Theorem satfv0 35345

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 7865	. . . 4 ⊢ ∅ ∈ ω
2		elelsuc 6407	. . . 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 35344	. . 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 1464	. 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 35335	. . . . . . . . . 10 ⊢ ((𝑖 ∈ ω ∧ 𝑗 ∈ ω) → (𝑖∈𝑔𝑗) ∈ (ω × (ω × ω)))
8		eleq1 2816	. . . . . . . . . 10 ⊢ (𝑥 = (𝑖∈𝑔𝑗) → (𝑥 ∈ (ω × (ω × ω)) ↔ (𝑖∈𝑔𝑗) ∈ (ω × (ω × ω))))
9	7, 8	syl5ibrcom 247	. . . . . . . . 9 ⊢ ((𝑖 ∈ ω ∧ 𝑗 ∈ ω) → (𝑥 = (𝑖∈𝑔𝑗) → 𝑥 ∈ (ω × (ω × ω))))
10	9	adantrd 491	. . . . . . . 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 561	. . . . . . 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 3198	. . . . . 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 3207	. . . . . 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 460	. . . . . 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 297	. . . . 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 5174	. . . 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 9596	. . . . 5 ⊢ ω ∈ V
18	17, 17	xpex 7729	. . . . 5 ⊢ (ω × ω) ∈ V
19		xpexg 7726	. . . . . 6 ⊢ ((ω ∈ V ∧ (ω × ω) ∈ V) → (ω × (ω × ω)) ∈ V)
20		oveq1 7394	. . . . . . . . . . . . . 14 ⊢ (𝑖 = 𝑚 → (𝑖∈𝑔𝑗) = (𝑚∈𝑔𝑗))
21	20	eqeq2d 2740	. . . . . . . . . . . . 13 ⊢ (𝑖 = 𝑚 → (𝑥 = (𝑖∈𝑔𝑗) ↔ 𝑥 = (𝑚∈𝑔𝑗)))
22		fveq2 6858	. . . . . . . . . . . . . . . 16 ⊢ (𝑖 = 𝑚 → (𝑎‘𝑖) = (𝑎‘𝑚))
23	22	breq1d 5117	. . . . . . . . . . . . . . 15 ⊢ (𝑖 = 𝑚 → ((𝑎‘𝑖)𝐸(𝑎‘𝑗) ↔ (𝑎‘𝑚)𝐸(𝑎‘𝑗)))
24	23	rabbidv 3413	. . . . . . . . . . . . . 14 ⊢ (𝑖 = 𝑚 → {𝑎 ∈ (𝑀 ↑m ω) ∣ (𝑎‘𝑖)𝐸(𝑎‘𝑗)} = {𝑎 ∈ (𝑀 ↑m ω) ∣ (𝑎‘𝑚)𝐸(𝑎‘𝑗)})
25	24	eqeq2d 2740	. . . . . . . . . . . . 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 7395	. . . . . . . . . . . . . 14 ⊢ (𝑗 = 𝑛 → (𝑚∈𝑔𝑗) = (𝑚∈𝑔𝑛))
28	27	eqeq2d 2740	. . . . . . . . . . . . 13 ⊢ (𝑗 = 𝑛 → (𝑥 = (𝑚∈𝑔𝑗) ↔ 𝑥 = (𝑚∈𝑔𝑛)))
29		fveq2 6858	. . . . . . . . . . . . . . . 16 ⊢ (𝑗 = 𝑛 → (𝑎‘𝑗) = (𝑎‘𝑛))
30	29	breq2d 5119	. . . . . . . . . . . . . . 15 ⊢ (𝑗 = 𝑛 → ((𝑎‘𝑚)𝐸(𝑎‘𝑗) ↔ (𝑎‘𝑚)𝐸(𝑎‘𝑛)))
31	30	rabbidv 3413	. . . . . . . . . . . . . 14 ⊢ (𝑗 = 𝑛 → {𝑎 ∈ (𝑀 ↑m ω) ∣ (𝑎‘𝑚)𝐸(𝑎‘𝑗)} = {𝑎 ∈ (𝑀 ↑m ω) ∣ (𝑎‘𝑚)𝐸(𝑎‘𝑛)})
32	31	eqeq2d 2740	. . . . . . . . . . . . 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 3220	. . . . . . . . . . 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 2733	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 = (𝑖∈𝑔𝑗) → (𝑥 = (𝑚∈𝑔𝑛) ↔ (𝑖∈𝑔𝑗) = (𝑚∈𝑔𝑛)))
36	35	adantl 481	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝑚 ∈ ω ∧ 𝑛 ∈ ω) ∧ (𝑖 ∈ ω ∧ 𝑗 ∈ ω)) ∧ 𝑥 = (𝑖∈𝑔𝑗)) → (𝑥 = (𝑚∈𝑔𝑛) ↔ (𝑖∈𝑔𝑗) = (𝑚∈𝑔𝑛)))
37		goeleq12bg 35336	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝑚 ∈ ω ∧ 𝑛 ∈ ω) ∧ (𝑖 ∈ ω ∧ 𝑗 ∈ ω)) → ((𝑖∈𝑔𝑗) = (𝑚∈𝑔𝑛) ↔ (𝑖 = 𝑚 ∧ 𝑗 = 𝑛)))
38	22	eqcomd 2735	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑖 = 𝑚 → (𝑎‘𝑚) = (𝑎‘𝑖))
39	29	eqcomd 2735	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑗 = 𝑛 → (𝑎‘𝑛) = (𝑎‘𝑗))
40	38, 39	breqan12d 5123	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑖 = 𝑚 ∧ 𝑗 = 𝑛) → ((𝑎‘𝑚)𝐸(𝑎‘𝑛) ↔ (𝑎‘𝑖)𝐸(𝑎‘𝑗)))
41	40	rabbidv 3413	. . . . . . . . . . . . . . . . . . . . . . 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	biimtrdi 253	. . . . . . . . . . . . . . . . . . . . . 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 406	. . . . . . . . . . . . . . . . . . . . 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 2746	. . . . . . . . . . . . . . . . . . . . 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 247	. . . . . . . . . . . . . . . . . . . 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 430	. . . . . . . . . . . . . . . . . . 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 480	. . . . . . . . . . . . . . . . . 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 240	. . . . . . . . . . . . . . . . 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 410	. . . . . . . . . . . . . . . 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 453	. . . . . . . . . . . . . 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 3194	. . . . . . . . . . . . 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 3179	. . . . . . . . . . 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 217	. . . . . . . . . 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 406	. . . . . . . . 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 1796	. . . . . . . 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 2733	. . . . . . . . . . 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 3202	. . . . . . . . 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 2559	. . . . . . . 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 231	. . . . . . 7 ⊢ ∃*𝑦∃𝑖 ∈ ω ∃𝑗 ∈ ω (𝑥 = (𝑖∈𝑔𝑗) ∧ 𝑦 = {𝑎 ∈ (𝑀 ↑m ω) ∣ (𝑎‘𝑖)𝐸(𝑎‘𝑗)})
63		moabex 5419	. . . . . . 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 7944	. . . . 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 692	. . . 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 2824	. . 3 ⊢ {⟨𝑥, 𝑦⟩ ∣ ∃𝑖 ∈ ω ∃𝑗 ∈ ω (𝑥 = (𝑖∈𝑔𝑗) ∧ 𝑦 = {𝑎 ∈ (𝑀 ↑m ω) ∣ (𝑎‘𝑖)𝐸(𝑎‘𝑗)})} ∈ V
68	67	rdg0 8389	. 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	eqtrdi 2780	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 206   ∧ wa 395   ∨ wo 847  ∀wal 1538   = wceq 1540   ∈ wcel 2109  ∃*wmo 2531  {cab 2707  ∀wral 3044  ∃wrex 3053  {crab 3405  Vcvv 3447   ∖ cdif 3911   ∪ cun 3912   ∩ cin 3913  ∅c0 4296  {csn 4589  ⟨cop 4595   class class class wbr 5107  {copab 5169   ↦ cmpt 5188   × cxp 5636   ↾ cres 5640  suc csuc 6334  ‘cfv 6511  (class class class)co 7387  ωcom 7842  1^st c1st 7966  2^nd c2nd 7967  reccrdg 8377   ↑_m cmap 8799  ∈_𝑔cgoe 35320  ⊼_𝑔cgna 35321  ∀_𝑔cgol 35322   Sat csat 35323

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5234  ax-sep 5251  ax-nul 5261  ax-pow 5320  ax-pr 5387  ax-un 7711  ax-inf2 9594

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-reu 3355  df-rab 3406  df-v 3449  df-sbc 3754  df-csb 3863  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-pss 3934  df-nul 4297  df-if 4489  df-pw 4565  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-iun 4957  df-br 5108  df-opab 5170  df-mpt 5189  df-tr 5215  df-id 5533  df-eprel 5538  df-po 5546  df-so 5547  df-fr 5591  df-we 5593  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-pred 6274  df-ord 6335  df-on 6336  df-lim 6337  df-suc 6338  df-iota 6464  df-fun 6513  df-fn 6514  df-f 6515  df-f1 6516  df-fo 6517  df-f1o 6518  df-fv 6519  df-ov 7390  df-oprab 7391  df-mpo 7392  df-om 7843  df-2nd 7969  df-frecs 8260  df-wrecs 8291  df-recs 8340  df-rdg 8378  df-goel 35327  df-sat 35330

This theorem is referenced by:  satfv1  35350  satfrel  35354  satfdm  35356  satfrnmapom  35357  satfv0fun  35358  satfv0fvfmla0  35400