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Theorem satf 35569

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 35559	. . 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 7375	. . . . . . . . . . . . . . 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 4079	. . . . . . . . . . . . 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 2748	. . . . . . . . . . . 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 631	. . . . . . . . . . 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 3162	. . . . . . . . . 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 3298	. . . . . . . . . . . . . 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 3419	. . . . . . . . . . . . 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 2748	. . . . . . . . . . . 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 631	. . . . . . . . . . 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 3162	. . . . . . . . . 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 919	. . . . . . . . 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 3162	. . . . . . . 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 5166	. . . . . . 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 4122	. . . . . 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 5194	. . . . 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 5102	. . . . . . . . . . 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 3419	. . . . . . . . 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 2748	. . . . . . . 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 631	. . . . . . 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 3203	. . . . . 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 5166	. . . . 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 8354	. . . . 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 585	. . . 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 5945	. . 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 3463	. . 3 ⊢ (𝑀 ∈ 𝑉 → 𝑀 ∈ V)
32	31	adantr 480	. 2 ⊢ ((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) → 𝑀 ∈ V)
33		elex 3463	. . 3 ⊢ (𝐸 ∈ 𝑊 → 𝐸 ∈ V)
34	33	adantl 481	. 2 ⊢ ((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) → 𝐸 ∈ V)
35		rdgfun 8357	. . 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 9564	. . . . 5 ⊢ ω ∈ V
37	36	sucex 7761	. . . 4 ⊢ suc ω ∈ V
38	37	a1i 11	. . 3 ⊢ ((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) → suc ω ∈ V)
39		resfunexg 7171	. . 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 588	. 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 7520	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 848 = wceq 1542 ∈ wcel 2114 ∀wral 3052 ∃wrex 3062 {crab 3401 Vcvv 3442 ∖ cdif 3900 ∪ cun 3901 ∩ cin 3902 {csn 4582 ⟨cop 4588 class class class wbr 5100 {copab 5162 ↦ cmpt 5181 ↾ cres 5634 suc csuc 6327 Fun wfun 6494 ‘cfv 6500 (class class class)co 7368 ∈ cmpo 7370 ωcom 7818 1^st c1st 7941 2^nd c2nd 7942 reccrdg 8350 ↑_m cmap 8775 ∈_𝑔cgoe 35549 ⊼_𝑔cgna 35550 ∀_𝑔cgol 35551 Sat csat 35552

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 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-rep 5226 ax-sep 5243 ax-nul 5253 ax-pr 5379 ax-un 7690 ax-inf2 9562

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-ral 3053 df-rex 3063 df-reu 3353 df-rab 3402 df-v 3444 df-sbc 3743 df-csb 3852 df-dif 3906 df-un 3908 df-in 3910 df-ss 3920 df-pss 3923 df-nul 4288 df-if 4482 df-pw 4558 df-sn 4583 df-pr 4585 df-op 4589 df-uni 4866 df-iun 4950 df-br 5101 df-opab 5163 df-mpt 5182 df-tr 5208 df-id 5527 df-eprel 5532 df-po 5540 df-so 5541 df-fr 5585 df-we 5587 df-xp 5638 df-rel 5639 df-cnv 5640 df-co 5641 df-dm 5642 df-rn 5643 df-res 5644 df-ima 5645 df-pred 6267 df-ord 6328 df-on 6329 df-lim 6330 df-suc 6331 df-iota 6456 df-fun 6502 df-fn 6503 df-f 6504 df-f1 6505 df-fo 6506 df-f1o 6507 df-fv 6508 df-ov 7371 df-oprab 7372 df-mpo 7373 df-om 7819 df-2nd 7944 df-frecs 8233 df-wrecs 8264 df-recs 8313 df-rdg 8351 df-sat 35559

This theorem is referenced by: satfsucom 35570 satfn 35571 satom 35572 satfvsucom 35573 satf0 35588

W3C validator