satf0 - Mathbox for Mario Carneiro

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Theorem satf0 32732

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 5175	. . 3 ⊢ ∅ ∈ V
2		satf 32713	. . 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 7581	. . . . . . . . . . . . . . . . . . 19 ⊢ ∅ ∈ ω
5	4	ne0ii 4253	. . . . . . . . . . . . . . . . . 18 ⊢ ω ≠ ∅
6		map0b 8430	. . . . . . . . . . . . . . . . . 18 ⊢ (ω ≠ ∅ → (∅ ↑_m ω) = ∅)
7	5, 6	ax-mp 5	. . . . . . . . . . . . . . . . 17 ⊢ (∅ ↑_m ω) = ∅
8	7	difeq1i 4046	. . . . . . . . . . . . . . . 16 ⊢ ((∅ ↑_m ω) ∖ ((2^nd ‘𝑢) ∩ (2^nd ‘𝑣))) = (∅ ∖ ((2^nd ‘𝑢) ∩ (2^nd ‘𝑣)))
9		0dif 4309	. . . . . . . . . . . . . . . 16 ⊢ (∅ ∖ ((2^nd ‘𝑢) ∩ (2^nd ‘𝑣))) = ∅
10	8, 9	eqtri 2821	. . . . . . . . . . . . . . 15 ⊢ ((∅ ↑_m ω) ∖ ((2^nd ‘𝑢) ∩ (2^nd ‘𝑣))) = ∅
11	10	eqeq2i 2811	. . . . . . . . . . . . . 14 ⊢ (𝑦 = ((∅ ↑_m ω) ∖ ((2^nd ‘𝑢) ∩ (2^nd ‘𝑣))) ↔ 𝑦 = ∅)
12	11	anbi2i 625	. . . . . . . . . . . . 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 3210	. . . . . . . . . . . 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 3300	. . . . . . . . . . . 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 278	. . . . . . . . . . 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 3429	. . . . . . . . . . . . . . . 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 4291	. . . . . . . . . . . . . . . 16 ⊢ {𝑎 ∈ ∅ ∣ ∀𝑧 ∈ ∅ ({⟨𝑖, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑖}))) ∈ (2^nd ‘𝑢)} = ∅
18	16, 17	eqtri 2821	. . . . . . . . . . . . . . 15 ⊢ {𝑎 ∈ (∅ ↑_m ω) ∣ ∀𝑧 ∈ ∅ ({⟨𝑖, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑖}))) ∈ (2^nd ‘𝑢)} = ∅
19	18	eqeq2i 2811	. . . . . . . . . . . . . 14 ⊢ (𝑦 = {𝑎 ∈ (∅ ↑_m ω) ∣ ∀𝑧 ∈ ∅ ({⟨𝑖, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑖}))) ∈ (2^nd ‘𝑢)} ↔ 𝑦 = ∅)
20	19	anbi2i 625	. . . . . . . . . . . . 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 3210	. . . . . . . . . . . 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 3300	. . . . . . . . . . . 12 ⊢ (∃𝑖 ∈ ω (𝑥 = ∀_𝑔𝑖(1^st ‘𝑢) ∧ 𝑦 = ∅) ↔ (∃𝑖 ∈ ω 𝑥 = ∀_𝑔𝑖(1^st ‘𝑢) ∧ 𝑦 = ∅))
23	21, 22	bitri 278	. . . . . . . . . . 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 912	. . . . . . . . . 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 3210	. . . . . . . . 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 1006	. . . . . . . . . . 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 227	. . . . . . . . . 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 3210	. . . . . . . . 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 3300	. . . . . . . . 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 300	. . . . . . . 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 466	. . . . . . 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 5097	. . . . . 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 4087	. . . . 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 5122	. . . 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 3429	. . . . . . . . . . 11 ⊢ {𝑎 ∈ (∅ ↑_m ω) ∣ (𝑎‘𝑖)∅(𝑎‘𝑗)} = {𝑎 ∈ ∅ ∣ (𝑎‘𝑖)∅(𝑎‘𝑗)}
36		rab0 4291	. . . . . . . . . . 11 ⊢ {𝑎 ∈ ∅ ∣ (𝑎‘𝑖)∅(𝑎‘𝑗)} = ∅
37	35, 36	eqtri 2821	. . . . . . . . . 10 ⊢ {𝑎 ∈ (∅ ↑_m ω) ∣ (𝑎‘𝑖)∅(𝑎‘𝑗)} = ∅
38	37	eqeq2i 2811	. . . . . . . . 9 ⊢ (𝑦 = {𝑎 ∈ (∅ ↑_m ω) ∣ (𝑎‘𝑖)∅(𝑎‘𝑗)} ↔ 𝑦 = ∅)
39	38	anbi2i 625	. . . . . . . 8 ⊢ ((𝑥 = (𝑖∈_𝑔𝑗) ∧ 𝑦 = {𝑎 ∈ (∅ ↑_m ω) ∣ (𝑎‘𝑖)∅(𝑎‘𝑗)}) ↔ (𝑥 = (𝑖∈_𝑔𝑗) ∧ 𝑦 = ∅))
40	39	2rexbii 3211	. . . . . . 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 3302	. . . . . . 7 ⊢ (∃𝑖 ∈ ω ∃𝑗 ∈ ω (𝑥 = (𝑖∈_𝑔𝑗) ∧ 𝑦 = ∅) ↔ (∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑥 = (𝑖∈_𝑔𝑗) ∧ 𝑦 = ∅))
42	40, 41	bitri 278	. . . . . 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 466	. . . . 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 5097	. . . 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 8032	. . . 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 5814	. 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 2821	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 399 ∨ wo 844 = wceq 1538 ∈ wcel 2111 ≠ wne 2987 ∀wral 3106 ∃wrex 3107 {crab 3110 Vcvv 3441 ∖ cdif 3878 ∪ cun 3879 ∩ cin 3880 ∅c0 4243 {csn 4525 ⟨cop 4531 class class class wbr 5030 {copab 5092 ↦ cmpt 5110 ↾ cres 5521 suc csuc 6161 ‘cfv 6324 (class class class)co 7135 ωcom 7560 1^st c1st 7669 2^nd c2nd 7670 reccrdg 8028 ↑_m cmap 8389 ∈_𝑔cgoe 32693 ⊼_𝑔cgna 32694 ∀_𝑔cgol 32695 Sat csat 32696

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 ax-inf2 9088

This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 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-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-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-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-map 8391 df-sat 32703

This theorem is referenced by: satf0sucom 32733

W3C validator