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Theorem satf 35547

Description: The satisfaction predicate as function over wff codes in the model 𝑀 and the binary relation 𝐸 on 𝑀. (Contributed by AV, 14-Sep-2023.)

**Assertion**
Ref	Expression
satf	⊢ ((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) → (𝑀 Sat 𝐸) = (rec((𝑓 ∈ V ↦ (𝑓 ∪ {⟨𝑥, 𝑦⟩ ∣ ∃𝑢 ∈ 𝑓 (∃𝑣 ∈ 𝑓 (𝑥 = ((1^st ‘𝑢)⊼_𝑔(1^st ‘𝑣)) ∧ 𝑦 = ((𝑀 ↑_m ω) ∖ ((2^nd ‘𝑢) ∩ (2^nd ‘𝑣)))) ∨ ∃𝑖 ∈ ω (𝑥 = ∀_𝑔𝑖(1^st ‘𝑢) ∧ 𝑦 = {𝑎 ∈ (𝑀 ↑_m ω) ∣ ∀𝑧 ∈ 𝑀 ({⟨𝑖, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑖}))) ∈ (2^nd ‘𝑢)}))})), {⟨𝑥, 𝑦⟩ ∣ ∃𝑖 ∈ ω ∃𝑗 ∈ ω (𝑥 = (𝑖∈_𝑔𝑗) ∧ 𝑦 = {𝑎 ∈ (𝑀 ↑_m ω) ∣ (𝑎‘𝑖)𝐸(𝑎‘𝑗)})}) ↾ suc ω))

Distinct variable groups: 𝐸,𝑎,𝑓,𝑖,𝑗,𝑢,𝑣,𝑥,𝑦 𝑀,𝑎,𝑓,𝑖,𝑗,𝑢,𝑣,𝑥,𝑦,𝑧 Allowed substitution hints: 𝐸(𝑧) 𝑉(𝑥,𝑦,𝑧,𝑣,𝑢,𝑓,𝑖,𝑗,𝑎) 𝑊(𝑥,𝑦,𝑧,𝑣,𝑢,𝑓,𝑖,𝑗,𝑎)

Proof of Theorem satf Dummy variables 𝑒 𝑚 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-sat 35537	. . 3 ⊢ Sat = (𝑚 ∈ V, 𝑒 ∈ V ↦ (rec((𝑓 ∈ V ↦ (𝑓 ∪ {⟨𝑥, 𝑦⟩ ∣ ∃𝑢 ∈ 𝑓 (∃𝑣 ∈ 𝑓 (𝑥 = ((1^st ‘𝑢)⊼_𝑔(1^st ‘𝑣)) ∧ 𝑦 = ((𝑚 ↑_m ω) ∖ ((2^nd ‘𝑢) ∩ (2^nd ‘𝑣)))) ∨ ∃𝑖 ∈ ω (𝑥 = ∀_𝑔𝑖(1^st ‘𝑢) ∧ 𝑦 = {𝑎 ∈ (𝑚 ↑_m ω) ∣ ∀𝑧 ∈ 𝑚 ({⟨𝑖, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑖}))) ∈ (2^nd ‘𝑢)}))})), {⟨𝑥, 𝑦⟩ ∣ ∃𝑖 ∈ ω ∃𝑗 ∈ ω (𝑥 = (𝑖∈_𝑔𝑗) ∧ 𝑦 = {𝑎 ∈ (𝑚 ↑_m ω) ∣ (𝑎‘𝑖)𝑒(𝑎‘𝑗)})}) ↾ suc ω))
2	1	a1i 11	. 2 ⊢ ((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) → Sat = (𝑚 ∈ V, 𝑒 ∈ V ↦ (rec((𝑓 ∈ V ↦ (𝑓 ∪ {⟨𝑥, 𝑦⟩ ∣ ∃𝑢 ∈ 𝑓 (∃𝑣 ∈ 𝑓 (𝑥 = ((1^st ‘𝑢)⊼_𝑔(1^st ‘𝑣)) ∧ 𝑦 = ((𝑚 ↑_m ω) ∖ ((2^nd ‘𝑢) ∩ (2^nd ‘𝑣)))) ∨ ∃𝑖 ∈ ω (𝑥 = ∀_𝑔𝑖(1^st ‘𝑢) ∧ 𝑦 = {𝑎 ∈ (𝑚 ↑_m ω) ∣ ∀𝑧 ∈ 𝑚 ({⟨𝑖, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑖}))) ∈ (2^nd ‘𝑢)}))})), {⟨𝑥, 𝑦⟩ ∣ ∃𝑖 ∈ ω ∃𝑗 ∈ ω (𝑥 = (𝑖∈_𝑔𝑗) ∧ 𝑦 = {𝑎 ∈ (𝑚 ↑_m ω) ∣ (𝑎‘𝑖)𝑒(𝑎‘𝑗)})}) ↾ suc ω)))
3		oveq1 7365	. . . . . . . . . . . . . . 15 ⊢ (𝑚 = 𝑀 → (𝑚 ↑_m ω) = (𝑀 ↑_m ω))
4	3	adantr 480	. . . . . . . . . . . . . 14 ⊢ ((𝑚 = 𝑀 ∧ 𝑒 = 𝐸) → (𝑚 ↑_m ω) = (𝑀 ↑_m ω))
5	4	difeq1d 4077	. . . . . . . . . . . . 13 ⊢ ((𝑚 = 𝑀 ∧ 𝑒 = 𝐸) → ((𝑚 ↑_m ω) ∖ ((2^nd ‘𝑢) ∩ (2^nd ‘𝑣))) = ((𝑀 ↑_m ω) ∖ ((2^nd ‘𝑢) ∩ (2^nd ‘𝑣))))
6	5	eqeq2d 2747	. . . . . . . . . . . 12 ⊢ ((𝑚 = 𝑀 ∧ 𝑒 = 𝐸) → (𝑦 = ((𝑚 ↑_m ω) ∖ ((2^nd ‘𝑢) ∩ (2^nd ‘𝑣))) ↔ 𝑦 = ((𝑀 ↑_m ω) ∖ ((2^nd ‘𝑢) ∩ (2^nd ‘𝑣)))))
7	6	anbi2d 630	. . . . . . . . . . 11 ⊢ ((𝑚 = 𝑀 ∧ 𝑒 = 𝐸) → ((𝑥 = ((1^st ‘𝑢)⊼_𝑔(1^st ‘𝑣)) ∧ 𝑦 = ((𝑚 ↑_m ω) ∖ ((2^nd ‘𝑢) ∩ (2^nd ‘𝑣)))) ↔ (𝑥 = ((1^st ‘𝑢)⊼_𝑔(1^st ‘𝑣)) ∧ 𝑦 = ((𝑀 ↑_m ω) ∖ ((2^nd ‘𝑢) ∩ (2^nd ‘𝑣))))))
8	7	rexbidv 3160	. . . . . . . . . 10 ⊢ ((𝑚 = 𝑀 ∧ 𝑒 = 𝐸) → (∃𝑣 ∈ 𝑓 (𝑥 = ((1^st ‘𝑢)⊼_𝑔(1^st ‘𝑣)) ∧ 𝑦 = ((𝑚 ↑_m ω) ∖ ((2^nd ‘𝑢) ∩ (2^nd ‘𝑣)))) ↔ ∃𝑣 ∈ 𝑓 (𝑥 = ((1^st ‘𝑢)⊼_𝑔(1^st ‘𝑣)) ∧ 𝑦 = ((𝑀 ↑_m ω) ∖ ((2^nd ‘𝑢) ∩ (2^nd ‘𝑣))))))
9		simpl 482	. . . . . . . . . . . . . . 15 ⊢ ((𝑚 = 𝑀 ∧ 𝑒 = 𝐸) → 𝑚 = 𝑀)
10	9	raleqdv 3296	. . . . . . . . . . . . . 14 ⊢ ((𝑚 = 𝑀 ∧ 𝑒 = 𝐸) → (∀𝑧 ∈ 𝑚 ({⟨𝑖, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑖}))) ∈ (2^nd ‘𝑢) ↔ ∀𝑧 ∈ 𝑀 ({⟨𝑖, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑖}))) ∈ (2^nd ‘𝑢)))
11	4, 10	rabeqbidv 3417	. . . . . . . . . . . . 13 ⊢ ((𝑚 = 𝑀 ∧ 𝑒 = 𝐸) → {𝑎 ∈ (𝑚 ↑_m ω) ∣ ∀𝑧 ∈ 𝑚 ({⟨𝑖, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑖}))) ∈ (2^nd ‘𝑢)} = {𝑎 ∈ (𝑀 ↑_m ω) ∣ ∀𝑧 ∈ 𝑀 ({⟨𝑖, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑖}))) ∈ (2^nd ‘𝑢)})
12	11	eqeq2d 2747	. . . . . . . . . . . 12 ⊢ ((𝑚 = 𝑀 ∧ 𝑒 = 𝐸) → (𝑦 = {𝑎 ∈ (𝑚 ↑_m ω) ∣ ∀𝑧 ∈ 𝑚 ({⟨𝑖, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑖}))) ∈ (2^nd ‘𝑢)} ↔ 𝑦 = {𝑎 ∈ (𝑀 ↑_m ω) ∣ ∀𝑧 ∈ 𝑀 ({⟨𝑖, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑖}))) ∈ (2^nd ‘𝑢)}))
13	12	anbi2d 630	. . . . . . . . . . 11 ⊢ ((𝑚 = 𝑀 ∧ 𝑒 = 𝐸) → ((𝑥 = ∀_𝑔𝑖(1^st ‘𝑢) ∧ 𝑦 = {𝑎 ∈ (𝑚 ↑_m ω) ∣ ∀𝑧 ∈ 𝑚 ({⟨𝑖, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑖}))) ∈ (2^nd ‘𝑢)}) ↔ (𝑥 = ∀_𝑔𝑖(1^st ‘𝑢) ∧ 𝑦 = {𝑎 ∈ (𝑀 ↑_m ω) ∣ ∀𝑧 ∈ 𝑀 ({⟨𝑖, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑖}))) ∈ (2^nd ‘𝑢)})))
14	13	rexbidv 3160	. . . . . . . . . 10 ⊢ ((𝑚 = 𝑀 ∧ 𝑒 = 𝐸) → (∃𝑖 ∈ ω (𝑥 = ∀_𝑔𝑖(1^st ‘𝑢) ∧ 𝑦 = {𝑎 ∈ (𝑚 ↑_m ω) ∣ ∀𝑧 ∈ 𝑚 ({⟨𝑖, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑖}))) ∈ (2^nd ‘𝑢)}) ↔ ∃𝑖 ∈ ω (𝑥 = ∀_𝑔𝑖(1^st ‘𝑢) ∧ 𝑦 = {𝑎 ∈ (𝑀 ↑_m ω) ∣ ∀𝑧 ∈ 𝑀 ({⟨𝑖, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑖}))) ∈ (2^nd ‘𝑢)})))
15	8, 14	orbi12d 918	. . . . . . . . 9 ⊢ ((𝑚 = 𝑀 ∧ 𝑒 = 𝐸) → ((∃𝑣 ∈ 𝑓 (𝑥 = ((1^st ‘𝑢)⊼_𝑔(1^st ‘𝑣)) ∧ 𝑦 = ((𝑚 ↑_m ω) ∖ ((2^nd ‘𝑢) ∩ (2^nd ‘𝑣)))) ∨ ∃𝑖 ∈ ω (𝑥 = ∀_𝑔𝑖(1^st ‘𝑢) ∧ 𝑦 = {𝑎 ∈ (𝑚 ↑_m ω) ∣ ∀𝑧 ∈ 𝑚 ({⟨𝑖, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑖}))) ∈ (2^nd ‘𝑢)})) ↔ (∃𝑣 ∈ 𝑓 (𝑥 = ((1^st ‘𝑢)⊼_𝑔(1^st ‘𝑣)) ∧ 𝑦 = ((𝑀 ↑_m ω) ∖ ((2^nd ‘𝑢) ∩ (2^nd ‘𝑣)))) ∨ ∃𝑖 ∈ ω (𝑥 = ∀_𝑔𝑖(1^st ‘𝑢) ∧ 𝑦 = {𝑎 ∈ (𝑀 ↑_m ω) ∣ ∀𝑧 ∈ 𝑀 ({⟨𝑖, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑖}))) ∈ (2^nd ‘𝑢)}))))
16	15	rexbidv 3160	. . . . . . . 8 ⊢ ((𝑚 = 𝑀 ∧ 𝑒 = 𝐸) → (∃𝑢 ∈ 𝑓 (∃𝑣 ∈ 𝑓 (𝑥 = ((1^st ‘𝑢)⊼_𝑔(1^st ‘𝑣)) ∧ 𝑦 = ((𝑚 ↑_m ω) ∖ ((2^nd ‘𝑢) ∩ (2^nd ‘𝑣)))) ∨ ∃𝑖 ∈ ω (𝑥 = ∀_𝑔𝑖(1^st ‘𝑢) ∧ 𝑦 = {𝑎 ∈ (𝑚 ↑_m ω) ∣ ∀𝑧 ∈ 𝑚 ({⟨𝑖, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑖}))) ∈ (2^nd ‘𝑢)})) ↔ ∃𝑢 ∈ 𝑓 (∃𝑣 ∈ 𝑓 (𝑥 = ((1^st ‘𝑢)⊼_𝑔(1^st ‘𝑣)) ∧ 𝑦 = ((𝑀 ↑_m ω) ∖ ((2^nd ‘𝑢) ∩ (2^nd ‘𝑣)))) ∨ ∃𝑖 ∈ ω (𝑥 = ∀_𝑔𝑖(1^st ‘𝑢) ∧ 𝑦 = {𝑎 ∈ (𝑀 ↑_m ω) ∣ ∀𝑧 ∈ 𝑀 ({⟨𝑖, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑖}))) ∈ (2^nd ‘𝑢)}))))
17	16	opabbidv 5164	. . . . . . 7 ⊢ ((𝑚 = 𝑀 ∧ 𝑒 = 𝐸) → {⟨𝑥, 𝑦⟩ ∣ ∃𝑢 ∈ 𝑓 (∃𝑣 ∈ 𝑓 (𝑥 = ((1^st ‘𝑢)⊼_𝑔(1^st ‘𝑣)) ∧ 𝑦 = ((𝑚 ↑_m ω) ∖ ((2^nd ‘𝑢) ∩ (2^nd ‘𝑣)))) ∨ ∃𝑖 ∈ ω (𝑥 = ∀_𝑔𝑖(1^st ‘𝑢) ∧ 𝑦 = {𝑎 ∈ (𝑚 ↑_m ω) ∣ ∀𝑧 ∈ 𝑚 ({⟨𝑖, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑖}))) ∈ (2^nd ‘𝑢)}))} = {⟨𝑥, 𝑦⟩ ∣ ∃𝑢 ∈ 𝑓 (∃𝑣 ∈ 𝑓 (𝑥 = ((1^st ‘𝑢)⊼_𝑔(1^st ‘𝑣)) ∧ 𝑦 = ((𝑀 ↑_m ω) ∖ ((2^nd ‘𝑢) ∩ (2^nd ‘𝑣)))) ∨ ∃𝑖 ∈ ω (𝑥 = ∀_𝑔𝑖(1^st ‘𝑢) ∧ 𝑦 = {𝑎 ∈ (𝑀 ↑_m ω) ∣ ∀𝑧 ∈ 𝑀 ({⟨𝑖, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑖}))) ∈ (2^nd ‘𝑢)}))})
18	17	uneq2d 4120	. . . . . 6 ⊢ ((𝑚 = 𝑀 ∧ 𝑒 = 𝐸) → (𝑓 ∪ {⟨𝑥, 𝑦⟩ ∣ ∃𝑢 ∈ 𝑓 (∃𝑣 ∈ 𝑓 (𝑥 = ((1^st ‘𝑢)⊼_𝑔(1^st ‘𝑣)) ∧ 𝑦 = ((𝑚 ↑_m ω) ∖ ((2^nd ‘𝑢) ∩ (2^nd ‘𝑣)))) ∨ ∃𝑖 ∈ ω (𝑥 = ∀_𝑔𝑖(1^st ‘𝑢) ∧ 𝑦 = {𝑎 ∈ (𝑚 ↑_m ω) ∣ ∀𝑧 ∈ 𝑚 ({⟨𝑖, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑖}))) ∈ (2^nd ‘𝑢)}))}) = (𝑓 ∪ {⟨𝑥, 𝑦⟩ ∣ ∃𝑢 ∈ 𝑓 (∃𝑣 ∈ 𝑓 (𝑥 = ((1^st ‘𝑢)⊼_𝑔(1^st ‘𝑣)) ∧ 𝑦 = ((𝑀 ↑_m ω) ∖ ((2^nd ‘𝑢) ∩ (2^nd ‘𝑣)))) ∨ ∃𝑖 ∈ ω (𝑥 = ∀_𝑔𝑖(1^st ‘𝑢) ∧ 𝑦 = {𝑎 ∈ (𝑀 ↑_m ω) ∣ ∀𝑧 ∈ 𝑀 ({⟨𝑖, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑖}))) ∈ (2^nd ‘𝑢)}))}))
19	18	mpteq2dv 5192	. . . . 5 ⊢ ((𝑚 = 𝑀 ∧ 𝑒 = 𝐸) → (𝑓 ∈ V ↦ (𝑓 ∪ {⟨𝑥, 𝑦⟩ ∣ ∃𝑢 ∈ 𝑓 (∃𝑣 ∈ 𝑓 (𝑥 = ((1^st ‘𝑢)⊼_𝑔(1^st ‘𝑣)) ∧ 𝑦 = ((𝑚 ↑_m ω) ∖ ((2^nd ‘𝑢) ∩ (2^nd ‘𝑣)))) ∨ ∃𝑖 ∈ ω (𝑥 = ∀_𝑔𝑖(1^st ‘𝑢) ∧ 𝑦 = {𝑎 ∈ (𝑚 ↑_m ω) ∣ ∀𝑧 ∈ 𝑚 ({⟨𝑖, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑖}))) ∈ (2^nd ‘𝑢)}))})) = (𝑓 ∈ V ↦ (𝑓 ∪ {⟨𝑥, 𝑦⟩ ∣ ∃𝑢 ∈ 𝑓 (∃𝑣 ∈ 𝑓 (𝑥 = ((1^st ‘𝑢)⊼_𝑔(1^st ‘𝑣)) ∧ 𝑦 = ((𝑀 ↑_m ω) ∖ ((2^nd ‘𝑢) ∩ (2^nd ‘𝑣)))) ∨ ∃𝑖 ∈ ω (𝑥 = ∀_𝑔𝑖(1^st ‘𝑢) ∧ 𝑦 = {𝑎 ∈ (𝑀 ↑_m ω) ∣ ∀𝑧 ∈ 𝑀 ({⟨𝑖, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑖}))) ∈ (2^nd ‘𝑢)}))})))
20		breq 5100	. . . . . . . . . . 11 ⊢ (𝑒 = 𝐸 → ((𝑎‘𝑖)𝑒(𝑎‘𝑗) ↔ (𝑎‘𝑖)𝐸(𝑎‘𝑗)))
21	20	adantl 481	. . . . . . . . . 10 ⊢ ((𝑚 = 𝑀 ∧ 𝑒 = 𝐸) → ((𝑎‘𝑖)𝑒(𝑎‘𝑗) ↔ (𝑎‘𝑖)𝐸(𝑎‘𝑗)))
22	4, 21	rabeqbidv 3417	. . . . . . . . 9 ⊢ ((𝑚 = 𝑀 ∧ 𝑒 = 𝐸) → {𝑎 ∈ (𝑚 ↑_m ω) ∣ (𝑎‘𝑖)𝑒(𝑎‘𝑗)} = {𝑎 ∈ (𝑀 ↑_m ω) ∣ (𝑎‘𝑖)𝐸(𝑎‘𝑗)})
23	22	eqeq2d 2747	. . . . . . . 8 ⊢ ((𝑚 = 𝑀 ∧ 𝑒 = 𝐸) → (𝑦 = {𝑎 ∈ (𝑚 ↑_m ω) ∣ (𝑎‘𝑖)𝑒(𝑎‘𝑗)} ↔ 𝑦 = {𝑎 ∈ (𝑀 ↑_m ω) ∣ (𝑎‘𝑖)𝐸(𝑎‘𝑗)}))
24	23	anbi2d 630	. . . . . . 7 ⊢ ((𝑚 = 𝑀 ∧ 𝑒 = 𝐸) → ((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)})))
25	24	2rexbidv 3201	. . . . . 6 ⊢ ((𝑚 = 𝑀 ∧ 𝑒 = 𝐸) → (∃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)})))
26	25	opabbidv 5164	. . . . 5 ⊢ ((𝑚 = 𝑀 ∧ 𝑒 = 𝐸) → {⟨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)})})
27		rdgeq12 8344	. . . . 5 ⊢ (((𝑓 ∈ V ↦ (𝑓 ∪ {⟨𝑥, 𝑦⟩ ∣ ∃𝑢 ∈ 𝑓 (∃𝑣 ∈ 𝑓 (𝑥 = ((1^st ‘𝑢)⊼_𝑔(1^st ‘𝑣)) ∧ 𝑦 = ((𝑚 ↑_m ω) ∖ ((2^nd ‘𝑢) ∩ (2^nd ‘𝑣)))) ∨ ∃𝑖 ∈ ω (𝑥 = ∀_𝑔𝑖(1^st ‘𝑢) ∧ 𝑦 = {𝑎 ∈ (𝑚 ↑_m ω) ∣ ∀𝑧 ∈ 𝑚 ({⟨𝑖, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑖}))) ∈ (2^nd ‘𝑢)}))})) = (𝑓 ∈ V ↦ (𝑓 ∪ {⟨𝑥, 𝑦⟩ ∣ ∃𝑢 ∈ 𝑓 (∃𝑣 ∈ 𝑓 (𝑥 = ((1^st ‘𝑢)⊼_𝑔(1^st ‘𝑣)) ∧ 𝑦 = ((𝑀 ↑_m ω) ∖ ((2^nd ‘𝑢) ∩ (2^nd ‘𝑣)))) ∨ ∃𝑖 ∈ ω (𝑥 = ∀_𝑔𝑖(1^st ‘𝑢) ∧ 𝑦 = {𝑎 ∈ (𝑀 ↑_m ω) ∣ ∀𝑧 ∈ 𝑀 ({⟨𝑖, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑖}))) ∈ (2^nd ‘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)})}) → rec((𝑓 ∈ V ↦ (𝑓 ∪ {⟨𝑥, 𝑦⟩ ∣ ∃𝑢 ∈ 𝑓 (∃𝑣 ∈ 𝑓 (𝑥 = ((1^st ‘𝑢)⊼_𝑔(1^st ‘𝑣)) ∧ 𝑦 = ((𝑚 ↑_m ω) ∖ ((2^nd ‘𝑢) ∩ (2^nd ‘𝑣)))) ∨ ∃𝑖 ∈ ω (𝑥 = ∀_𝑔𝑖(1^st ‘𝑢) ∧ 𝑦 = {𝑎 ∈ (𝑚 ↑_m ω) ∣ ∀𝑧 ∈ 𝑚 ({⟨𝑖, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑖}))) ∈ (2^nd ‘𝑢)}))})), {⟨𝑥, 𝑦⟩ ∣ ∃𝑖 ∈ ω ∃𝑗 ∈ ω (𝑥 = (𝑖∈_𝑔𝑗) ∧ 𝑦 = {𝑎 ∈ (𝑚 ↑_m ω) ∣ (𝑎‘𝑖)𝑒(𝑎‘𝑗)})}) = rec((𝑓 ∈ V ↦ (𝑓 ∪ {⟨𝑥, 𝑦⟩ ∣ ∃𝑢 ∈ 𝑓 (∃𝑣 ∈ 𝑓 (𝑥 = ((1^st ‘𝑢)⊼_𝑔(1^st ‘𝑣)) ∧ 𝑦 = ((𝑀 ↑_m ω) ∖ ((2^nd ‘𝑢) ∩ (2^nd ‘𝑣)))) ∨ ∃𝑖 ∈ ω (𝑥 = ∀_𝑔𝑖(1^st ‘𝑢) ∧ 𝑦 = {𝑎 ∈ (𝑀 ↑_m ω) ∣ ∀𝑧 ∈ 𝑀 ({⟨𝑖, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑖}))) ∈ (2^nd ‘𝑢)}))})), {⟨𝑥, 𝑦⟩ ∣ ∃𝑖 ∈ ω ∃𝑗 ∈ ω (𝑥 = (𝑖∈_𝑔𝑗) ∧ 𝑦 = {𝑎 ∈ (𝑀 ↑_m ω) ∣ (𝑎‘𝑖)𝐸(𝑎‘𝑗)})}))
28	19, 26, 27	syl2anc 584	. . . 4 ⊢ ((𝑚 = 𝑀 ∧ 𝑒 = 𝐸) → rec((𝑓 ∈ V ↦ (𝑓 ∪ {⟨𝑥, 𝑦⟩ ∣ ∃𝑢 ∈ 𝑓 (∃𝑣 ∈ 𝑓 (𝑥 = ((1^st ‘𝑢)⊼_𝑔(1^st ‘𝑣)) ∧ 𝑦 = ((𝑚 ↑_m ω) ∖ ((2^nd ‘𝑢) ∩ (2^nd ‘𝑣)))) ∨ ∃𝑖 ∈ ω (𝑥 = ∀_𝑔𝑖(1^st ‘𝑢) ∧ 𝑦 = {𝑎 ∈ (𝑚 ↑_m ω) ∣ ∀𝑧 ∈ 𝑚 ({⟨𝑖, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑖}))) ∈ (2^nd ‘𝑢)}))})), {⟨𝑥, 𝑦⟩ ∣ ∃𝑖 ∈ ω ∃𝑗 ∈ ω (𝑥 = (𝑖∈_𝑔𝑗) ∧ 𝑦 = {𝑎 ∈ (𝑚 ↑_m ω) ∣ (𝑎‘𝑖)𝑒(𝑎‘𝑗)})}) = rec((𝑓 ∈ V ↦ (𝑓 ∪ {⟨𝑥, 𝑦⟩ ∣ ∃𝑢 ∈ 𝑓 (∃𝑣 ∈ 𝑓 (𝑥 = ((1^st ‘𝑢)⊼_𝑔(1^st ‘𝑣)) ∧ 𝑦 = ((𝑀 ↑_m ω) ∖ ((2^nd ‘𝑢) ∩ (2^nd ‘𝑣)))) ∨ ∃𝑖 ∈ ω (𝑥 = ∀_𝑔𝑖(1^st ‘𝑢) ∧ 𝑦 = {𝑎 ∈ (𝑀 ↑_m ω) ∣ ∀𝑧 ∈ 𝑀 ({⟨𝑖, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑖}))) ∈ (2^nd ‘𝑢)}))})), {⟨𝑥, 𝑦⟩ ∣ ∃𝑖 ∈ ω ∃𝑗 ∈ ω (𝑥 = (𝑖∈_𝑔𝑗) ∧ 𝑦 = {𝑎 ∈ (𝑀 ↑_m ω) ∣ (𝑎‘𝑖)𝐸(𝑎‘𝑗)})}))
29	28	reseq1d 5937	. . 3 ⊢ ((𝑚 = 𝑀 ∧ 𝑒 = 𝐸) → (rec((𝑓 ∈ V ↦ (𝑓 ∪ {⟨𝑥, 𝑦⟩ ∣ ∃𝑢 ∈ 𝑓 (∃𝑣 ∈ 𝑓 (𝑥 = ((1^st ‘𝑢)⊼_𝑔(1^st ‘𝑣)) ∧ 𝑦 = ((𝑚 ↑_m ω) ∖ ((2^nd ‘𝑢) ∩ (2^nd ‘𝑣)))) ∨ ∃𝑖 ∈ ω (𝑥 = ∀_𝑔𝑖(1^st ‘𝑢) ∧ 𝑦 = {𝑎 ∈ (𝑚 ↑_m ω) ∣ ∀𝑧 ∈ 𝑚 ({⟨𝑖, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑖}))) ∈ (2^nd ‘𝑢)}))})), {⟨𝑥, 𝑦⟩ ∣ ∃𝑖 ∈ ω ∃𝑗 ∈ ω (𝑥 = (𝑖∈_𝑔𝑗) ∧ 𝑦 = {𝑎 ∈ (𝑚 ↑_m ω) ∣ (𝑎‘𝑖)𝑒(𝑎‘𝑗)})}) ↾ suc ω) = (rec((𝑓 ∈ V ↦ (𝑓 ∪ {⟨𝑥, 𝑦⟩ ∣ ∃𝑢 ∈ 𝑓 (∃𝑣 ∈ 𝑓 (𝑥 = ((1^st ‘𝑢)⊼_𝑔(1^st ‘𝑣)) ∧ 𝑦 = ((𝑀 ↑_m ω) ∖ ((2^nd ‘𝑢) ∩ (2^nd ‘𝑣)))) ∨ ∃𝑖 ∈ ω (𝑥 = ∀_𝑔𝑖(1^st ‘𝑢) ∧ 𝑦 = {𝑎 ∈ (𝑀 ↑_m ω) ∣ ∀𝑧 ∈ 𝑀 ({⟨𝑖, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑖}))) ∈ (2^nd ‘𝑢)}))})), {⟨𝑥, 𝑦⟩ ∣ ∃𝑖 ∈ ω ∃𝑗 ∈ ω (𝑥 = (𝑖∈_𝑔𝑗) ∧ 𝑦 = {𝑎 ∈ (𝑀 ↑_m ω) ∣ (𝑎‘𝑖)𝐸(𝑎‘𝑗)})}) ↾ suc ω))
30	29	adantl 481	. 2 ⊢ (((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) ∧ (𝑚 = 𝑀 ∧ 𝑒 = 𝐸)) → (rec((𝑓 ∈ V ↦ (𝑓 ∪ {⟨𝑥, 𝑦⟩ ∣ ∃𝑢 ∈ 𝑓 (∃𝑣 ∈ 𝑓 (𝑥 = ((1^st ‘𝑢)⊼_𝑔(1^st ‘𝑣)) ∧ 𝑦 = ((𝑚 ↑_m ω) ∖ ((2^nd ‘𝑢) ∩ (2^nd ‘𝑣)))) ∨ ∃𝑖 ∈ ω (𝑥 = ∀_𝑔𝑖(1^st ‘𝑢) ∧ 𝑦 = {𝑎 ∈ (𝑚 ↑_m ω) ∣ ∀𝑧 ∈ 𝑚 ({⟨𝑖, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑖}))) ∈ (2^nd ‘𝑢)}))})), {⟨𝑥, 𝑦⟩ ∣ ∃𝑖 ∈ ω ∃𝑗 ∈ ω (𝑥 = (𝑖∈_𝑔𝑗) ∧ 𝑦 = {𝑎 ∈ (𝑚 ↑_m ω) ∣ (𝑎‘𝑖)𝑒(𝑎‘𝑗)})}) ↾ suc ω) = (rec((𝑓 ∈ V ↦ (𝑓 ∪ {⟨𝑥, 𝑦⟩ ∣ ∃𝑢 ∈ 𝑓 (∃𝑣 ∈ 𝑓 (𝑥 = ((1^st ‘𝑢)⊼_𝑔(1^st ‘𝑣)) ∧ 𝑦 = ((𝑀 ↑_m ω) ∖ ((2^nd ‘𝑢) ∩ (2^nd ‘𝑣)))) ∨ ∃𝑖 ∈ ω (𝑥 = ∀_𝑔𝑖(1^st ‘𝑢) ∧ 𝑦 = {𝑎 ∈ (𝑀 ↑_m ω) ∣ ∀𝑧 ∈ 𝑀 ({⟨𝑖, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑖}))) ∈ (2^nd ‘𝑢)}))})), {⟨𝑥, 𝑦⟩ ∣ ∃𝑖 ∈ ω ∃𝑗 ∈ ω (𝑥 = (𝑖∈_𝑔𝑗) ∧ 𝑦 = {𝑎 ∈ (𝑀 ↑_m ω) ∣ (𝑎‘𝑖)𝐸(𝑎‘𝑗)})}) ↾ suc ω))
31		elex 3461	. . 3 ⊢ (𝑀 ∈ 𝑉 → 𝑀 ∈ V)
32	31	adantr 480	. 2 ⊢ ((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) → 𝑀 ∈ V)
33		elex 3461	. . 3 ⊢ (𝐸 ∈ 𝑊 → 𝐸 ∈ V)
34	33	adantl 481	. 2 ⊢ ((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) → 𝐸 ∈ V)
35		rdgfun 8347	. . 3 ⊢ Fun rec((𝑓 ∈ V ↦ (𝑓 ∪ {⟨𝑥, 𝑦⟩ ∣ ∃𝑢 ∈ 𝑓 (∃𝑣 ∈ 𝑓 (𝑥 = ((1^st ‘𝑢)⊼_𝑔(1^st ‘𝑣)) ∧ 𝑦 = ((𝑀 ↑_m ω) ∖ ((2^nd ‘𝑢) ∩ (2^nd ‘𝑣)))) ∨ ∃𝑖 ∈ ω (𝑥 = ∀_𝑔𝑖(1^st ‘𝑢) ∧ 𝑦 = {𝑎 ∈ (𝑀 ↑_m ω) ∣ ∀𝑧 ∈ 𝑀 ({⟨𝑖, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑖}))) ∈ (2^nd ‘𝑢)}))})), {⟨𝑥, 𝑦⟩ ∣ ∃𝑖 ∈ ω ∃𝑗 ∈ ω (𝑥 = (𝑖∈_𝑔𝑗) ∧ 𝑦 = {𝑎 ∈ (𝑀 ↑_m ω) ∣ (𝑎‘𝑖)𝐸(𝑎‘𝑗)})})
36		omex 9552	. . . . 5 ⊢ ω ∈ V
37	36	sucex 7751	. . . 4 ⊢ suc ω ∈ V
38	37	a1i 11	. . 3 ⊢ ((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) → suc ω ∈ V)
39		resfunexg 7161	. . 3 ⊢ ((Fun rec((𝑓 ∈ V ↦ (𝑓 ∪ {⟨𝑥, 𝑦⟩ ∣ ∃𝑢 ∈ 𝑓 (∃𝑣 ∈ 𝑓 (𝑥 = ((1^st ‘𝑢)⊼_𝑔(1^st ‘𝑣)) ∧ 𝑦 = ((𝑀 ↑_m ω) ∖ ((2^nd ‘𝑢) ∩ (2^nd ‘𝑣)))) ∨ ∃𝑖 ∈ ω (𝑥 = ∀_𝑔𝑖(1^st ‘𝑢) ∧ 𝑦 = {𝑎 ∈ (𝑀 ↑_m ω) ∣ ∀𝑧 ∈ 𝑀 ({⟨𝑖, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑖}))) ∈ (2^nd ‘𝑢)}))})), {⟨𝑥, 𝑦⟩ ∣ ∃𝑖 ∈ ω ∃𝑗 ∈ ω (𝑥 = (𝑖∈_𝑔𝑗) ∧ 𝑦 = {𝑎 ∈ (𝑀 ↑_m ω) ∣ (𝑎‘𝑖)𝐸(𝑎‘𝑗)})}) ∧ suc ω ∈ V) → (rec((𝑓 ∈ V ↦ (𝑓 ∪ {⟨𝑥, 𝑦⟩ ∣ ∃𝑢 ∈ 𝑓 (∃𝑣 ∈ 𝑓 (𝑥 = ((1^st ‘𝑢)⊼_𝑔(1^st ‘𝑣)) ∧ 𝑦 = ((𝑀 ↑_m ω) ∖ ((2^nd ‘𝑢) ∩ (2^nd ‘𝑣)))) ∨ ∃𝑖 ∈ ω (𝑥 = ∀_𝑔𝑖(1^st ‘𝑢) ∧ 𝑦 = {𝑎 ∈ (𝑀 ↑_m ω) ∣ ∀𝑧 ∈ 𝑀 ({⟨𝑖, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑖}))) ∈ (2^nd ‘𝑢)}))})), {⟨𝑥, 𝑦⟩ ∣ ∃𝑖 ∈ ω ∃𝑗 ∈ ω (𝑥 = (𝑖∈_𝑔𝑗) ∧ 𝑦 = {𝑎 ∈ (𝑀 ↑_m ω) ∣ (𝑎‘𝑖)𝐸(𝑎‘𝑗)})}) ↾ suc ω) ∈ V)
40	35, 38, 39	sylancr 587	. 2 ⊢ ((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) → (rec((𝑓 ∈ V ↦ (𝑓 ∪ {⟨𝑥, 𝑦⟩ ∣ ∃𝑢 ∈ 𝑓 (∃𝑣 ∈ 𝑓 (𝑥 = ((1^st ‘𝑢)⊼_𝑔(1^st ‘𝑣)) ∧ 𝑦 = ((𝑀 ↑_m ω) ∖ ((2^nd ‘𝑢) ∩ (2^nd ‘𝑣)))) ∨ ∃𝑖 ∈ ω (𝑥 = ∀_𝑔𝑖(1^st ‘𝑢) ∧ 𝑦 = {𝑎 ∈ (𝑀 ↑_m ω) ∣ ∀𝑧 ∈ 𝑀 ({⟨𝑖, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑖}))) ∈ (2^nd ‘𝑢)}))})), {⟨𝑥, 𝑦⟩ ∣ ∃𝑖 ∈ ω ∃𝑗 ∈ ω (𝑥 = (𝑖∈_𝑔𝑗) ∧ 𝑦 = {𝑎 ∈ (𝑀 ↑_m ω) ∣ (𝑎‘𝑖)𝐸(𝑎‘𝑗)})}) ↾ suc ω) ∈ V)
41	2, 30, 32, 34, 40	ovmpod 7510	1 ⊢ ((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) → (𝑀 Sat 𝐸) = (rec((𝑓 ∈ V ↦ (𝑓 ∪ {⟨𝑥, 𝑦⟩ ∣ ∃𝑢 ∈ 𝑓 (∃𝑣 ∈ 𝑓 (𝑥 = ((1^st ‘𝑢)⊼_𝑔(1^st ‘𝑣)) ∧ 𝑦 = ((𝑀 ↑_m ω) ∖ ((2^nd ‘𝑢) ∩ (2^nd ‘𝑣)))) ∨ ∃𝑖 ∈ ω (𝑥 = ∀_𝑔𝑖(1^st ‘𝑢) ∧ 𝑦 = {𝑎 ∈ (𝑀 ↑_m ω) ∣ ∀𝑧 ∈ 𝑀 ({⟨𝑖, 𝑧⟩} ∪ (𝑎 ↾ (ω ∖ {𝑖}))) ∈ (2^nd ‘𝑢)}))})), {⟨𝑥, 𝑦⟩ ∣ ∃𝑖 ∈ ω ∃𝑗 ∈ ω (𝑥 = (𝑖∈_𝑔𝑗) ∧ 𝑦 = {𝑎 ∈ (𝑀 ↑_m ω) ∣ (𝑎‘𝑖)𝐸(𝑎‘𝑗)})}) ↾ suc ω))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∨ wo 847 = wceq 1541 ∈ wcel 2113 ∀wral 3051 ∃wrex 3060 {crab 3399 Vcvv 3440 ∖ cdif 3898 ∪ cun 3899 ∩ cin 3900 {csn 4580 ⟨cop 4586 class class class wbr 5098 {copab 5160 ↦ cmpt 5179 ↾ cres 5626 suc csuc 6319 Fun wfun 6486 ‘cfv 6492 (class class class)co 7358 ∈ cmpo 7360 ωcom 7808 1^st c1st 7931 2^nd c2nd 7932 reccrdg 8340 ↑_m cmap 8763 ∈_𝑔cgoe 35527 ⊼_𝑔cgna 35528 ∀_𝑔cgol 35529 Sat csat 35530

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 1968 ax-7 2009 ax-8 2115 ax-9 2123 ax-10 2146 ax-11 2162 ax-12 2184 ax-ext 2708 ax-rep 5224 ax-sep 5241 ax-nul 5251 ax-pr 5377 ax-un 7680 ax-inf2 9550

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2539 df-eu 2569 df-clab 2715 df-cleq 2728 df-clel 2811 df-nfc 2885 df-ne 2933 df-ral 3052 df-rex 3061 df-reu 3351 df-rab 3400 df-v 3442 df-sbc 3741 df-csb 3850 df-dif 3904 df-un 3906 df-in 3908 df-ss 3918 df-pss 3921 df-nul 4286 df-if 4480 df-pw 4556 df-sn 4581 df-pr 4583 df-op 4587 df-uni 4864 df-iun 4948 df-br 5099 df-opab 5161 df-mpt 5180 df-tr 5206 df-id 5519 df-eprel 5524 df-po 5532 df-so 5533 df-fr 5577 df-we 5579 df-xp 5630 df-rel 5631 df-cnv 5632 df-co 5633 df-dm 5634 df-rn 5635 df-res 5636 df-ima 5637 df-pred 6259 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6494 df-fn 6495 df-f 6496 df-f1 6497 df-fo 6498 df-f1o 6499 df-fv 6500 df-ov 7361 df-oprab 7362 df-mpo 7363 df-om 7809 df-2nd 7934 df-frecs 8223 df-wrecs 8254 df-recs 8303 df-rdg 8341 df-sat 35537

This theorem is referenced by: satfsucom 35548 satfn 35549 satom 35550 satfvsucom 35551 satf0 35566

W3C validator