satffunlem2 - Mathbox for Mario Carneiro

	Mathbox for Mario Carneiro		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > Mathboxes > satffunlem2		Structured version Visualization version GIF version

Theorem satffunlem2 35444

Description: Lemma 2 for satffun 35445: 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 7815	. . . . . . . . 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 6383	. . . . . . . 8 ⊢ 𝑁 ⊆ suc 𝑁
7	6	a1i 11	. . . . . . 7 ⊢ ((𝑁 ∈ ω ∧ (𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊)) → 𝑁 ⊆ suc 𝑁)
8		eqid 2731	. . . . . . . . 9 ⊢ (𝑀 Sat 𝐸) = (𝑀 Sat 𝐸)
9	8	satfsschain 35400	. . . . . . . 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 2731	. . . . . . . 8 ⊢ ((𝑀 ↑_m ω) ∖ ((2^nd ‘𝑢) ∩ (2^nd ‘𝑣))) = ((𝑀 ↑_m ω) ∖ ((2^nd ‘𝑢) ∩ (2^nd ‘𝑣)))
13		eqid 2731	. . . . . . . 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 35440	. . . . . . 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 35442	. . . 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 6522	. . . 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 35401	. . . . . 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 1425	. . . . 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 6498	. . . 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 206 ∧ wa 395 ∨ wo 847 = wceq 1541 ∈ wcel 2111 ∀wral 3047 ∃wrex 3056 {crab 3395 ∖ cdif 3894 ∪ cun 3895 ∩ cin 3896 ⊆ wss 3897 ∅c0 4278 {csn 4571 ⟨cop 4577 {copab 5148 dom cdm 5611 ↾ cres 5613 suc csuc 6303 Fun wfun 6470 ‘cfv 6476 (class class class)co 7341 ωcom 7791 1^st c1st 7914 2^nd c2nd 7915 ↑_m cmap 8745 ⊼_𝑔cgna 35370 ∀_𝑔cgol 35371 Sat csat 35372

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 2113 ax-9 2121 ax-10 2144 ax-11 2160 ax-12 2180 ax-ext 2703 ax-rep 5212 ax-sep 5229 ax-nul 5239 ax-pow 5298 ax-pr 5365 ax-un 7663 ax-inf2 9526

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 2535 df-eu 2564 df-clab 2710 df-cleq 2723 df-clel 2806 df-nfc 2881 df-ne 2929 df-nel 3033 df-ral 3048 df-rex 3057 df-reu 3347 df-rab 3396 df-v 3438 df-sbc 3737 df-csb 3846 df-dif 3900 df-un 3902 df-in 3904 df-ss 3914 df-pss 3917 df-nul 4279 df-if 4471 df-pw 4547 df-sn 4572 df-pr 4574 df-op 4578 df-uni 4855 df-int 4893 df-iun 4938 df-br 5087 df-opab 5149 df-mpt 5168 df-tr 5194 df-id 5506 df-eprel 5511 df-po 5519 df-so 5520 df-fr 5564 df-we 5566 df-xp 5617 df-rel 5618 df-cnv 5619 df-co 5620 df-dm 5621 df-rn 5622 df-res 5623 df-ima 5624 df-pred 6243 df-ord 6304 df-on 6305 df-lim 6306 df-suc 6307 df-iota 6432 df-fun 6478 df-fn 6479 df-f 6480 df-f1 6481 df-fo 6482 df-f1o 6483 df-fv 6484 df-ov 7344 df-oprab 7345 df-mpo 7346 df-om 7792 df-1st 7916 df-2nd 7917 df-frecs 8206 df-wrecs 8237 df-recs 8286 df-rdg 8324 df-1o 8380 df-2o 8381 df-map 8747 df-goel 35376 df-gona 35377 df-goal 35378 df-sat 35379 df-fmla 35381

This theorem is referenced by: satffun 35445

W3C validator