (𝑚∈𝑔𝑛) ∧ 𝑦 = {𝑎 ∈ (𝑀 ↑m ω) ∣ (𝑎‘𝑚)𝐸(𝑎‘𝑛)}) → 𝑦 = 𝑧)))
53	52	com23 86	. . . . . . . . . . . 12 ⊢ ((𝑚 ∈ ω ∧ 𝑛 ∈ ω) → ((𝑥 = (𝑚∈𝑔𝑛) ∧ 𝑦 = {𝑎 ∈ (𝑀 ↑m ω) ∣ (𝑎‘𝑚)𝐸(𝑎‘𝑛)}) → (∃𝑖 ∈ ω ∃𝑗 ∈ ω (𝑥 = (𝑖∈𝑔𝑗) ∧ 𝑧 = {𝑎 ∈ (𝑀 ↑m ω) ∣ (𝑎‘𝑖)𝐸(𝑎‘𝑗)}) → 𝑦 = 𝑧)))
54	53	rexlimivv 3292	. . . . . . . . . . 11 ⊢ (∃𝑚 ∈ ω ∃𝑛 ∈ ω (𝑥 = (𝑚∈𝑔𝑛) ∧ 𝑦 = {𝑎 ∈ (𝑀 ↑m ω) ∣ (𝑎‘𝑚)𝐸(𝑎‘𝑛)}) → (∃𝑖 ∈ ω ∃𝑗 ∈ ω (𝑥 = (𝑖∈𝑔𝑗) ∧ 𝑧 = {𝑎 ∈ (𝑀 ↑m ω) ∣ (𝑎‘𝑖)𝐸(𝑎‘𝑗)}) → 𝑦 = 𝑧))
55	34, 54	sylbi 219	. . . . . . . . . 10 ⊢ (∃𝑖 ∈ ω ∃𝑗 ∈ ω (𝑥 = (𝑖∈𝑔𝑗) ∧ 𝑦 = {𝑎 ∈ (𝑀 ↑m ω) ∣ (𝑎‘𝑖)𝐸(𝑎‘𝑗)}) → (∃𝑖 ∈ ω ∃𝑗 ∈ ω (𝑥 = (𝑖∈𝑔𝑗) ∧ 𝑧 = {𝑎 ∈ (𝑀 ↑m ω) ∣ (𝑎‘𝑖)𝐸(𝑎‘𝑗)}) → 𝑦 = 𝑧))
56	55	imp 409	. . . . . . . . 9 ⊢ ((∃𝑖 ∈ ω ∃𝑗 ∈ ω (𝑥 = (𝑖∈𝑔𝑗) ∧ 𝑦 = {𝑎 ∈ (𝑀 ↑m ω) ∣ (𝑎‘𝑖)𝐸(𝑎‘𝑗)}) ∧ ∃𝑖 ∈ ω ∃𝑗 ∈ ω (𝑥 = (𝑖∈𝑔𝑗) ∧ 𝑧 = {𝑎 ∈ (𝑀 ↑m ω) ∣ (𝑎‘𝑖)𝐸(𝑎‘𝑗)})) → 𝑦 = 𝑧)
57	56	gen2 1797	. . . . . . . 8 ⊢ ∀𝑦∀𝑧((∃𝑖 ∈ ω ∃𝑗 ∈ ω (𝑥 = (𝑖∈𝑔𝑗) ∧ 𝑦 = {𝑎 ∈ (𝑀 ↑m ω) ∣ (𝑎‘𝑖)𝐸(𝑎‘𝑗)}) ∧ ∃𝑖 ∈ ω ∃𝑗 ∈ ω (𝑥 = (𝑖∈𝑔𝑗) ∧ 𝑧 = {𝑎 ∈ (𝑀 ↑m ω) ∣ (𝑎‘𝑖)𝐸(𝑎‘𝑗)})) → 𝑦 = 𝑧)
58		eqeq1 2825	. . . . . . . . . . 11 ⊢ (𝑦 = 𝑧 → (𝑦 = {𝑎 ∈ (𝑀 ↑m ω) ∣ (𝑎‘𝑖)𝐸(𝑎‘𝑗)} ↔ 𝑧 = {𝑎 ∈ (𝑀 ↑m ω) ∣ (𝑎‘𝑖)𝐸(𝑎‘𝑗)}))
59	58	anbi2d 630	. . . . . . . . . 10 ⊢ (𝑦 = 𝑧 → ((𝑥 = (𝑖∈𝑔𝑗) ∧ 𝑦 = {𝑎 ∈ (𝑀 ↑m ω) ∣ (𝑎‘𝑖)𝐸(𝑎‘𝑗)}) ↔ (𝑥 = (𝑖∈𝑔𝑗) ∧ 𝑧 = {𝑎 ∈ (𝑀 ↑m ω) ∣ (𝑎‘𝑖)𝐸(𝑎‘𝑗)})))
60	59	2rexbidv 3300	. . . . . . . . 9 ⊢ (𝑦 = 𝑧 → (∃𝑖 ∈ ω ∃𝑗 ∈ ω (𝑥 = (𝑖∈𝑔𝑗) ∧ 𝑦 = {𝑎 ∈ (𝑀 ↑m ω) ∣ (𝑎‘𝑖)𝐸(𝑎‘𝑗)}) ↔ ∃𝑖 ∈ ω ∃𝑗 ∈ ω (𝑥 = (𝑖∈𝑔𝑗) ∧ 𝑧 = {𝑎 ∈ (𝑀 ↑m ω) ∣ (𝑎‘𝑖)𝐸(𝑎‘𝑗)})))
61	60	mo4 2650	. . . . . . . 8 ⊢ (∃*𝑦∃𝑖 ∈ ω ∃𝑗 ∈ ω (𝑥 = (𝑖∈𝑔𝑗) ∧ 𝑦 = {𝑎 ∈ (𝑀 ↑m ω) ∣ (𝑎‘𝑖)𝐸(𝑎‘𝑗)}) ↔ ∀𝑦∀𝑧((∃𝑖 ∈ ω ∃𝑗 ∈ ω (𝑥 = (𝑖∈𝑔𝑗) ∧ 𝑦 = {𝑎 ∈ (𝑀 ↑m ω) ∣ (𝑎‘𝑖)𝐸(𝑎‘𝑗)}) ∧ ∃𝑖 ∈ ω ∃𝑗 ∈ ω (𝑥 = (𝑖∈𝑔𝑗) ∧ 𝑧 = {𝑎 ∈ (𝑀 ↑m ω) ∣ (𝑎‘𝑖)𝐸(𝑎‘𝑗)})) → 𝑦 = 𝑧))
62	57, 61	mpbir 233	. . . . . . 7 ⊢ ∃*𝑦∃𝑖 ∈ ω ∃𝑗 ∈ ω (𝑥 = (𝑖∈𝑔𝑗) ∧ 𝑦 = {𝑎 ∈ (𝑀 ↑m ω) ∣ (𝑎‘𝑖)𝐸(𝑎‘𝑗)})
63		moabex 5351	. . . . . . 7 ⊢ (∃*𝑦∃𝑖 ∈ ω ∃𝑗 ∈ ω (𝑥 = (𝑖∈𝑔𝑗) ∧ 𝑦 = {𝑎 ∈ (𝑀 ↑m ω) ∣ (𝑎‘𝑖)𝐸(𝑎‘𝑗)}) → {𝑦 ∣ ∃𝑖 ∈ ω ∃𝑗 ∈ ω (𝑥 = (𝑖∈𝑔𝑗) ∧ 𝑦 = {𝑎 ∈ (𝑀 ↑m ω) ∣ (𝑎‘𝑖)𝐸(𝑎‘𝑗)})} ∈ V)
64	62, 63	mp1i 13	. . . . . 6 ⊢ (((ω ∈ V ∧ (ω × ω) ∈ V) ∧ 𝑥 ∈ (ω × (ω × ω))) → {𝑦 ∣ ∃𝑖 ∈ ω ∃𝑗 ∈ ω (𝑥 = (𝑖∈𝑔𝑗) ∧ 𝑦 = {𝑎 ∈ (𝑀 ↑m ω) ∣ (𝑎‘𝑖)𝐸(𝑎‘𝑗)})} ∈ V)
65	19, 64	opabex3d 7666	. . . . 5 ⊢ ((ω ∈ V ∧ (ω × ω) ∈ V) → {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ (ω × (ω × ω)) ∧ ∃𝑖 ∈ ω ∃𝑗 ∈ ω (𝑥 = (𝑖∈𝑔𝑗) ∧ 𝑦 = {𝑎 ∈ (𝑀 ↑m ω) ∣ (𝑎‘𝑖)𝐸(𝑎‘𝑗)}))} ∈ V)
66	17, 18, 65	mp2an 690	. . . 4 ⊢ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ (ω × (ω × ω)) ∧ ∃𝑖 ∈ ω ∃𝑗 ∈ ω (𝑥 = (𝑖∈𝑔𝑗) ∧ 𝑦 = {𝑎 ∈ (𝑀 ↑m ω) ∣ (𝑎‘𝑖)𝐸(𝑎‘𝑗)}))} ∈ V
67	16, 66	eqeltri 2909	. . 3 ⊢ {⟨𝑥, 𝑦⟩ ∣ ∃𝑖 ∈ ω ∃𝑗 ∈ ω (𝑥 = (𝑖∈𝑔𝑗) ∧ 𝑦 = {𝑎 ∈ (𝑀 ↑m ω) ∣ (𝑎‘𝑖)𝐸(𝑎‘𝑗)})} ∈ V
68	67	rdg0 8057	. 2 ⊢ (rec((𝑓 ∈ V ↦ (𝑓 ∪ {⟨𝑥, 𝑦⟩ ∣ ∃𝑢 ∈ 𝑓 (∃𝑣 ∈ 𝑓 (𝑥 = ((1st ‘𝑢)⊼𝑔(1st ‘𝑣)) ∧ 𝑦 = ((𝑀 ↑m ω) ∖ ((2nd ‘𝑢) ∩ (2nd ‘𝑣)))) ∨ ∃𝑖 ∈ ω (𝑥 = ∀𝑔𝑖(1st ‘𝑢) ∧ 𝑦 = {𝑎 ∈ (𝑀 ↑m ω) ∣ ∀𝑧 ∈ 𝑀 ({⟨𝑖, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑖}))) ∈ (2nd ‘𝑢)}))})), {⟨𝑥, 𝑦⟩ ∣ ∃𝑖 ∈ ω ∃𝑗 ∈ ω (𝑥 = (𝑖∈𝑔𝑗) ∧ 𝑦 = {𝑎 ∈ (𝑀 ↑m ω) ∣ (𝑎‘𝑖)𝐸(𝑎‘𝑗)})})‘∅) = {⟨𝑥, 𝑦⟩ ∣ ∃𝑖 ∈ ω ∃𝑗 ∈ ω (𝑥 = (𝑖∈𝑔𝑗) ∧ 𝑦 = {𝑎 ∈ (𝑀 ↑m ω) ∣ (𝑎‘𝑖)𝐸(𝑎‘𝑗)})}
69	6, 68	syl6eq 2872	1 ⊢ ((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) → (𝑆‘∅) = {⟨𝑥, 𝑦⟩ ∣ ∃𝑖 ∈ ω ∃𝑗 ∈ ω (𝑥 = (𝑖∈𝑔𝑗) ∧ 𝑦 = {𝑎 ∈ (𝑀 ↑m ω) ∣ (𝑎‘𝑖)𝐸(𝑎‘𝑗)})})

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398   ∨ wo 843  ∀wal 1535   = wceq 1537   ∈ wcel 2114  ∃*wmo 2620  {cab 2799  ∀wral 3138  ∃wrex 3139  {crab 3142  Vcvv 3494   ∖ cdif 3933   ∪ cun 3934   ∩ cin 3935  ∅c0 4291  {csn 4567  ⟨cop 4573   class class class wbr 5066  {copab 5128   ↦ cmpt 5146   × cxp 5553   ↾ cres 5557  suc csuc 6193  ‘cfv 6355  (class class class)co 7156  ωcom 7580  1^st c1st 7687  2^nd c2nd 7688  reccrdg 8045   ↑_m cmap 8406  ∈_𝑔cgoe 32580  ⊼_𝑔cgna 32581  ∀_𝑔cgol 32582   Sat csat 32583

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2793  ax-rep 5190  ax-sep 5203  ax-nul 5210  ax-pow 5266  ax-pr 5330  ax-un 7461  ax-inf2 9104

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-reu 3145  df-rab 3147  df-v 3496  df-sbc 3773  df-csb 3884  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-pss 3954  df-nul 4292  df-if 4468  df-pw 4541  df-sn 4568  df-pr 4570  df-tp 4572  df-op 4574  df-uni 4839  df-iun 4921  df-br 5067  df-opab 5129  df-mpt 5147  df-tr 5173  df-id 5460  df-eprel 5465  df-po 5474  df-so 5475  df-fr 5514  df-we 5516  df-xp 5561  df-rel 5562  df-cnv 5563  df-co 5564  df-dm 5565  df-rn 5566  df-res 5567  df-ima 5568  df-pred 6148  df-ord 6194  df-on 6195  df-lim 6196  df-suc 6197  df-iota 6314  df-fun 6357  df-fn 6358  df-f 6359  df-f1 6360  df-fo 6361  df-f1o 6362  df-fv 6363  df-ov 7159  df-oprab 7160  df-mpo 7161  df-om 7581  df-wrecs 7947  df-recs 8008  df-rdg 8046  df-goel 32587  df-sat 32590

This theorem is referenced by:  satfv1  32610  satfrel  32614  satfdm  32616  satfrnmapom  32617  satfv0fun  32618  satfv0fvfmla0  32660