satffunlem2 - Mathbox for Mario Carneiro

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Theorem satffunlem2 34941

Description: Lemma 2 for satffun 34942: induction step. (Contributed by AV, 28-Oct-2023.)

**Assertion**
Ref	Expression
satffunlem2	⊢ ((𝑁 ∈ ω ∧ (𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊)) → (Fun ((𝑀 Sat 𝐸)‘suc 𝑁) → Fun ((𝑀 Sat 𝐸)‘suc suc 𝑁)))

Proof of Theorem satffunlem2 Dummy variables 𝑓 𝑖 𝑗 𝑢 𝑣 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpr 484	. . . 4 ⊢ (((𝑁 ∈ ω ∧ (𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊)) ∧ Fun ((𝑀 Sat 𝐸)‘suc 𝑁)) → Fun ((𝑀 Sat 𝐸)‘suc 𝑁))
2		simpr 484	. . . . . . 7 ⊢ ((𝑁 ∈ ω ∧ (𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊)) → (𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊))
3		peano2 7888	. . . . . . . . 9 ⊢ (𝑁 ∈ ω → suc 𝑁 ∈ ω)
4	3	ancri 549	. . . . . . . 8 ⊢ (𝑁 ∈ ω → (suc 𝑁 ∈ ω ∧ 𝑁 ∈ ω))
5	4	adantr 480	. . . . . . 7 ⊢ ((𝑁 ∈ ω ∧ (𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊)) → (suc 𝑁 ∈ ω ∧ 𝑁 ∈ ω))
6		sssucid 6443	. . . . . . . 8 ⊢ 𝑁 ⊆ suc 𝑁
7	6	a1i 11	. . . . . . 7 ⊢ ((𝑁 ∈ ω ∧ (𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊)) → 𝑁 ⊆ suc 𝑁)
8		eqid 2727	. . . . . . . . 9 ⊢ (𝑀 Sat 𝐸) = (𝑀 Sat 𝐸)
9	8	satfsschain 34897	. . . . . . . 8 ⊢ (((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) ∧ (suc 𝑁 ∈ ω ∧ 𝑁 ∈ ω)) → (𝑁 ⊆ suc 𝑁 → ((𝑀 Sat 𝐸)‘𝑁) ⊆ ((𝑀 Sat 𝐸)‘suc 𝑁)))
10	9	imp 406	. . . . . . 7 ⊢ ((((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) ∧ (suc 𝑁 ∈ ω ∧ 𝑁 ∈ ω)) ∧ 𝑁 ⊆ suc 𝑁) → ((𝑀 Sat 𝐸)‘𝑁) ⊆ ((𝑀 Sat 𝐸)‘suc 𝑁))
11	2, 5, 7, 10	syl21anc 837	. . . . . 6 ⊢ ((𝑁 ∈ ω ∧ (𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊)) → ((𝑀 Sat 𝐸)‘𝑁) ⊆ ((𝑀 Sat 𝐸)‘suc 𝑁))
12		eqid 2727	. . . . . . . 8 ⊢ ((𝑀 ↑_m ω) ∖ ((2^nd ‘𝑢) ∩ (2^nd ‘𝑣))) = ((𝑀 ↑_m ω) ∖ ((2^nd ‘𝑢) ∩ (2^nd ‘𝑣)))
13		eqid 2727	. . . . . . . 8 ⊢ {𝑓 ∈ (𝑀 ↑_m ω) ∣ ∀𝑗 ∈ 𝑀 ({⟨𝑖, 𝑗⟩} ∪ (𝑓 ↾ (ω ∖ {𝑖}))) ∈ (2^nd ‘𝑢)} = {𝑓 ∈ (𝑀 ↑_m ω) ∣ ∀𝑗 ∈ 𝑀 ({⟨𝑖, 𝑗⟩} ∪ (𝑓 ↾ (ω ∖ {𝑖}))) ∈ (2^nd ‘𝑢)}
14	8, 12, 13	satffunlem2lem1 34937	. . . . . . 7 ⊢ ((Fun ((𝑀 Sat 𝐸)‘suc 𝑁) ∧ ((𝑀 Sat 𝐸)‘𝑁) ⊆ ((𝑀 Sat 𝐸)‘suc 𝑁)) → Fun {⟨𝑥, 𝑦⟩ ∣ (∃𝑢 ∈ (((𝑀 Sat 𝐸)‘suc 𝑁) ∖ ((𝑀 Sat 𝐸)‘𝑁))(∃𝑣 ∈ ((𝑀 Sat 𝐸)‘suc 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) ∣ ∀𝑗 ∈ 𝑀 ({⟨𝑖, 𝑗⟩} ∪ (𝑓 ↾ (ω ∖ {𝑖}))) ∈ (2^nd ‘𝑢)})) ∨ ∃𝑢 ∈ ((𝑀 Sat 𝐸)‘𝑁)∃𝑣 ∈ (((𝑀 Sat 𝐸)‘suc 𝑁) ∖ ((𝑀 Sat 𝐸)‘𝑁))(𝑥 = ((1^st ‘𝑢)⊼_𝑔(1^st ‘𝑣)) ∧ 𝑦 = ((𝑀 ↑_m ω) ∖ ((2^nd ‘𝑢) ∩ (2^nd ‘𝑣)))))})
15	14	expcom 413	. . . . . 6 ⊢ (((𝑀 Sat 𝐸)‘𝑁) ⊆ ((𝑀 Sat 𝐸)‘suc 𝑁) → (Fun ((𝑀 Sat 𝐸)‘suc 𝑁) → Fun {⟨𝑥, 𝑦⟩ ∣ (∃𝑢 ∈ (((𝑀 Sat 𝐸)‘suc 𝑁) ∖ ((𝑀 Sat 𝐸)‘𝑁))(∃𝑣 ∈ ((𝑀 Sat 𝐸)‘suc 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) ∣ ∀𝑗 ∈ 𝑀 ({⟨𝑖, 𝑗⟩} ∪ (𝑓 ↾ (ω ∖ {𝑖}))) ∈ (2^nd ‘𝑢)})) ∨ ∃𝑢 ∈ ((𝑀 Sat 𝐸)‘𝑁)∃𝑣 ∈ (((𝑀 Sat 𝐸)‘suc 𝑁) ∖ ((𝑀 Sat 𝐸)‘𝑁))(𝑥 = ((1^st ‘𝑢)⊼_𝑔(1^st ‘𝑣)) ∧ 𝑦 = ((𝑀 ↑_m ω) ∖ ((2^nd ‘𝑢) ∩ (2^nd ‘𝑣)))))}))
16	11, 15	syl 17	. . . . 5 ⊢ ((𝑁 ∈ ω ∧ (𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊)) → (Fun ((𝑀 Sat 𝐸)‘suc 𝑁) → Fun {⟨𝑥, 𝑦⟩ ∣ (∃𝑢 ∈ (((𝑀 Sat 𝐸)‘suc 𝑁) ∖ ((𝑀 Sat 𝐸)‘𝑁))(∃𝑣 ∈ ((𝑀 Sat 𝐸)‘suc 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) ∣ ∀𝑗 ∈ 𝑀 ({⟨𝑖, 𝑗⟩} ∪ (𝑓 ↾ (ω ∖ {𝑖}))) ∈ (2^nd ‘𝑢)})) ∨ ∃𝑢 ∈ ((𝑀 Sat 𝐸)‘𝑁)∃𝑣 ∈ (((𝑀 Sat 𝐸)‘suc 𝑁) ∖ ((𝑀 Sat 𝐸)‘𝑁))(𝑥 = ((1^st ‘𝑢)⊼_𝑔(1^st ‘𝑣)) ∧ 𝑦 = ((𝑀 ↑_m ω) ∖ ((2^nd ‘𝑢) ∩ (2^nd ‘𝑣)))))}))
17	16	imp 406	. . . 4 ⊢ (((𝑁 ∈ ω ∧ (𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊)) ∧ Fun ((𝑀 Sat 𝐸)‘suc 𝑁)) → Fun {⟨𝑥, 𝑦⟩ ∣ (∃𝑢 ∈ (((𝑀 Sat 𝐸)‘suc 𝑁) ∖ ((𝑀 Sat 𝐸)‘𝑁))(∃𝑣 ∈ ((𝑀 Sat 𝐸)‘suc 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) ∣ ∀𝑗 ∈ 𝑀 ({⟨𝑖, 𝑗⟩} ∪ (𝑓 ↾ (ω ∖ {𝑖}))) ∈ (2^nd ‘𝑢)})) ∨ ∃𝑢 ∈ ((𝑀 Sat 𝐸)‘𝑁)∃𝑣 ∈ (((𝑀 Sat 𝐸)‘suc 𝑁) ∖ ((𝑀 Sat 𝐸)‘𝑁))(𝑥 = ((1^st ‘𝑢)⊼_𝑔(1^st ‘𝑣)) ∧ 𝑦 = ((𝑀 ↑_m ω) ∖ ((2^nd ‘𝑢) ∩ (2^nd ‘𝑣)))))})
18	8, 12, 13	satffunlem2lem2 34939	. . . 4 ⊢ (((𝑁 ∈ ω ∧ (𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊)) ∧ Fun ((𝑀 Sat 𝐸)‘suc 𝑁)) → (dom ((𝑀 Sat 𝐸)‘suc 𝑁) ∩ dom {⟨𝑥, 𝑦⟩ ∣ (∃𝑢 ∈ (((𝑀 Sat 𝐸)‘suc 𝑁) ∖ ((𝑀 Sat 𝐸)‘𝑁))(∃𝑣 ∈ ((𝑀 Sat 𝐸)‘suc 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) ∣ ∀𝑗 ∈ 𝑀 ({⟨𝑖, 𝑗⟩} ∪ (𝑓 ↾ (ω ∖ {𝑖}))) ∈ (2^nd ‘𝑢)})) ∨ ∃𝑢 ∈ ((𝑀 Sat 𝐸)‘𝑁)∃𝑣 ∈ (((𝑀 Sat 𝐸)‘suc 𝑁) ∖ ((𝑀 Sat 𝐸)‘𝑁))(𝑥 = ((1^st ‘𝑢)⊼_𝑔(1^st ‘𝑣)) ∧ 𝑦 = ((𝑀 ↑_m ω) ∖ ((2^nd ‘𝑢) ∩ (2^nd ‘𝑣)))))}) = ∅)
19		funun 6593	. . . 4 ⊢ (((Fun ((𝑀 Sat 𝐸)‘suc 𝑁) ∧ Fun {⟨𝑥, 𝑦⟩ ∣ (∃𝑢 ∈ (((𝑀 Sat 𝐸)‘suc 𝑁) ∖ ((𝑀 Sat 𝐸)‘𝑁))(∃𝑣 ∈ ((𝑀 Sat 𝐸)‘suc 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) ∣ ∀𝑗 ∈ 𝑀 ({⟨𝑖, 𝑗⟩} ∪ (𝑓 ↾ (ω ∖ {𝑖}))) ∈ (2^nd ‘𝑢)})) ∨ ∃𝑢 ∈ ((𝑀 Sat 𝐸)‘𝑁)∃𝑣 ∈ (((𝑀 Sat 𝐸)‘suc 𝑁) ∖ ((𝑀 Sat 𝐸)‘𝑁))(𝑥 = ((1^st ‘𝑢)⊼_𝑔(1^st ‘𝑣)) ∧ 𝑦 = ((𝑀 ↑_m ω) ∖ ((2^nd ‘𝑢) ∩ (2^nd ‘𝑣)))))}) ∧ (dom ((𝑀 Sat 𝐸)‘suc 𝑁) ∩ dom {⟨𝑥, 𝑦⟩ ∣ (∃𝑢 ∈ (((𝑀 Sat 𝐸)‘suc 𝑁) ∖ ((𝑀 Sat 𝐸)‘𝑁))(∃𝑣 ∈ ((𝑀 Sat 𝐸)‘suc 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) ∣ ∀𝑗 ∈ 𝑀 ({⟨𝑖, 𝑗⟩} ∪ (𝑓 ↾ (ω ∖ {𝑖}))) ∈ (2^nd ‘𝑢)})) ∨ ∃𝑢 ∈ ((𝑀 Sat 𝐸)‘𝑁)∃𝑣 ∈ (((𝑀 Sat 𝐸)‘suc 𝑁) ∖ ((𝑀 Sat 𝐸)‘𝑁))(𝑥 = ((1^st ‘𝑢)⊼_𝑔(1^st ‘𝑣)) ∧ 𝑦 = ((𝑀 ↑_m ω) ∖ ((2^nd ‘𝑢) ∩ (2^nd ‘𝑣)))))}) = ∅) → Fun (((𝑀 Sat 𝐸)‘suc 𝑁) ∪ {⟨𝑥, 𝑦⟩ ∣ (∃𝑢 ∈ (((𝑀 Sat 𝐸)‘suc 𝑁) ∖ ((𝑀 Sat 𝐸)‘𝑁))(∃𝑣 ∈ ((𝑀 Sat 𝐸)‘suc 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) ∣ ∀𝑗 ∈ 𝑀 ({⟨𝑖, 𝑗⟩} ∪ (𝑓 ↾ (ω ∖ {𝑖}))) ∈ (2^nd ‘𝑢)})) ∨ ∃𝑢 ∈ ((𝑀 Sat 𝐸)‘𝑁)∃𝑣 ∈ (((𝑀 Sat 𝐸)‘suc 𝑁) ∖ ((𝑀 Sat 𝐸)‘𝑁))(𝑥 = ((1^st ‘𝑢)⊼_𝑔(1^st ‘𝑣)) ∧ 𝑦 = ((𝑀 ↑_m ω) ∖ ((2^nd ‘𝑢) ∩ (2^nd ‘𝑣)))))}))
20	1, 17, 18, 19	syl21anc 837	. . 3 ⊢ (((𝑁 ∈ ω ∧ (𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊)) ∧ Fun ((𝑀 Sat 𝐸)‘suc 𝑁)) → Fun (((𝑀 Sat 𝐸)‘suc 𝑁) ∪ {⟨𝑥, 𝑦⟩ ∣ (∃𝑢 ∈ (((𝑀 Sat 𝐸)‘suc 𝑁) ∖ ((𝑀 Sat 𝐸)‘𝑁))(∃𝑣 ∈ ((𝑀 Sat 𝐸)‘suc 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) ∣ ∀𝑗 ∈ 𝑀 ({⟨𝑖, 𝑗⟩} ∪ (𝑓 ↾ (ω ∖ {𝑖}))) ∈ (2^nd ‘𝑢)})) ∨ ∃𝑢 ∈ ((𝑀 Sat 𝐸)‘𝑁)∃𝑣 ∈ (((𝑀 Sat 𝐸)‘suc 𝑁) ∖ ((𝑀 Sat 𝐸)‘𝑁))(𝑥 = ((1^st ‘𝑢)⊼_𝑔(1^st ‘𝑣)) ∧ 𝑦 = ((𝑀 ↑_m ω) ∖ ((2^nd ‘𝑢) ∩ (2^nd ‘𝑣)))))}))
21		simpl 482	. . . . . 6 ⊢ ((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) → 𝑀 ∈ 𝑉)
22		simpr 484	. . . . . 6 ⊢ ((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) → 𝐸 ∈ 𝑊)
23		simpl 482	. . . . . 6 ⊢ ((𝑁 ∈ ω ∧ (𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊)) → 𝑁 ∈ ω)
24	8, 12, 13	satfvsucsuc 34898	. . . . . 6 ⊢ ((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ∧ 𝑁 ∈ ω) → ((𝑀 Sat 𝐸)‘suc suc 𝑁) = (((𝑀 Sat 𝐸)‘suc 𝑁) ∪ {⟨𝑥, 𝑦⟩ ∣ (∃𝑢 ∈ (((𝑀 Sat 𝐸)‘suc 𝑁) ∖ ((𝑀 Sat 𝐸)‘𝑁))(∃𝑣 ∈ ((𝑀 Sat 𝐸)‘suc 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) ∣ ∀𝑗 ∈ 𝑀 ({⟨𝑖, 𝑗⟩} ∪ (𝑓 ↾ (ω ∖ {𝑖}))) ∈ (2^nd ‘𝑢)})) ∨ ∃𝑢 ∈ ((𝑀 Sat 𝐸)‘𝑁)∃𝑣 ∈ (((𝑀 Sat 𝐸)‘suc 𝑁) ∖ ((𝑀 Sat 𝐸)‘𝑁))(𝑥 = ((1^st ‘𝑢)⊼_𝑔(1^st ‘𝑣)) ∧ 𝑦 = ((𝑀 ↑_m ω) ∖ ((2^nd ‘𝑢) ∩ (2^nd ‘𝑣)))))}))
25	21, 22, 23, 24	syl2an23an 1421	. . . . 5 ⊢ ((𝑁 ∈ ω ∧ (𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊)) → ((𝑀 Sat 𝐸)‘suc suc 𝑁) = (((𝑀 Sat 𝐸)‘suc 𝑁) ∪ {⟨𝑥, 𝑦⟩ ∣ (∃𝑢 ∈ (((𝑀 Sat 𝐸)‘suc 𝑁) ∖ ((𝑀 Sat 𝐸)‘𝑁))(∃𝑣 ∈ ((𝑀 Sat 𝐸)‘suc 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) ∣ ∀𝑗 ∈ 𝑀 ({⟨𝑖, 𝑗⟩} ∪ (𝑓 ↾ (ω ∖ {𝑖}))) ∈ (2^nd ‘𝑢)})) ∨ ∃𝑢 ∈ ((𝑀 Sat 𝐸)‘𝑁)∃𝑣 ∈ (((𝑀 Sat 𝐸)‘suc 𝑁) ∖ ((𝑀 Sat 𝐸)‘𝑁))(𝑥 = ((1^st ‘𝑢)⊼_𝑔(1^st ‘𝑣)) ∧ 𝑦 = ((𝑀 ↑_m ω) ∖ ((2^nd ‘𝑢) ∩ (2^nd ‘𝑣)))))}))
26	25	funeqd 6569	. . . 4 ⊢ ((𝑁 ∈ ω ∧ (𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊)) → (Fun ((𝑀 Sat 𝐸)‘suc suc 𝑁) ↔ Fun (((𝑀 Sat 𝐸)‘suc 𝑁) ∪ {⟨𝑥, 𝑦⟩ ∣ (∃𝑢 ∈ (((𝑀 Sat 𝐸)‘suc 𝑁) ∖ ((𝑀 Sat 𝐸)‘𝑁))(∃𝑣 ∈ ((𝑀 Sat 𝐸)‘suc 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) ∣ ∀𝑗 ∈ 𝑀 ({⟨𝑖, 𝑗⟩} ∪ (𝑓 ↾ (ω ∖ {𝑖}))) ∈ (2^nd ‘𝑢)})) ∨ ∃𝑢 ∈ ((𝑀 Sat 𝐸)‘𝑁)∃𝑣 ∈ (((𝑀 Sat 𝐸)‘suc 𝑁) ∖ ((𝑀 Sat 𝐸)‘𝑁))(𝑥 = ((1^st ‘𝑢)⊼_𝑔(1^st ‘𝑣)) ∧ 𝑦 = ((𝑀 ↑_m ω) ∖ ((2^nd ‘𝑢) ∩ (2^nd ‘𝑣)))))})))
27	26	adantr 480	. . 3 ⊢ (((𝑁 ∈ ω ∧ (𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊)) ∧ Fun ((𝑀 Sat 𝐸)‘suc 𝑁)) → (Fun ((𝑀 Sat 𝐸)‘suc suc 𝑁) ↔ Fun (((𝑀 Sat 𝐸)‘suc 𝑁) ∪ {⟨𝑥, 𝑦⟩ ∣ (∃𝑢 ∈ (((𝑀 Sat 𝐸)‘suc 𝑁) ∖ ((𝑀 Sat 𝐸)‘𝑁))(∃𝑣 ∈ ((𝑀 Sat 𝐸)‘suc 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) ∣ ∀𝑗 ∈ 𝑀 ({⟨𝑖, 𝑗⟩} ∪ (𝑓 ↾ (ω ∖ {𝑖}))) ∈ (2^nd ‘𝑢)})) ∨ ∃𝑢 ∈ ((𝑀 Sat 𝐸)‘𝑁)∃𝑣 ∈ (((𝑀 Sat 𝐸)‘suc 𝑁) ∖ ((𝑀 Sat 𝐸)‘𝑁))(𝑥 = ((1^st ‘𝑢)⊼_𝑔(1^st ‘𝑣)) ∧ 𝑦 = ((𝑀 ↑_m ω) ∖ ((2^nd ‘𝑢) ∩ (2^nd ‘𝑣)))))})))
28	20, 27	mpbird 257	. 2 ⊢ (((𝑁 ∈ ω ∧ (𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊)) ∧ Fun ((𝑀 Sat 𝐸)‘suc 𝑁)) → Fun ((𝑀 Sat 𝐸)‘suc suc 𝑁))
29	28	ex 412	1 ⊢ ((𝑁 ∈ ω ∧ (𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊)) → (Fun ((𝑀 Sat 𝐸)‘suc 𝑁) → Fun ((𝑀 Sat 𝐸)‘suc suc 𝑁)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 ∨ wo 846 = wceq 1534 ∈ wcel 2099 ∀wral 3056 ∃wrex 3065 {crab 3427 ∖ cdif 3941 ∪ cun 3942 ∩ cin 3943 ⊆ wss 3944 ∅c0 4318 {csn 4624 ⟨cop 4630 {copab 5204 dom cdm 5672 ↾ cres 5674 suc csuc 6365 Fun wfun 6536 ‘cfv 6542 (class class class)co 7414 ωcom 7862 1^st c1st 7983 2^nd c2nd 7984 ↑_m cmap 8834 ⊼_𝑔cgna 34867 ∀_𝑔cgol 34868 Sat csat 34869

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2164 ax-ext 2698 ax-rep 5279 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7732 ax-inf2 9650

This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2936 df-nel 3042 df-ral 3057 df-rex 3066 df-reu 3372 df-rab 3428 df-v 3471 df-sbc 3775 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3963 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-int 4945 df-iun 4993 df-br 5143 df-opab 5205 df-mpt 5226 df-tr 5260 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6299 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-ov 7417 df-oprab 7418 df-mpo 7419 df-om 7863 df-1st 7985 df-2nd 7986 df-frecs 8278 df-wrecs 8309 df-recs 8383 df-rdg 8422 df-1o 8478 df-2o 8479 df-map 8836 df-goel 34873 df-gona 34874 df-goal 34875 df-sat 34876 df-fmla 34878

This theorem is referenced by: satffun 34942

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