satf0 - Mathbox for Mario Carneiro

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Theorem satf0 35340

Description: The satisfaction predicate as function over wff codes in the empty model with an empty binary relation. (Contributed by AV, 14-Sep-2023.)

**Assertion**
Ref	Expression
satf0	⊢ (∅ Sat ∅) = (rec((𝑓 ∈ V ↦ (𝑓 ∪ {⟨𝑥, 𝑦⟩ ∣ (𝑦 = ∅ ∧ ∃𝑢 ∈ 𝑓 (∃𝑣 ∈ 𝑓 𝑥 = ((1^st ‘𝑢)⊼_𝑔(1^st ‘𝑣)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀_𝑔𝑖(1^st ‘𝑢)))})), {⟨𝑥, 𝑦⟩ ∣ (𝑦 = ∅ ∧ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑥 = (𝑖∈_𝑔𝑗))}) ↾ suc ω)

Distinct variable group: 𝑓,𝑖,𝑗,𝑢,𝑣,𝑥,𝑦

Proof of Theorem satf0 Dummy variables 𝑎 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		0ex 5325	. . 3 ⊢ ∅ ∈ V
2		satf 35321	. . 3 ⊢ ((∅ ∈ V ∧ ∅ ∈ V) → (∅ Sat ∅) = (rec((𝑓 ∈ V ↦ (𝑓 ∪ {⟨𝑥, 𝑦⟩ ∣ ∃𝑢 ∈ 𝑓 (∃𝑣 ∈ 𝑓 (𝑥 = ((1^st ‘𝑢)⊼_𝑔(1^st ‘𝑣)) ∧ 𝑦 = ((∅ ↑_m ω) ∖ ((2^nd ‘𝑢) ∩ (2^nd ‘𝑣)))) ∨ ∃𝑖 ∈ ω (𝑥 = ∀_𝑔𝑖(1^st ‘𝑢) ∧ 𝑦 = {𝑎 ∈ (∅ ↑_m ω) ∣ ∀𝑧 ∈ ∅ ({⟨𝑖, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑖}))) ∈ (2^nd ‘𝑢)}))})), {⟨𝑥, 𝑦⟩ ∣ ∃𝑖 ∈ ω ∃𝑗 ∈ ω (𝑥 = (𝑖∈_𝑔𝑗) ∧ 𝑦 = {𝑎 ∈ (∅ ↑_m ω) ∣ (𝑎‘𝑖)∅(𝑎‘𝑗)})}) ↾ suc ω))
3	1, 1, 2	mp2an 691	. 2 ⊢ (∅ Sat ∅) = (rec((𝑓 ∈ V ↦ (𝑓 ∪ {⟨𝑥, 𝑦⟩ ∣ ∃𝑢 ∈ 𝑓 (∃𝑣 ∈ 𝑓 (𝑥 = ((1^st ‘𝑢)⊼_𝑔(1^st ‘𝑣)) ∧ 𝑦 = ((∅ ↑_m ω) ∖ ((2^nd ‘𝑢) ∩ (2^nd ‘𝑣)))) ∨ ∃𝑖 ∈ ω (𝑥 = ∀_𝑔𝑖(1^st ‘𝑢) ∧ 𝑦 = {𝑎 ∈ (∅ ↑_m ω) ∣ ∀𝑧 ∈ ∅ ({⟨𝑖, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑖}))) ∈ (2^nd ‘𝑢)}))})), {⟨𝑥, 𝑦⟩ ∣ ∃𝑖 ∈ ω ∃𝑗 ∈ ω (𝑥 = (𝑖∈_𝑔𝑗) ∧ 𝑦 = {𝑎 ∈ (∅ ↑_m ω) ∣ (𝑎‘𝑖)∅(𝑎‘𝑗)})}) ↾ suc ω)
4		peano1 7927	. . . . . . . . . . . . . . . . . . 19 ⊢ ∅ ∈ ω
5	4	ne0ii 4367	. . . . . . . . . . . . . . . . . 18 ⊢ ω ≠ ∅
6		map0b 8941	. . . . . . . . . . . . . . . . . 18 ⊢ (ω ≠ ∅ → (∅ ↑_m ω) = ∅)
7	5, 6	ax-mp 5	. . . . . . . . . . . . . . . . 17 ⊢ (∅ ↑_m ω) = ∅
8	7	difeq1i 4145	. . . . . . . . . . . . . . . 16 ⊢ ((∅ ↑_m ω) ∖ ((2^nd ‘𝑢) ∩ (2^nd ‘𝑣))) = (∅ ∖ ((2^nd ‘𝑢) ∩ (2^nd ‘𝑣)))
9		0dif 4428	. . . . . . . . . . . . . . . 16 ⊢ (∅ ∖ ((2^nd ‘𝑢) ∩ (2^nd ‘𝑣))) = ∅
10	8, 9	eqtri 2768	. . . . . . . . . . . . . . 15 ⊢ ((∅ ↑_m ω) ∖ ((2^nd ‘𝑢) ∩ (2^nd ‘𝑣))) = ∅
11	10	eqeq2i 2753	. . . . . . . . . . . . . 14 ⊢ (𝑦 = ((∅ ↑_m ω) ∖ ((2^nd ‘𝑢) ∩ (2^nd ‘𝑣))) ↔ 𝑦 = ∅)
12	11	anbi2i 622	. . . . . . . . . . . . 13 ⊢ ((𝑥 = ((1^st ‘𝑢)⊼_𝑔(1^st ‘𝑣)) ∧ 𝑦 = ((∅ ↑_m ω) ∖ ((2^nd ‘𝑢) ∩ (2^nd ‘𝑣)))) ↔ (𝑥 = ((1^st ‘𝑢)⊼_𝑔(1^st ‘𝑣)) ∧ 𝑦 = ∅))
13	12	rexbii 3100	. . . . . . . . . . . 12 ⊢ (∃𝑣 ∈ 𝑓 (𝑥 = ((1^st ‘𝑢)⊼_𝑔(1^st ‘𝑣)) ∧ 𝑦 = ((∅ ↑_m ω) ∖ ((2^nd ‘𝑢) ∩ (2^nd ‘𝑣)))) ↔ ∃𝑣 ∈ 𝑓 (𝑥 = ((1^st ‘𝑢)⊼_𝑔(1^st ‘𝑣)) ∧ 𝑦 = ∅))
14		r19.41v 3195	. . . . . . . . . . . 12 ⊢ (∃𝑣 ∈ 𝑓 (𝑥 = ((1^st ‘𝑢)⊼_𝑔(1^st ‘𝑣)) ∧ 𝑦 = ∅) ↔ (∃𝑣 ∈ 𝑓 𝑥 = ((1^st ‘𝑢)⊼_𝑔(1^st ‘𝑣)) ∧ 𝑦 = ∅))
15	13, 14	bitri 275	. . . . . . . . . . 11 ⊢ (∃𝑣 ∈ 𝑓 (𝑥 = ((1^st ‘𝑢)⊼_𝑔(1^st ‘𝑣)) ∧ 𝑦 = ((∅ ↑_m ω) ∖ ((2^nd ‘𝑢) ∩ (2^nd ‘𝑣)))) ↔ (∃𝑣 ∈ 𝑓 𝑥 = ((1^st ‘𝑢)⊼_𝑔(1^st ‘𝑣)) ∧ 𝑦 = ∅))
16	7	rabeqi 3457	. . . . . . . . . . . . . . . 16 ⊢ {𝑎 ∈ (∅ ↑_m ω) ∣ ∀𝑧 ∈ ∅ ({⟨𝑖, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑖}))) ∈ (2^nd ‘𝑢)} = {𝑎 ∈ ∅ ∣ ∀𝑧 ∈ ∅ ({⟨𝑖, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑖}))) ∈ (2^nd ‘𝑢)}
17		rab0 4409	. . . . . . . . . . . . . . . 16 ⊢ {𝑎 ∈ ∅ ∣ ∀𝑧 ∈ ∅ ({⟨𝑖, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑖}))) ∈ (2^nd ‘𝑢)} = ∅
18	16, 17	eqtri 2768	. . . . . . . . . . . . . . 15 ⊢ {𝑎 ∈ (∅ ↑_m ω) ∣ ∀𝑧 ∈ ∅ ({⟨𝑖, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑖}))) ∈ (2^nd ‘𝑢)} = ∅
19	18	eqeq2i 2753	. . . . . . . . . . . . . 14 ⊢ (𝑦 = {𝑎 ∈ (∅ ↑_m ω) ∣ ∀𝑧 ∈ ∅ ({⟨𝑖, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑖}))) ∈ (2^nd ‘𝑢)} ↔ 𝑦 = ∅)
20	19	anbi2i 622	. . . . . . . . . . . . 13 ⊢ ((𝑥 = ∀_𝑔𝑖(1^st ‘𝑢) ∧ 𝑦 = {𝑎 ∈ (∅ ↑_m ω) ∣ ∀𝑧 ∈ ∅ ({⟨𝑖, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑖}))) ∈ (2^nd ‘𝑢)}) ↔ (𝑥 = ∀_𝑔𝑖(1^st ‘𝑢) ∧ 𝑦 = ∅))
21	20	rexbii 3100	. . . . . . . . . . . 12 ⊢ (∃𝑖 ∈ ω (𝑥 = ∀_𝑔𝑖(1^st ‘𝑢) ∧ 𝑦 = {𝑎 ∈ (∅ ↑_m ω) ∣ ∀𝑧 ∈ ∅ ({⟨𝑖, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑖}))) ∈ (2^nd ‘𝑢)}) ↔ ∃𝑖 ∈ ω (𝑥 = ∀_𝑔𝑖(1^st ‘𝑢) ∧ 𝑦 = ∅))
22		r19.41v 3195	. . . . . . . . . . . 12 ⊢ (∃𝑖 ∈ ω (𝑥 = ∀_𝑔𝑖(1^st ‘𝑢) ∧ 𝑦 = ∅) ↔ (∃𝑖 ∈ ω 𝑥 = ∀_𝑔𝑖(1^st ‘𝑢) ∧ 𝑦 = ∅))
23	21, 22	bitri 275	. . . . . . . . . . 11 ⊢ (∃𝑖 ∈ ω (𝑥 = ∀_𝑔𝑖(1^st ‘𝑢) ∧ 𝑦 = {𝑎 ∈ (∅ ↑_m ω) ∣ ∀𝑧 ∈ ∅ ({⟨𝑖, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑖}))) ∈ (2^nd ‘𝑢)}) ↔ (∃𝑖 ∈ ω 𝑥 = ∀_𝑔𝑖(1^st ‘𝑢) ∧ 𝑦 = ∅))
24	15, 23	orbi12i 913	. . . . . . . . . 10 ⊢ ((∃𝑣 ∈ 𝑓 (𝑥 = ((1^st ‘𝑢)⊼_𝑔(1^st ‘𝑣)) ∧ 𝑦 = ((∅ ↑_m ω) ∖ ((2^nd ‘𝑢) ∩ (2^nd ‘𝑣)))) ∨ ∃𝑖 ∈ ω (𝑥 = ∀_𝑔𝑖(1^st ‘𝑢) ∧ 𝑦 = {𝑎 ∈ (∅ ↑_m ω) ∣ ∀𝑧 ∈ ∅ ({⟨𝑖, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑖}))) ∈ (2^nd ‘𝑢)})) ↔ ((∃𝑣 ∈ 𝑓 𝑥 = ((1^st ‘𝑢)⊼_𝑔(1^st ‘𝑣)) ∧ 𝑦 = ∅) ∨ (∃𝑖 ∈ ω 𝑥 = ∀_𝑔𝑖(1^st ‘𝑢) ∧ 𝑦 = ∅)))
25	24	rexbii 3100	. . . . . . . . 9 ⊢ (∃𝑢 ∈ 𝑓 (∃𝑣 ∈ 𝑓 (𝑥 = ((1^st ‘𝑢)⊼_𝑔(1^st ‘𝑣)) ∧ 𝑦 = ((∅ ↑_m ω) ∖ ((2^nd ‘𝑢) ∩ (2^nd ‘𝑣)))) ∨ ∃𝑖 ∈ ω (𝑥 = ∀_𝑔𝑖(1^st ‘𝑢) ∧ 𝑦 = {𝑎 ∈ (∅ ↑_m ω) ∣ ∀𝑧 ∈ ∅ ({⟨𝑖, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑖}))) ∈ (2^nd ‘𝑢)})) ↔ ∃𝑢 ∈ 𝑓 ((∃𝑣 ∈ 𝑓 𝑥 = ((1^st ‘𝑢)⊼_𝑔(1^st ‘𝑣)) ∧ 𝑦 = ∅) ∨ (∃𝑖 ∈ ω 𝑥 = ∀_𝑔𝑖(1^st ‘𝑢) ∧ 𝑦 = ∅)))
26		andir 1009	. . . . . . . . . . 11 ⊢ (((∃𝑣 ∈ 𝑓 𝑥 = ((1^st ‘𝑢)⊼_𝑔(1^st ‘𝑣)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀_𝑔𝑖(1^st ‘𝑢)) ∧ 𝑦 = ∅) ↔ ((∃𝑣 ∈ 𝑓 𝑥 = ((1^st ‘𝑢)⊼_𝑔(1^st ‘𝑣)) ∧ 𝑦 = ∅) ∨ (∃𝑖 ∈ ω 𝑥 = ∀_𝑔𝑖(1^st ‘𝑢) ∧ 𝑦 = ∅)))
27	26	bicomi 224	. . . . . . . . . 10 ⊢ (((∃𝑣 ∈ 𝑓 𝑥 = ((1^st ‘𝑢)⊼_𝑔(1^st ‘𝑣)) ∧ 𝑦 = ∅) ∨ (∃𝑖 ∈ ω 𝑥 = ∀_𝑔𝑖(1^st ‘𝑢) ∧ 𝑦 = ∅)) ↔ ((∃𝑣 ∈ 𝑓 𝑥 = ((1^st ‘𝑢)⊼_𝑔(1^st ‘𝑣)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀_𝑔𝑖(1^st ‘𝑢)) ∧ 𝑦 = ∅))
28	27	rexbii 3100	. . . . . . . . 9 ⊢ (∃𝑢 ∈ 𝑓 ((∃𝑣 ∈ 𝑓 𝑥 = ((1^st ‘𝑢)⊼_𝑔(1^st ‘𝑣)) ∧ 𝑦 = ∅) ∨ (∃𝑖 ∈ ω 𝑥 = ∀_𝑔𝑖(1^st ‘𝑢) ∧ 𝑦 = ∅)) ↔ ∃𝑢 ∈ 𝑓 ((∃𝑣 ∈ 𝑓 𝑥 = ((1^st ‘𝑢)⊼_𝑔(1^st ‘𝑣)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀_𝑔𝑖(1^st ‘𝑢)) ∧ 𝑦 = ∅))
29		r19.41v 3195	. . . . . . . . 9 ⊢ (∃𝑢 ∈ 𝑓 ((∃𝑣 ∈ 𝑓 𝑥 = ((1^st ‘𝑢)⊼_𝑔(1^st ‘𝑣)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀_𝑔𝑖(1^st ‘𝑢)) ∧ 𝑦 = ∅) ↔ (∃𝑢 ∈ 𝑓 (∃𝑣 ∈ 𝑓 𝑥 = ((1^st ‘𝑢)⊼_𝑔(1^st ‘𝑣)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀_𝑔𝑖(1^st ‘𝑢)) ∧ 𝑦 = ∅))
30	25, 28, 29	3bitri 297	. . . . . . . 8 ⊢ (∃𝑢 ∈ 𝑓 (∃𝑣 ∈ 𝑓 (𝑥 = ((1^st ‘𝑢)⊼_𝑔(1^st ‘𝑣)) ∧ 𝑦 = ((∅ ↑_m ω) ∖ ((2^nd ‘𝑢) ∩ (2^nd ‘𝑣)))) ∨ ∃𝑖 ∈ ω (𝑥 = ∀_𝑔𝑖(1^st ‘𝑢) ∧ 𝑦 = {𝑎 ∈ (∅ ↑_m ω) ∣ ∀𝑧 ∈ ∅ ({⟨𝑖, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑖}))) ∈ (2^nd ‘𝑢)})) ↔ (∃𝑢 ∈ 𝑓 (∃𝑣 ∈ 𝑓 𝑥 = ((1^st ‘𝑢)⊼_𝑔(1^st ‘𝑣)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀_𝑔𝑖(1^st ‘𝑢)) ∧ 𝑦 = ∅))
31	30	biancomi 462	. . . . . . 7 ⊢ (∃𝑢 ∈ 𝑓 (∃𝑣 ∈ 𝑓 (𝑥 = ((1^st ‘𝑢)⊼_𝑔(1^st ‘𝑣)) ∧ 𝑦 = ((∅ ↑_m ω) ∖ ((2^nd ‘𝑢) ∩ (2^nd ‘𝑣)))) ∨ ∃𝑖 ∈ ω (𝑥 = ∀_𝑔𝑖(1^st ‘𝑢) ∧ 𝑦 = {𝑎 ∈ (∅ ↑_m ω) ∣ ∀𝑧 ∈ ∅ ({⟨𝑖, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑖}))) ∈ (2^nd ‘𝑢)})) ↔ (𝑦 = ∅ ∧ ∃𝑢 ∈ 𝑓 (∃𝑣 ∈ 𝑓 𝑥 = ((1^st ‘𝑢)⊼_𝑔(1^st ‘𝑣)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀_𝑔𝑖(1^st ‘𝑢))))
32	31	opabbii 5233	. . . . . 6 ⊢ {⟨𝑥, 𝑦⟩ ∣ ∃𝑢 ∈ 𝑓 (∃𝑣 ∈ 𝑓 (𝑥 = ((1^st ‘𝑢)⊼_𝑔(1^st ‘𝑣)) ∧ 𝑦 = ((∅ ↑_m ω) ∖ ((2^nd ‘𝑢) ∩ (2^nd ‘𝑣)))) ∨ ∃𝑖 ∈ ω (𝑥 = ∀_𝑔𝑖(1^st ‘𝑢) ∧ 𝑦 = {𝑎 ∈ (∅ ↑_m ω) ∣ ∀𝑧 ∈ ∅ ({⟨𝑖, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑖}))) ∈ (2^nd ‘𝑢)}))} = {⟨𝑥, 𝑦⟩ ∣ (𝑦 = ∅ ∧ ∃𝑢 ∈ 𝑓 (∃𝑣 ∈ 𝑓 𝑥 = ((1^st ‘𝑢)⊼_𝑔(1^st ‘𝑣)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀_𝑔𝑖(1^st ‘𝑢)))}
33	32	uneq2i 4188	. . . . 5 ⊢ (𝑓 ∪ {⟨𝑥, 𝑦⟩ ∣ ∃𝑢 ∈ 𝑓 (∃𝑣 ∈ 𝑓 (𝑥 = ((1^st ‘𝑢)⊼_𝑔(1^st ‘𝑣)) ∧ 𝑦 = ((∅ ↑_m ω) ∖ ((2^nd ‘𝑢) ∩ (2^nd ‘𝑣)))) ∨ ∃𝑖 ∈ ω (𝑥 = ∀_𝑔𝑖(1^st ‘𝑢) ∧ 𝑦 = {𝑎 ∈ (∅ ↑_m ω) ∣ ∀𝑧 ∈ ∅ ({⟨𝑖, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑖}))) ∈ (2^nd ‘𝑢)}))}) = (𝑓 ∪ {⟨𝑥, 𝑦⟩ ∣ (𝑦 = ∅ ∧ ∃𝑢 ∈ 𝑓 (∃𝑣 ∈ 𝑓 𝑥 = ((1^st ‘𝑢)⊼_𝑔(1^st ‘𝑣)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀_𝑔𝑖(1^st ‘𝑢)))})
34	33	mpteq2i 5271	. . . 4 ⊢ (𝑓 ∈ V ↦ (𝑓 ∪ {⟨𝑥, 𝑦⟩ ∣ ∃𝑢 ∈ 𝑓 (∃𝑣 ∈ 𝑓 (𝑥 = ((1^st ‘𝑢)⊼_𝑔(1^st ‘𝑣)) ∧ 𝑦 = ((∅ ↑_m ω) ∖ ((2^nd ‘𝑢) ∩ (2^nd ‘𝑣)))) ∨ ∃𝑖 ∈ ω (𝑥 = ∀_𝑔𝑖(1^st ‘𝑢) ∧ 𝑦 = {𝑎 ∈ (∅ ↑_m ω) ∣ ∀𝑧 ∈ ∅ ({⟨𝑖, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑖}))) ∈ (2^nd ‘𝑢)}))})) = (𝑓 ∈ V ↦ (𝑓 ∪ {⟨𝑥, 𝑦⟩ ∣ (𝑦 = ∅ ∧ ∃𝑢 ∈ 𝑓 (∃𝑣 ∈ 𝑓 𝑥 = ((1^st ‘𝑢)⊼_𝑔(1^st ‘𝑣)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀_𝑔𝑖(1^st ‘𝑢)))}))
35	7	rabeqi 3457	. . . . . . . . . . 11 ⊢ {𝑎 ∈ (∅ ↑_m ω) ∣ (𝑎‘𝑖)∅(𝑎‘𝑗)} = {𝑎 ∈ ∅ ∣ (𝑎‘𝑖)∅(𝑎‘𝑗)}
36		rab0 4409	. . . . . . . . . . 11 ⊢ {𝑎 ∈ ∅ ∣ (𝑎‘𝑖)∅(𝑎‘𝑗)} = ∅
37	35, 36	eqtri 2768	. . . . . . . . . 10 ⊢ {𝑎 ∈ (∅ ↑_m ω) ∣ (𝑎‘𝑖)∅(𝑎‘𝑗)} = ∅
38	37	eqeq2i 2753	. . . . . . . . 9 ⊢ (𝑦 = {𝑎 ∈ (∅ ↑_m ω) ∣ (𝑎‘𝑖)∅(𝑎‘𝑗)} ↔ 𝑦 = ∅)
39	38	anbi2i 622	. . . . . . . 8 ⊢ ((𝑥 = (𝑖∈_𝑔𝑗) ∧ 𝑦 = {𝑎 ∈ (∅ ↑_m ω) ∣ (𝑎‘𝑖)∅(𝑎‘𝑗)}) ↔ (𝑥 = (𝑖∈_𝑔𝑗) ∧ 𝑦 = ∅))
40	39	2rexbii 3135	. . . . . . 7 ⊢ (∃𝑖 ∈ ω ∃𝑗 ∈ ω (𝑥 = (𝑖∈_𝑔𝑗) ∧ 𝑦 = {𝑎 ∈ (∅ ↑_m ω) ∣ (𝑎‘𝑖)∅(𝑎‘𝑗)}) ↔ ∃𝑖 ∈ ω ∃𝑗 ∈ ω (𝑥 = (𝑖∈_𝑔𝑗) ∧ 𝑦 = ∅))
41		r19.41vv 3233	. . . . . . 7 ⊢ (∃𝑖 ∈ ω ∃𝑗 ∈ ω (𝑥 = (𝑖∈_𝑔𝑗) ∧ 𝑦 = ∅) ↔ (∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑥 = (𝑖∈_𝑔𝑗) ∧ 𝑦 = ∅))
42	40, 41	bitri 275	. . . . . 6 ⊢ (∃𝑖 ∈ ω ∃𝑗 ∈ ω (𝑥 = (𝑖∈_𝑔𝑗) ∧ 𝑦 = {𝑎 ∈ (∅ ↑_m ω) ∣ (𝑎‘𝑖)∅(𝑎‘𝑗)}) ↔ (∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑥 = (𝑖∈_𝑔𝑗) ∧ 𝑦 = ∅))
43	42	biancomi 462	. . . . 5 ⊢ (∃𝑖 ∈ ω ∃𝑗 ∈ ω (𝑥 = (𝑖∈_𝑔𝑗) ∧ 𝑦 = {𝑎 ∈ (∅ ↑_m ω) ∣ (𝑎‘𝑖)∅(𝑎‘𝑗)}) ↔ (𝑦 = ∅ ∧ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑥 = (𝑖∈_𝑔𝑗)))
44	43	opabbii 5233	. . . 4 ⊢ {⟨𝑥, 𝑦⟩ ∣ ∃𝑖 ∈ ω ∃𝑗 ∈ ω (𝑥 = (𝑖∈_𝑔𝑗) ∧ 𝑦 = {𝑎 ∈ (∅ ↑_m ω) ∣ (𝑎‘𝑖)∅(𝑎‘𝑗)})} = {⟨𝑥, 𝑦⟩ ∣ (𝑦 = ∅ ∧ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑥 = (𝑖∈_𝑔𝑗))}
45		rdgeq12 8469	. . . 4 ⊢ (((𝑓 ∈ V ↦ (𝑓 ∪ {⟨𝑥, 𝑦⟩ ∣ ∃𝑢 ∈ 𝑓 (∃𝑣 ∈ 𝑓 (𝑥 = ((1^st ‘𝑢)⊼_𝑔(1^st ‘𝑣)) ∧ 𝑦 = ((∅ ↑_m ω) ∖ ((2^nd ‘𝑢) ∩ (2^nd ‘𝑣)))) ∨ ∃𝑖 ∈ ω (𝑥 = ∀_𝑔𝑖(1^st ‘𝑢) ∧ 𝑦 = {𝑎 ∈ (∅ ↑_m ω) ∣ ∀𝑧 ∈ ∅ ({⟨𝑖, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑖}))) ∈ (2^nd ‘𝑢)}))})) = (𝑓 ∈ V ↦ (𝑓 ∪ {⟨𝑥, 𝑦⟩ ∣ (𝑦 = ∅ ∧ ∃𝑢 ∈ 𝑓 (∃𝑣 ∈ 𝑓 𝑥 = ((1^st ‘𝑢)⊼_𝑔(1^st ‘𝑣)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀_𝑔𝑖(1^st ‘𝑢)))})) ∧ {⟨𝑥, 𝑦⟩ ∣ ∃𝑖 ∈ ω ∃𝑗 ∈ ω (𝑥 = (𝑖∈_𝑔𝑗) ∧ 𝑦 = {𝑎 ∈ (∅ ↑_m ω) ∣ (𝑎‘𝑖)∅(𝑎‘𝑗)})} = {⟨𝑥, 𝑦⟩ ∣ (𝑦 = ∅ ∧ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑥 = (𝑖∈_𝑔𝑗))}) → rec((𝑓 ∈ V ↦ (𝑓 ∪ {⟨𝑥, 𝑦⟩ ∣ ∃𝑢 ∈ 𝑓 (∃𝑣 ∈ 𝑓 (𝑥 = ((1^st ‘𝑢)⊼_𝑔(1^st ‘𝑣)) ∧ 𝑦 = ((∅ ↑_m ω) ∖ ((2^nd ‘𝑢) ∩ (2^nd ‘𝑣)))) ∨ ∃𝑖 ∈ ω (𝑥 = ∀_𝑔𝑖(1^st ‘𝑢) ∧ 𝑦 = {𝑎 ∈ (∅ ↑_m ω) ∣ ∀𝑧 ∈ ∅ ({⟨𝑖, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑖}))) ∈ (2^nd ‘𝑢)}))})), {⟨𝑥, 𝑦⟩ ∣ ∃𝑖 ∈ ω ∃𝑗 ∈ ω (𝑥 = (𝑖∈_𝑔𝑗) ∧ 𝑦 = {𝑎 ∈ (∅ ↑_m ω) ∣ (𝑎‘𝑖)∅(𝑎‘𝑗)})}) = rec((𝑓 ∈ V ↦ (𝑓 ∪ {⟨𝑥, 𝑦⟩ ∣ (𝑦 = ∅ ∧ ∃𝑢 ∈ 𝑓 (∃𝑣 ∈ 𝑓 𝑥 = ((1^st ‘𝑢)⊼_𝑔(1^st ‘𝑣)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀_𝑔𝑖(1^st ‘𝑢)))})), {⟨𝑥, 𝑦⟩ ∣ (𝑦 = ∅ ∧ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑥 = (𝑖∈_𝑔𝑗))}))
46	34, 44, 45	mp2an 691	. . 3 ⊢ rec((𝑓 ∈ V ↦ (𝑓 ∪ {⟨𝑥, 𝑦⟩ ∣ ∃𝑢 ∈ 𝑓 (∃𝑣 ∈ 𝑓 (𝑥 = ((1^st ‘𝑢)⊼_𝑔(1^st ‘𝑣)) ∧ 𝑦 = ((∅ ↑_m ω) ∖ ((2^nd ‘𝑢) ∩ (2^nd ‘𝑣)))) ∨ ∃𝑖 ∈ ω (𝑥 = ∀_𝑔𝑖(1^st ‘𝑢) ∧ 𝑦 = {𝑎 ∈ (∅ ↑_m ω) ∣ ∀𝑧 ∈ ∅ ({⟨𝑖, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑖}))) ∈ (2^nd ‘𝑢)}))})), {⟨𝑥, 𝑦⟩ ∣ ∃𝑖 ∈ ω ∃𝑗 ∈ ω (𝑥 = (𝑖∈_𝑔𝑗) ∧ 𝑦 = {𝑎 ∈ (∅ ↑_m ω) ∣ (𝑎‘𝑖)∅(𝑎‘𝑗)})}) = rec((𝑓 ∈ V ↦ (𝑓 ∪ {⟨𝑥, 𝑦⟩ ∣ (𝑦 = ∅ ∧ ∃𝑢 ∈ 𝑓 (∃𝑣 ∈ 𝑓 𝑥 = ((1^st ‘𝑢)⊼_𝑔(1^st ‘𝑣)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀_𝑔𝑖(1^st ‘𝑢)))})), {⟨𝑥, 𝑦⟩ ∣ (𝑦 = ∅ ∧ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑥 = (𝑖∈_𝑔𝑗))})
47	46	reseq1i 6005	. 2 ⊢ (rec((𝑓 ∈ V ↦ (𝑓 ∪ {⟨𝑥, 𝑦⟩ ∣ ∃𝑢 ∈ 𝑓 (∃𝑣 ∈ 𝑓 (𝑥 = ((1^st ‘𝑢)⊼_𝑔(1^st ‘𝑣)) ∧ 𝑦 = ((∅ ↑_m ω) ∖ ((2^nd ‘𝑢) ∩ (2^nd ‘𝑣)))) ∨ ∃𝑖 ∈ ω (𝑥 = ∀_𝑔𝑖(1^st ‘𝑢) ∧ 𝑦 = {𝑎 ∈ (∅ ↑_m ω) ∣ ∀𝑧 ∈ ∅ ({⟨𝑖, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑖}))) ∈ (2^nd ‘𝑢)}))})), {⟨𝑥, 𝑦⟩ ∣ ∃𝑖 ∈ ω ∃𝑗 ∈ ω (𝑥 = (𝑖∈_𝑔𝑗) ∧ 𝑦 = {𝑎 ∈ (∅ ↑_m ω) ∣ (𝑎‘𝑖)∅(𝑎‘𝑗)})}) ↾ suc ω) = (rec((𝑓 ∈ V ↦ (𝑓 ∪ {⟨𝑥, 𝑦⟩ ∣ (𝑦 = ∅ ∧ ∃𝑢 ∈ 𝑓 (∃𝑣 ∈ 𝑓 𝑥 = ((1^st ‘𝑢)⊼_𝑔(1^st ‘𝑣)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀_𝑔𝑖(1^st ‘𝑢)))})), {⟨𝑥, 𝑦⟩ ∣ (𝑦 = ∅ ∧ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑥 = (𝑖∈_𝑔𝑗))}) ↾ suc ω)
48	3, 47	eqtri 2768	1 ⊢ (∅ Sat ∅) = (rec((𝑓 ∈ V ↦ (𝑓 ∪ {⟨𝑥, 𝑦⟩ ∣ (𝑦 = ∅ ∧ ∃𝑢 ∈ 𝑓 (∃𝑣 ∈ 𝑓 𝑥 = ((1^st ‘𝑢)⊼_𝑔(1^st ‘𝑣)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀_𝑔𝑖(1^st ‘𝑢)))})), {⟨𝑥, 𝑦⟩ ∣ (𝑦 = ∅ ∧ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑥 = (𝑖∈_𝑔𝑗))}) ↾ suc ω)

Colors of variables: wff setvar class

Syntax hints: ∧ wa 395 ∨ wo 846 = wceq 1537 ∈ wcel 2108 ≠ wne 2946 ∀wral 3067 ∃wrex 3076 {crab 3443 Vcvv 3488 ∖ cdif 3973 ∪ cun 3974 ∩ cin 3975 ∅c0 4352 {csn 4648 ⟨cop 4654 class class class wbr 5166 {copab 5228 ↦ cmpt 5249 ↾ cres 5702 suc csuc 6397 ‘cfv 6573 (class class class)co 7448 ωcom 7903 1^st c1st 8028 2^nd c2nd 8029 reccrdg 8465 ↑_m cmap 8884 ∈_𝑔cgoe 35301 ⊼_𝑔cgna 35302 ∀_𝑔cgol 35303 Sat csat 35304

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2158 ax-12 2178 ax-ext 2711 ax-rep 5303 ax-sep 5317 ax-nul 5324 ax-pow 5383 ax-pr 5447 ax-un 7770 ax-inf2 9710

This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3or 1088 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-mo 2543 df-eu 2572 df-clab 2718 df-cleq 2732 df-clel 2819 df-nfc 2895 df-ne 2947 df-ral 3068 df-rex 3077 df-reu 3389 df-rab 3444 df-v 3490 df-sbc 3805 df-csb 3922 df-dif 3979 df-un 3981 df-in 3983 df-ss 3993 df-pss 3996 df-nul 4353 df-if 4549 df-pw 4624 df-sn 4649 df-pr 4651 df-op 4655 df-uni 4932 df-iun 5017 df-br 5167 df-opab 5229 df-mpt 5250 df-tr 5284 df-id 5593 df-eprel 5599 df-po 5607 df-so 5608 df-fr 5652 df-we 5654 df-xp 5706 df-rel 5707 df-cnv 5708 df-co 5709 df-dm 5710 df-rn 5711 df-res 5712 df-ima 5713 df-pred 6332 df-ord 6398 df-on 6399 df-lim 6400 df-suc 6401 df-iota 6525 df-fun 6575 df-fn 6576 df-f 6577 df-f1 6578 df-fo 6579 df-f1o 6580 df-fv 6581 df-ov 7451 df-oprab 7452 df-mpo 7453 df-om 7904 df-1st 8030 df-2nd 8031 df-frecs 8322 df-wrecs 8353 df-recs 8427 df-rdg 8466 df-map 8886 df-sat 35311

This theorem is referenced by: satf0sucom 35341

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