satfbrsuc - Mathbox for Mario Carneiro

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Theorem satfbrsuc 35948

Description: The binary relation of a satisfaction predicate as function over wff codes at a successor. (Contributed by AV, 13-Oct-2023.)

**Hypotheses**
Ref	Expression
satfbrsuc.s	⊢ 𝑆 = (𝑀 Sat 𝐸)
satfbrsuc.p	⊢ 𝑃 = (𝑆‘𝑁)

**Assertion**
Ref	Expression
satfbrsuc	⊢ (((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) ∧ 𝑁 ∈ ω ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌)) → (𝐴(𝑆‘suc 𝑁)𝐵 ↔ (𝐴𝑃𝐵 ∨ ∃𝑢 ∈ 𝑃 (∃𝑣 ∈ 𝑃 (𝐴 = ((1^st ‘𝑢)⊼_𝑔(1^st ‘𝑣)) ∧ 𝐵 = ((𝑀 ↑_m ω) ∖ ((2^nd ‘𝑢) ∩ (2^nd ‘𝑣)))) ∨ ∃𝑖 ∈ ω (𝐴 = ∀_𝑔𝑖(1^st ‘𝑢) ∧ 𝐵 = {𝑓 ∈ (𝑀 ↑_m ω) ∣ ∀𝑧 ∈ 𝑀 ({⟨𝑖, 𝑧⟩} ∪ (𝑓 ↾ (ω ∖ {𝑖}))) ∈ (2^nd ‘𝑢)})))))

Distinct variable groups: 𝐴,𝑖,𝑢,𝑣 𝐵,𝑖,𝑢,𝑣 𝑓,𝐸,𝑖,𝑢,𝑣,𝑧 𝑓,𝑀,𝑖,𝑢,𝑣,𝑧 𝑢,𝑁,𝑣 𝑣,𝑃 𝑢,𝑆,𝑣 𝑢,𝑉 𝑢,𝑊 Allowed substitution hints: 𝐴(𝑧, 𝑓) 𝐵(𝑧, 𝑓) 𝑃(𝑧, 𝑢, 𝑓, 𝑖) 𝑆(𝑧, 𝑓, 𝑖) 𝑁(𝑧, 𝑓, 𝑖) 𝑉(𝑧, 𝑣, 𝑓, 𝑖) 𝑊(𝑧, 𝑣, 𝑓, 𝑖) 𝑋(𝑧, 𝑣, 𝑢, 𝑓, 𝑖) 𝑌(𝑧, 𝑣, 𝑢, 𝑓, 𝑖)

Proof of Theorem satfbrsuc Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		satfbrsuc.s	. . . . . 6 ⊢ 𝑆 = (𝑀 Sat 𝐸)
2	1	satfvsuc 35943	. . . . 5 ⊢ ((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ∧ 𝑁 ∈ ω) → (𝑆‘suc 𝑁) = ((𝑆‘𝑁) ∪ {⟨𝑥, 𝑦⟩ ∣ ∃𝑢 ∈ (𝑆‘𝑁)(∃𝑣 ∈ (𝑆‘𝑁)(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 = {𝑓 ∈ (𝑀 ↑_m ω) ∣ ∀𝑧 ∈ 𝑀 ({⟨𝑖, 𝑧⟩} ∪ (𝑓 ↾ (ω ∖ {𝑖}))) ∈ (2^nd ‘𝑢)}))}))
3	2	3expa 1136	. . . 4 ⊢ (((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) ∧ 𝑁 ∈ ω) → (𝑆‘suc 𝑁) = ((𝑆‘𝑁) ∪ {⟨𝑥, 𝑦⟩ ∣ ∃𝑢 ∈ (𝑆‘𝑁)(∃𝑣 ∈ (𝑆‘𝑁)(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 = {𝑓 ∈ (𝑀 ↑_m ω) ∣ ∀𝑧 ∈ 𝑀 ({⟨𝑖, 𝑧⟩} ∪ (𝑓 ↾ (ω ∖ {𝑖}))) ∈ (2^nd ‘𝑢)}))}))
4	3	3adant3 1150	. . 3 ⊢ (((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) ∧ 𝑁 ∈ ω ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌)) → (𝑆‘suc 𝑁) = ((𝑆‘𝑁) ∪ {⟨𝑥, 𝑦⟩ ∣ ∃𝑢 ∈ (𝑆‘𝑁)(∃𝑣 ∈ (𝑆‘𝑁)(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 = {𝑓 ∈ (𝑀 ↑_m ω) ∣ ∀𝑧 ∈ 𝑀 ({⟨𝑖, 𝑧⟩} ∪ (𝑓 ↾ (ω ∖ {𝑖}))) ∈ (2^nd ‘𝑢)}))}))
5	4	breqd 5114	. 2 ⊢ (((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) ∧ 𝑁 ∈ ω ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌)) → (𝐴(𝑆‘suc 𝑁)𝐵 ↔ 𝐴((𝑆‘𝑁) ∪ {⟨𝑥, 𝑦⟩ ∣ ∃𝑢 ∈ (𝑆‘𝑁)(∃𝑣 ∈ (𝑆‘𝑁)(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 = {𝑓 ∈ (𝑀 ↑_m ω) ∣ ∀𝑧 ∈ 𝑀 ({⟨𝑖, 𝑧⟩} ∪ (𝑓 ↾ (ω ∖ {𝑖}))) ∈ (2^nd ‘𝑢)}))})𝐵))
6		brun 5156	. . . 4 ⊢ (𝐴((𝑆‘𝑁) ∪ {⟨𝑥, 𝑦⟩ ∣ ∃𝑢 ∈ (𝑆‘𝑁)(∃𝑣 ∈ (𝑆‘𝑁)(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 = {𝑓 ∈ (𝑀 ↑_m ω) ∣ ∀𝑧 ∈ 𝑀 ({⟨𝑖, 𝑧⟩} ∪ (𝑓 ↾ (ω ∖ {𝑖}))) ∈ (2^nd ‘𝑢)}))})𝐵 ↔ (𝐴(𝑆‘𝑁)𝐵 ∨ 𝐴{⟨𝑥, 𝑦⟩ ∣ ∃𝑢 ∈ (𝑆‘𝑁)(∃𝑣 ∈ (𝑆‘𝑁)(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 = {𝑓 ∈ (𝑀 ↑_m ω) ∣ ∀𝑧 ∈ 𝑀 ({⟨𝑖, 𝑧⟩} ∪ (𝑓 ↾ (ω ∖ {𝑖}))) ∈ (2^nd ‘𝑢)}))}𝐵))
7		satfbrsuc.p	. . . . . . . 8 ⊢ 𝑃 = (𝑆‘𝑁)
8	7	eqcomi 2769	. . . . . . 7 ⊢ (𝑆‘𝑁) = 𝑃
9	8	breqi 5109	. . . . . 6 ⊢ (𝐴(𝑆‘𝑁)𝐵 ↔ 𝐴𝑃𝐵)
10	9	a1i 11	. . . . 5 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌) → (𝐴(𝑆‘𝑁)𝐵 ↔ 𝐴𝑃𝐵))
11		eqeq1 2764	. . . . . . . . . 10 ⊢ (𝑥 = 𝐴 → (𝑥 = ((1^st ‘𝑢)⊼_𝑔(1^st ‘𝑣)) ↔ 𝐴 = ((1^st ‘𝑢)⊼_𝑔(1^st ‘𝑣))))
12		eqeq1 2764	. . . . . . . . . 10 ⊢ (𝑦 = 𝐵 → (𝑦 = ((𝑀 ↑_m ω) ∖ ((2^nd ‘𝑢) ∩ (2^nd ‘𝑣))) ↔ 𝐵 = ((𝑀 ↑_m ω) ∖ ((2^nd ‘𝑢) ∩ (2^nd ‘𝑣)))))
13	11, 12	bi2anan9 650	. . . . . . . . 9 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → ((𝑥 = ((1^st ‘𝑢)⊼_𝑔(1^st ‘𝑣)) ∧ 𝑦 = ((𝑀 ↑_m ω) ∖ ((2^nd ‘𝑢) ∩ (2^nd ‘𝑣)))) ↔ (𝐴 = ((1^st ‘𝑢)⊼_𝑔(1^st ‘𝑣)) ∧ 𝐵 = ((𝑀 ↑_m ω) ∖ ((2^nd ‘𝑢) ∩ (2^nd ‘𝑣))))))
14	13	rexbidv 3186	. . . . . . . 8 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (∃𝑣 ∈ 𝑃 (𝑥 = ((1^st ‘𝑢)⊼_𝑔(1^st ‘𝑣)) ∧ 𝑦 = ((𝑀 ↑_m ω) ∖ ((2^nd ‘𝑢) ∩ (2^nd ‘𝑣)))) ↔ ∃𝑣 ∈ 𝑃 (𝐴 = ((1^st ‘𝑢)⊼_𝑔(1^st ‘𝑣)) ∧ 𝐵 = ((𝑀 ↑_m ω) ∖ ((2^nd ‘𝑢) ∩ (2^nd ‘𝑣))))))
15		eqeq1 2764	. . . . . . . . . 10 ⊢ (𝑥 = 𝐴 → (𝑥 = ∀_𝑔𝑖(1^st ‘𝑢) ↔ 𝐴 = ∀_𝑔𝑖(1^st ‘𝑢)))
16		eqeq1 2764	. . . . . . . . . 10 ⊢ (𝑦 = 𝐵 → (𝑦 = {𝑓 ∈ (𝑀 ↑_m ω) ∣ ∀𝑧 ∈ 𝑀 ({⟨𝑖, 𝑧⟩} ∪ (𝑓 ↾ (ω ∖ {𝑖}))) ∈ (2^nd ‘𝑢)} ↔ 𝐵 = {𝑓 ∈ (𝑀 ↑_m ω) ∣ ∀𝑧 ∈ 𝑀 ({⟨𝑖, 𝑧⟩} ∪ (𝑓 ↾ (ω ∖ {𝑖}))) ∈ (2^nd ‘𝑢)}))
17	15, 16	bi2anan9 650	. . . . . . . . 9 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → ((𝑥 = ∀_𝑔𝑖(1^st ‘𝑢) ∧ 𝑦 = {𝑓 ∈ (𝑀 ↑_m ω) ∣ ∀𝑧 ∈ 𝑀 ({⟨𝑖, 𝑧⟩} ∪ (𝑓 ↾ (ω ∖ {𝑖}))) ∈ (2^nd ‘𝑢)}) ↔ (𝐴 = ∀_𝑔𝑖(1^st ‘𝑢) ∧ 𝐵 = {𝑓 ∈ (𝑀 ↑_m ω) ∣ ∀𝑧 ∈ 𝑀 ({⟨𝑖, 𝑧⟩} ∪ (𝑓 ↾ (ω ∖ {𝑖}))) ∈ (2^nd ‘𝑢)})))
18	17	rexbidv 3186	. . . . . . . 8 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (∃𝑖 ∈ ω (𝑥 = ∀_𝑔𝑖(1^st ‘𝑢) ∧ 𝑦 = {𝑓 ∈ (𝑀 ↑_m ω) ∣ ∀𝑧 ∈ 𝑀 ({⟨𝑖, 𝑧⟩} ∪ (𝑓 ↾ (ω ∖ {𝑖}))) ∈ (2^nd ‘𝑢)}) ↔ ∃𝑖 ∈ ω (𝐴 = ∀_𝑔𝑖(1^st ‘𝑢) ∧ 𝐵 = {𝑓 ∈ (𝑀 ↑_m ω) ∣ ∀𝑧 ∈ 𝑀 ({⟨𝑖, 𝑧⟩} ∪ (𝑓 ↾ (ω ∖ {𝑖}))) ∈ (2^nd ‘𝑢)})))
19	14, 18	orbi12d 932	. . . . . . 7 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → ((∃𝑣 ∈ 𝑃 (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 = {𝑓 ∈ (𝑀 ↑_m ω) ∣ ∀𝑧 ∈ 𝑀 ({⟨𝑖, 𝑧⟩} ∪ (𝑓 ↾ (ω ∖ {𝑖}))) ∈ (2^nd ‘𝑢)})) ↔ (∃𝑣 ∈ 𝑃 (𝐴 = ((1^st ‘𝑢)⊼_𝑔(1^st ‘𝑣)) ∧ 𝐵 = ((𝑀 ↑_m ω) ∖ ((2^nd ‘𝑢) ∩ (2^nd ‘𝑣)))) ∨ ∃𝑖 ∈ ω (𝐴 = ∀_𝑔𝑖(1^st ‘𝑢) ∧ 𝐵 = {𝑓 ∈ (𝑀 ↑_m ω) ∣ ∀𝑧 ∈ 𝑀 ({⟨𝑖, 𝑧⟩} ∪ (𝑓 ↾ (ω ∖ {𝑖}))) ∈ (2^nd ‘𝑢)}))))
20	19	rexbidv 3186	. . . . . 6 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (∃𝑢 ∈ 𝑃 (∃𝑣 ∈ 𝑃 (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 = {𝑓 ∈ (𝑀 ↑_m ω) ∣ ∀𝑧 ∈ 𝑀 ({⟨𝑖, 𝑧⟩} ∪ (𝑓 ↾ (ω ∖ {𝑖}))) ∈ (2^nd ‘𝑢)})) ↔ ∃𝑢 ∈ 𝑃 (∃𝑣 ∈ 𝑃 (𝐴 = ((1^st ‘𝑢)⊼_𝑔(1^st ‘𝑣)) ∧ 𝐵 = ((𝑀 ↑_m ω) ∖ ((2^nd ‘𝑢) ∩ (2^nd ‘𝑣)))) ∨ ∃𝑖 ∈ ω (𝐴 = ∀_𝑔𝑖(1^st ‘𝑢) ∧ 𝐵 = {𝑓 ∈ (𝑀 ↑_m ω) ∣ ∀𝑧 ∈ 𝑀 ({⟨𝑖, 𝑧⟩} ∪ (𝑓 ↾ (ω ∖ {𝑖}))) ∈ (2^nd ‘𝑢)}))))
21	8	rexeqi 3318	. . . . . . . . 9 ⊢ (∃𝑣 ∈ (𝑆‘𝑁)(𝑥 = ((1^st ‘𝑢)⊼_𝑔(1^st ‘𝑣)) ∧ 𝑦 = ((𝑀 ↑_m ω) ∖ ((2^nd ‘𝑢) ∩ (2^nd ‘𝑣)))) ↔ ∃𝑣 ∈ 𝑃 (𝑥 = ((1^st ‘𝑢)⊼_𝑔(1^st ‘𝑣)) ∧ 𝑦 = ((𝑀 ↑_m ω) ∖ ((2^nd ‘𝑢) ∩ (2^nd ‘𝑣)))))
22	21	orbi1i 927	. . . . . . . 8 ⊢ ((∃𝑣 ∈ (𝑆‘𝑁)(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 = {𝑓 ∈ (𝑀 ↑_m ω) ∣ ∀𝑧 ∈ 𝑀 ({⟨𝑖, 𝑧⟩} ∪ (𝑓 ↾ (ω ∖ {𝑖}))) ∈ (2^nd ‘𝑢)})) ↔ (∃𝑣 ∈ 𝑃 (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 = {𝑓 ∈ (𝑀 ↑_m ω) ∣ ∀𝑧 ∈ 𝑀 ({⟨𝑖, 𝑧⟩} ∪ (𝑓 ↾ (ω ∖ {𝑖}))) ∈ (2^nd ‘𝑢)})))
23	8, 22	rexeqbii 3333	. . . . . . 7 ⊢ (∃𝑢 ∈ (𝑆‘𝑁)(∃𝑣 ∈ (𝑆‘𝑁)(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 = {𝑓 ∈ (𝑀 ↑_m ω) ∣ ∀𝑧 ∈ 𝑀 ({⟨𝑖, 𝑧⟩} ∪ (𝑓 ↾ (ω ∖ {𝑖}))) ∈ (2^nd ‘𝑢)})) ↔ ∃𝑢 ∈ 𝑃 (∃𝑣 ∈ 𝑃 (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 = {𝑓 ∈ (𝑀 ↑_m ω) ∣ ∀𝑧 ∈ 𝑀 ({⟨𝑖, 𝑧⟩} ∪ (𝑓 ↾ (ω ∖ {𝑖}))) ∈ (2^nd ‘𝑢)})))
24	23	opabbii 5172	. . . . . 6 ⊢ {⟨𝑥, 𝑦⟩ ∣ ∃𝑢 ∈ (𝑆‘𝑁)(∃𝑣 ∈ (𝑆‘𝑁)(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 = {𝑓 ∈ (𝑀 ↑_m ω) ∣ ∀𝑧 ∈ 𝑀 ({⟨𝑖, 𝑧⟩} ∪ (𝑓 ↾ (ω ∖ {𝑖}))) ∈ (2^nd ‘𝑢)}))} = {⟨𝑥, 𝑦⟩ ∣ ∃𝑢 ∈ 𝑃 (∃𝑣 ∈ 𝑃 (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 = {𝑓 ∈ (𝑀 ↑_m ω) ∣ ∀𝑧 ∈ 𝑀 ({⟨𝑖, 𝑧⟩} ∪ (𝑓 ↾ (ω ∖ {𝑖}))) ∈ (2^nd ‘𝑢)}))}
25	20, 24	brabga 5512	. . . . 5 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌) → (𝐴{⟨𝑥, 𝑦⟩ ∣ ∃𝑢 ∈ (𝑆‘𝑁)(∃𝑣 ∈ (𝑆‘𝑁)(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 = {𝑓 ∈ (𝑀 ↑_m ω) ∣ ∀𝑧 ∈ 𝑀 ({⟨𝑖, 𝑧⟩} ∪ (𝑓 ↾ (ω ∖ {𝑖}))) ∈ (2^nd ‘𝑢)}))}𝐵 ↔ ∃𝑢 ∈ 𝑃 (∃𝑣 ∈ 𝑃 (𝐴 = ((1^st ‘𝑢)⊼_𝑔(1^st ‘𝑣)) ∧ 𝐵 = ((𝑀 ↑_m ω) ∖ ((2^nd ‘𝑢) ∩ (2^nd ‘𝑣)))) ∨ ∃𝑖 ∈ ω (𝐴 = ∀_𝑔𝑖(1^st ‘𝑢) ∧ 𝐵 = {𝑓 ∈ (𝑀 ↑_m ω) ∣ ∀𝑧 ∈ 𝑀 ({⟨𝑖, 𝑧⟩} ∪ (𝑓 ↾ (ω ∖ {𝑖}))) ∈ (2^nd ‘𝑢)}))))
26	10, 25	orbi12d 932	. . . 4 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌) → ((𝐴(𝑆‘𝑁)𝐵 ∨ 𝐴{⟨𝑥, 𝑦⟩ ∣ ∃𝑢 ∈ (𝑆‘𝑁)(∃𝑣 ∈ (𝑆‘𝑁)(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 = {𝑓 ∈ (𝑀 ↑_m ω) ∣ ∀𝑧 ∈ 𝑀 ({⟨𝑖, 𝑧⟩} ∪ (𝑓 ↾ (ω ∖ {𝑖}))) ∈ (2^nd ‘𝑢)}))}𝐵) ↔ (𝐴𝑃𝐵 ∨ ∃𝑢 ∈ 𝑃 (∃𝑣 ∈ 𝑃 (𝐴 = ((1^st ‘𝑢)⊼_𝑔(1^st ‘𝑣)) ∧ 𝐵 = ((𝑀 ↑_m ω) ∖ ((2^nd ‘𝑢) ∩ (2^nd ‘𝑣)))) ∨ ∃𝑖 ∈ ω (𝐴 = ∀_𝑔𝑖(1^st ‘𝑢) ∧ 𝐵 = {𝑓 ∈ (𝑀 ↑_m ω) ∣ ∀𝑧 ∈ 𝑀 ({⟨𝑖, 𝑧⟩} ∪ (𝑓 ↾ (ω ∖ {𝑖}))) ∈ (2^nd ‘𝑢)})))))
27	6, 26	bitrid 286	. . 3 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌) → (𝐴((𝑆‘𝑁) ∪ {⟨𝑥, 𝑦⟩ ∣ ∃𝑢 ∈ (𝑆‘𝑁)(∃𝑣 ∈ (𝑆‘𝑁)(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 = {𝑓 ∈ (𝑀 ↑_m ω) ∣ ∀𝑧 ∈ 𝑀 ({⟨𝑖, 𝑧⟩} ∪ (𝑓 ↾ (ω ∖ {𝑖}))) ∈ (2^nd ‘𝑢)}))})𝐵 ↔ (𝐴𝑃𝐵 ∨ ∃𝑢 ∈ 𝑃 (∃𝑣 ∈ 𝑃 (𝐴 = ((1^st ‘𝑢)⊼_𝑔(1^st ‘𝑣)) ∧ 𝐵 = ((𝑀 ↑_m ω) ∖ ((2^nd ‘𝑢) ∩ (2^nd ‘𝑣)))) ∨ ∃𝑖 ∈ ω (𝐴 = ∀_𝑔𝑖(1^st ‘𝑢) ∧ 𝐵 = {𝑓 ∈ (𝑀 ↑_m ω) ∣ ∀𝑧 ∈ 𝑀 ({⟨𝑖, 𝑧⟩} ∪ (𝑓 ↾ (ω ∖ {𝑖}))) ∈ (2^nd ‘𝑢)})))))
28	27	3ad2ant3 1153	. 2 ⊢ (((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) ∧ 𝑁 ∈ ω ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌)) → (𝐴((𝑆‘𝑁) ∪ {⟨𝑥, 𝑦⟩ ∣ ∃𝑢 ∈ (𝑆‘𝑁)(∃𝑣 ∈ (𝑆‘𝑁)(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 = {𝑓 ∈ (𝑀 ↑_m ω) ∣ ∀𝑧 ∈ 𝑀 ({⟨𝑖, 𝑧⟩} ∪ (𝑓 ↾ (ω ∖ {𝑖}))) ∈ (2^nd ‘𝑢)}))})𝐵 ↔ (𝐴𝑃𝐵 ∨ ∃𝑢 ∈ 𝑃 (∃𝑣 ∈ 𝑃 (𝐴 = ((1^st ‘𝑢)⊼_𝑔(1^st ‘𝑣)) ∧ 𝐵 = ((𝑀 ↑_m ω) ∖ ((2^nd ‘𝑢) ∩ (2^nd ‘𝑣)))) ∨ ∃𝑖 ∈ ω (𝐴 = ∀_𝑔𝑖(1^st ‘𝑢) ∧ 𝐵 = {𝑓 ∈ (𝑀 ↑_m ω) ∣ ∀𝑧 ∈ 𝑀 ({⟨𝑖, 𝑧⟩} ∪ (𝑓 ↾ (ω ∖ {𝑖}))) ∈ (2^nd ‘𝑢)})))))
29	5, 28	bitrd 282	1 ⊢ (((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) ∧ 𝑁 ∈ ω ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌)) → (𝐴(𝑆‘suc 𝑁)𝐵 ↔ (𝐴𝑃𝐵 ∨ ∃𝑢 ∈ 𝑃 (∃𝑣 ∈ 𝑃 (𝐴 = ((1^st ‘𝑢)⊼_𝑔(1^st ‘𝑣)) ∧ 𝐵 = ((𝑀 ↑_m ω) ∖ ((2^nd ‘𝑢) ∩ (2^nd ‘𝑣)))) ∨ ∃𝑖 ∈ ω (𝐴 = ∀_𝑔𝑖(1^st ‘𝑢) ∧ 𝐵 = {𝑓 ∈ (𝑀 ↑_m ω) ∣ ∀𝑧 ∈ 𝑀 ({⟨𝑖, 𝑧⟩} ∪ (𝑓 ↾ (ω ∖ {𝑖}))) ∈ (2^nd ‘𝑢)})))))

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ↔ wb 209 ∧ wa 401 ∨ wo 861 ∧ w3a 1103 = wceq 1570 ∈ wcel 2145 ∀wral 3076 ∃wrex 3086 {crab 3412 ∖ cdif 3896 ∪ cun 3897 ∩ cin 3898 {csn 4584 ⟨cop 4590 class class class wbr 5103 {copab 5167 ↾ cres 5657 suc csuc 6359 ‘cfv 6533 (class class class)co 7414 ωcom 7863 1^st c1st 7985 2^nd c2nd 7986 ↑_m cmap 8829 ⊼_𝑔cgna 35916 ∀_𝑔cgol 35917 Sat csat 35918

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2213 ax-ext 2732 ax-rep 5232 ax-sep 5251 ax-nul 5263 ax-pow 5330 ax-pr 5398 ax-un 7737 ax-inf2 9623

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2564 df-eu 2594 df-clab 2739 df-cleq 2752 df-clel 2835 df-nfc 2909 df-ne 2956 df-ral 3077 df-rex 3087 df-reu 3366 df-rab 3413 df-v 3452 df-sbc 3740 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-pss 3919 df-nul 4280 df-if 4483 df-pw 4559 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4868 df-iun 4953 df-br 5104 df-opab 5168 df-mpt 5187 df-tr 5213 df-id 5550 df-eprel 5555 df-po 5563 df-so 5564 df-fr 5608 df-we 5610 df-xp 5661 df-rel 5662 df-cnv 5663 df-co 5664 df-dm 5665 df-rn 5666 df-res 5667 df-ima 5668 df-pred 6299 df-ord 6360 df-on 6361 df-lim 6362 df-suc 6363 df-iota 6489 df-fun 6535 df-fn 6536 df-f 6537 df-f1 6538 df-fo 6539 df-f1o 6540 df-fv 6541 df-ov 7417 df-oprab 7418 df-mpo 7419 df-om 7864 df-2nd 7988 df-frecs 8281 df-wrecs 8312 df-recs 8361 df-rdg 8400 df-sat 35925

This theorem is used by: (None)

W3C validator