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Theorem satffunlem2 35414
Description: Lemma 2 for satffun 35415: induction step. (Contributed by AV, 28-Oct-2023.)
Assertion
Ref Expression
satffunlem2 ((𝑁 ∈ ω ∧ (𝑀𝑉𝐸𝑊)) → (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.
StepHypRef Expression
1 simpr 484 . . . 4 (((𝑁 ∈ ω ∧ (𝑀𝑉𝐸𝑊)) ∧ Fun ((𝑀 Sat 𝐸)‘suc 𝑁)) → Fun ((𝑀 Sat 𝐸)‘suc 𝑁))
2 simpr 484 . . . . . . 7 ((𝑁 ∈ ω ∧ (𝑀𝑉𝐸𝑊)) → (𝑀𝑉𝐸𝑊))
3 peano2 7913 . . . . . . . . 9 (𝑁 ∈ ω → suc 𝑁 ∈ ω)
43ancri 549 . . . . . . . 8 (𝑁 ∈ ω → (suc 𝑁 ∈ ω ∧ 𝑁 ∈ ω))
54adantr 480 . . . . . . 7 ((𝑁 ∈ ω ∧ (𝑀𝑉𝐸𝑊)) → (suc 𝑁 ∈ ω ∧ 𝑁 ∈ ω))
6 sssucid 6463 . . . . . . . 8 𝑁 ⊆ suc 𝑁
76a1i 11 . . . . . . 7 ((𝑁 ∈ ω ∧ (𝑀𝑉𝐸𝑊)) → 𝑁 ⊆ suc 𝑁)
8 eqid 2736 . . . . . . . . 9 (𝑀 Sat 𝐸) = (𝑀 Sat 𝐸)
98satfsschain 35370 . . . . . . . 8 (((𝑀𝑉𝐸𝑊) ∧ (suc 𝑁 ∈ ω ∧ 𝑁 ∈ ω)) → (𝑁 ⊆ suc 𝑁 → ((𝑀 Sat 𝐸)‘𝑁) ⊆ ((𝑀 Sat 𝐸)‘suc 𝑁)))
109imp 406 . . . . . . 7 ((((𝑀𝑉𝐸𝑊) ∧ (suc 𝑁 ∈ ω ∧ 𝑁 ∈ ω)) ∧ 𝑁 ⊆ suc 𝑁) → ((𝑀 Sat 𝐸)‘𝑁) ⊆ ((𝑀 Sat 𝐸)‘suc 𝑁))
112, 5, 7, 10syl21anc 837 . . . . . 6 ((𝑁 ∈ ω ∧ (𝑀𝑉𝐸𝑊)) → ((𝑀 Sat 𝐸)‘𝑁) ⊆ ((𝑀 Sat 𝐸)‘suc 𝑁))
12 eqid 2736 . . . . . . . 8 ((𝑀m ω) ∖ ((2nd𝑢) ∩ (2nd𝑣))) = ((𝑀m ω) ∖ ((2nd𝑢) ∩ (2nd𝑣)))
13 eqid 2736 . . . . . . . 8 {𝑓 ∈ (𝑀m ω) ∣ ∀𝑗𝑀 ({⟨𝑖, 𝑗⟩} ∪ (𝑓 ↾ (ω ∖ {𝑖}))) ∈ (2nd𝑢)} = {𝑓 ∈ (𝑀m ω) ∣ ∀𝑗𝑀 ({⟨𝑖, 𝑗⟩} ∪ (𝑓 ↾ (ω ∖ {𝑖}))) ∈ (2nd𝑢)}
148, 12, 13satffunlem2lem1 35410 . . . . . . 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 = ((1st𝑢)⊼𝑔(1st𝑣)) ∧ 𝑦 = ((𝑀m ω) ∖ ((2nd𝑢) ∩ (2nd𝑣)))) ∨ ∃𝑖 ∈ ω (𝑥 = ∀𝑔𝑖(1st𝑢) ∧ 𝑦 = {𝑓 ∈ (𝑀m ω) ∣ ∀𝑗𝑀 ({⟨𝑖, 𝑗⟩} ∪ (𝑓 ↾ (ω ∖ {𝑖}))) ∈ (2nd𝑢)})) ∨ ∃𝑢 ∈ ((𝑀 Sat 𝐸)‘𝑁)∃𝑣 ∈ (((𝑀 Sat 𝐸)‘suc 𝑁) ∖ ((𝑀 Sat 𝐸)‘𝑁))(𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ∧ 𝑦 = ((𝑀m ω) ∖ ((2nd𝑢) ∩ (2nd𝑣)))))})
1514expcom 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 = ((1st𝑢)⊼𝑔(1st𝑣)) ∧ 𝑦 = ((𝑀m ω) ∖ ((2nd𝑢) ∩ (2nd𝑣)))) ∨ ∃𝑖 ∈ ω (𝑥 = ∀𝑔𝑖(1st𝑢) ∧ 𝑦 = {𝑓 ∈ (𝑀m ω) ∣ ∀𝑗𝑀 ({⟨𝑖, 𝑗⟩} ∪ (𝑓 ↾ (ω ∖ {𝑖}))) ∈ (2nd𝑢)})) ∨ ∃𝑢 ∈ ((𝑀 Sat 𝐸)‘𝑁)∃𝑣 ∈ (((𝑀 Sat 𝐸)‘suc 𝑁) ∖ ((𝑀 Sat 𝐸)‘𝑁))(𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ∧ 𝑦 = ((𝑀m ω) ∖ ((2nd𝑢) ∩ (2nd𝑣)))))}))
1611, 15syl 17 . . . . 5 ((𝑁 ∈ ω ∧ (𝑀𝑉𝐸𝑊)) → (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 = ((1st𝑢)⊼𝑔(1st𝑣)) ∧ 𝑦 = ((𝑀m ω) ∖ ((2nd𝑢) ∩ (2nd𝑣)))) ∨ ∃𝑖 ∈ ω (𝑥 = ∀𝑔𝑖(1st𝑢) ∧ 𝑦 = {𝑓 ∈ (𝑀m ω) ∣ ∀𝑗𝑀 ({⟨𝑖, 𝑗⟩} ∪ (𝑓 ↾ (ω ∖ {𝑖}))) ∈ (2nd𝑢)})) ∨ ∃𝑢 ∈ ((𝑀 Sat 𝐸)‘𝑁)∃𝑣 ∈ (((𝑀 Sat 𝐸)‘suc 𝑁) ∖ ((𝑀 Sat 𝐸)‘𝑁))(𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ∧ 𝑦 = ((𝑀m ω) ∖ ((2nd𝑢) ∩ (2nd𝑣)))))}))
1716imp 406 . . . 4 (((𝑁 ∈ ω ∧ (𝑀𝑉𝐸𝑊)) ∧ 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 = ((1st𝑢)⊼𝑔(1st𝑣)) ∧ 𝑦 = ((𝑀m ω) ∖ ((2nd𝑢) ∩ (2nd𝑣)))) ∨ ∃𝑖 ∈ ω (𝑥 = ∀𝑔𝑖(1st𝑢) ∧ 𝑦 = {𝑓 ∈ (𝑀m ω) ∣ ∀𝑗𝑀 ({⟨𝑖, 𝑗⟩} ∪ (𝑓 ↾ (ω ∖ {𝑖}))) ∈ (2nd𝑢)})) ∨ ∃𝑢 ∈ ((𝑀 Sat 𝐸)‘𝑁)∃𝑣 ∈ (((𝑀 Sat 𝐸)‘suc 𝑁) ∖ ((𝑀 Sat 𝐸)‘𝑁))(𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ∧ 𝑦 = ((𝑀m ω) ∖ ((2nd𝑢) ∩ (2nd𝑣)))))})
188, 12, 13satffunlem2lem2 35412 . . . 4 (((𝑁 ∈ ω ∧ (𝑀𝑉𝐸𝑊)) ∧ 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 = ((1st𝑢)⊼𝑔(1st𝑣)) ∧ 𝑦 = ((𝑀m ω) ∖ ((2nd𝑢) ∩ (2nd𝑣)))) ∨ ∃𝑖 ∈ ω (𝑥 = ∀𝑔𝑖(1st𝑢) ∧ 𝑦 = {𝑓 ∈ (𝑀m ω) ∣ ∀𝑗𝑀 ({⟨𝑖, 𝑗⟩} ∪ (𝑓 ↾ (ω ∖ {𝑖}))) ∈ (2nd𝑢)})) ∨ ∃𝑢 ∈ ((𝑀 Sat 𝐸)‘𝑁)∃𝑣 ∈ (((𝑀 Sat 𝐸)‘suc 𝑁) ∖ ((𝑀 Sat 𝐸)‘𝑁))(𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ∧ 𝑦 = ((𝑀m ω) ∖ ((2nd𝑢) ∩ (2nd𝑣)))))}) = ∅)
19 funun 6611 . . . 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 = ((1st𝑢)⊼𝑔(1st𝑣)) ∧ 𝑦 = ((𝑀m ω) ∖ ((2nd𝑢) ∩ (2nd𝑣)))) ∨ ∃𝑖 ∈ ω (𝑥 = ∀𝑔𝑖(1st𝑢) ∧ 𝑦 = {𝑓 ∈ (𝑀m ω) ∣ ∀𝑗𝑀 ({⟨𝑖, 𝑗⟩} ∪ (𝑓 ↾ (ω ∖ {𝑖}))) ∈ (2nd𝑢)})) ∨ ∃𝑢 ∈ ((𝑀 Sat 𝐸)‘𝑁)∃𝑣 ∈ (((𝑀 Sat 𝐸)‘suc 𝑁) ∖ ((𝑀 Sat 𝐸)‘𝑁))(𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ∧ 𝑦 = ((𝑀m ω) ∖ ((2nd𝑢) ∩ (2nd𝑣)))))}) ∧ (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 = ((1st𝑢)⊼𝑔(1st𝑣)) ∧ 𝑦 = ((𝑀m ω) ∖ ((2nd𝑢) ∩ (2nd𝑣)))) ∨ ∃𝑖 ∈ ω (𝑥 = ∀𝑔𝑖(1st𝑢) ∧ 𝑦 = {𝑓 ∈ (𝑀m ω) ∣ ∀𝑗𝑀 ({⟨𝑖, 𝑗⟩} ∪ (𝑓 ↾ (ω ∖ {𝑖}))) ∈ (2nd𝑢)})) ∨ ∃𝑢 ∈ ((𝑀 Sat 𝐸)‘𝑁)∃𝑣 ∈ (((𝑀 Sat 𝐸)‘suc 𝑁) ∖ ((𝑀 Sat 𝐸)‘𝑁))(𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ∧ 𝑦 = ((𝑀m ω) ∖ ((2nd𝑢) ∩ (2nd𝑣)))))}) = ∅) → 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 = ((1st𝑢)⊼𝑔(1st𝑣)) ∧ 𝑦 = ((𝑀m ω) ∖ ((2nd𝑢) ∩ (2nd𝑣)))) ∨ ∃𝑖 ∈ ω (𝑥 = ∀𝑔𝑖(1st𝑢) ∧ 𝑦 = {𝑓 ∈ (𝑀m ω) ∣ ∀𝑗𝑀 ({⟨𝑖, 𝑗⟩} ∪ (𝑓 ↾ (ω ∖ {𝑖}))) ∈ (2nd𝑢)})) ∨ ∃𝑢 ∈ ((𝑀 Sat 𝐸)‘𝑁)∃𝑣 ∈ (((𝑀 Sat 𝐸)‘suc 𝑁) ∖ ((𝑀 Sat 𝐸)‘𝑁))(𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ∧ 𝑦 = ((𝑀m ω) ∖ ((2nd𝑢) ∩ (2nd𝑣)))))}))
201, 17, 18, 19syl21anc 837 . . 3 (((𝑁 ∈ ω ∧ (𝑀𝑉𝐸𝑊)) ∧ 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 = ((1st𝑢)⊼𝑔(1st𝑣)) ∧ 𝑦 = ((𝑀m ω) ∖ ((2nd𝑢) ∩ (2nd𝑣)))) ∨ ∃𝑖 ∈ ω (𝑥 = ∀𝑔𝑖(1st𝑢) ∧ 𝑦 = {𝑓 ∈ (𝑀m ω) ∣ ∀𝑗𝑀 ({⟨𝑖, 𝑗⟩} ∪ (𝑓 ↾ (ω ∖ {𝑖}))) ∈ (2nd𝑢)})) ∨ ∃𝑢 ∈ ((𝑀 Sat 𝐸)‘𝑁)∃𝑣 ∈ (((𝑀 Sat 𝐸)‘suc 𝑁) ∖ ((𝑀 Sat 𝐸)‘𝑁))(𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ∧ 𝑦 = ((𝑀m ω) ∖ ((2nd𝑢) ∩ (2nd𝑣)))))}))
21 simpl 482 . . . . . 6 ((𝑀𝑉𝐸𝑊) → 𝑀𝑉)
22 simpr 484 . . . . . 6 ((𝑀𝑉𝐸𝑊) → 𝐸𝑊)
23 simpl 482 . . . . . 6 ((𝑁 ∈ ω ∧ (𝑀𝑉𝐸𝑊)) → 𝑁 ∈ ω)
248, 12, 13satfvsucsuc 35371 . . . . . 6 ((𝑀𝑉𝐸𝑊𝑁 ∈ 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 = ((1st𝑢)⊼𝑔(1st𝑣)) ∧ 𝑦 = ((𝑀m ω) ∖ ((2nd𝑢) ∩ (2nd𝑣)))) ∨ ∃𝑖 ∈ ω (𝑥 = ∀𝑔𝑖(1st𝑢) ∧ 𝑦 = {𝑓 ∈ (𝑀m ω) ∣ ∀𝑗𝑀 ({⟨𝑖, 𝑗⟩} ∪ (𝑓 ↾ (ω ∖ {𝑖}))) ∈ (2nd𝑢)})) ∨ ∃𝑢 ∈ ((𝑀 Sat 𝐸)‘𝑁)∃𝑣 ∈ (((𝑀 Sat 𝐸)‘suc 𝑁) ∖ ((𝑀 Sat 𝐸)‘𝑁))(𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ∧ 𝑦 = ((𝑀m ω) ∖ ((2nd𝑢) ∩ (2nd𝑣)))))}))
2521, 22, 23, 24syl2an23an 1424 . . . . 5 ((𝑁 ∈ ω ∧ (𝑀𝑉𝐸𝑊)) → ((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 = ((1st𝑢)⊼𝑔(1st𝑣)) ∧ 𝑦 = ((𝑀m ω) ∖ ((2nd𝑢) ∩ (2nd𝑣)))) ∨ ∃𝑖 ∈ ω (𝑥 = ∀𝑔𝑖(1st𝑢) ∧ 𝑦 = {𝑓 ∈ (𝑀m ω) ∣ ∀𝑗𝑀 ({⟨𝑖, 𝑗⟩} ∪ (𝑓 ↾ (ω ∖ {𝑖}))) ∈ (2nd𝑢)})) ∨ ∃𝑢 ∈ ((𝑀 Sat 𝐸)‘𝑁)∃𝑣 ∈ (((𝑀 Sat 𝐸)‘suc 𝑁) ∖ ((𝑀 Sat 𝐸)‘𝑁))(𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ∧ 𝑦 = ((𝑀m ω) ∖ ((2nd𝑢) ∩ (2nd𝑣)))))}))
2625funeqd 6587 . . . 4 ((𝑁 ∈ ω ∧ (𝑀𝑉𝐸𝑊)) → (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 = ((1st𝑢)⊼𝑔(1st𝑣)) ∧ 𝑦 = ((𝑀m ω) ∖ ((2nd𝑢) ∩ (2nd𝑣)))) ∨ ∃𝑖 ∈ ω (𝑥 = ∀𝑔𝑖(1st𝑢) ∧ 𝑦 = {𝑓 ∈ (𝑀m ω) ∣ ∀𝑗𝑀 ({⟨𝑖, 𝑗⟩} ∪ (𝑓 ↾ (ω ∖ {𝑖}))) ∈ (2nd𝑢)})) ∨ ∃𝑢 ∈ ((𝑀 Sat 𝐸)‘𝑁)∃𝑣 ∈ (((𝑀 Sat 𝐸)‘suc 𝑁) ∖ ((𝑀 Sat 𝐸)‘𝑁))(𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ∧ 𝑦 = ((𝑀m ω) ∖ ((2nd𝑢) ∩ (2nd𝑣)))))})))
2726adantr 480 . . 3 (((𝑁 ∈ ω ∧ (𝑀𝑉𝐸𝑊)) ∧ 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 = ((1st𝑢)⊼𝑔(1st𝑣)) ∧ 𝑦 = ((𝑀m ω) ∖ ((2nd𝑢) ∩ (2nd𝑣)))) ∨ ∃𝑖 ∈ ω (𝑥 = ∀𝑔𝑖(1st𝑢) ∧ 𝑦 = {𝑓 ∈ (𝑀m ω) ∣ ∀𝑗𝑀 ({⟨𝑖, 𝑗⟩} ∪ (𝑓 ↾ (ω ∖ {𝑖}))) ∈ (2nd𝑢)})) ∨ ∃𝑢 ∈ ((𝑀 Sat 𝐸)‘𝑁)∃𝑣 ∈ (((𝑀 Sat 𝐸)‘suc 𝑁) ∖ ((𝑀 Sat 𝐸)‘𝑁))(𝑥 = ((1st𝑢)⊼𝑔(1st𝑣)) ∧ 𝑦 = ((𝑀m ω) ∖ ((2nd𝑢) ∩ (2nd𝑣)))))})))
2820, 27mpbird 257 . 2 (((𝑁 ∈ ω ∧ (𝑀𝑉𝐸𝑊)) ∧ Fun ((𝑀 Sat 𝐸)‘suc 𝑁)) → Fun ((𝑀 Sat 𝐸)‘suc suc 𝑁))
2928ex 412 1 ((𝑁 ∈ ω ∧ (𝑀𝑉𝐸𝑊)) → (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 1539  wcel 2107  wral 3060  wrex 3069  {crab 3435  cdif 3947  cun 3948  cin 3949  wss 3950  c0 4332  {csn 4625  cop 4631  {copab 5204  dom cdm 5684  cres 5686  suc csuc 6385  Fun wfun 6554  cfv 6560  (class class class)co 7432  ωcom 7888  1st c1st 8013  2nd c2nd 8014  m cmap 8867  𝑔cgna 35340  𝑔cgol 35341   Sat csat 35342
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-rep 5278  ax-sep 5295  ax-nul 5305  ax-pow 5364  ax-pr 5431  ax-un 7756  ax-inf2 9682
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-nel 3046  df-ral 3061  df-rex 3070  df-reu 3380  df-rab 3436  df-v 3481  df-sbc 3788  df-csb 3899  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-pss 3970  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-int 4946  df-iun 4992  df-br 5143  df-opab 5205  df-mpt 5225  df-tr 5259  df-id 5577  df-eprel 5583  df-po 5591  df-so 5592  df-fr 5636  df-we 5638  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-pred 6320  df-ord 6386  df-on 6387  df-lim 6388  df-suc 6389  df-iota 6513  df-fun 6562  df-fn 6563  df-f 6564  df-f1 6565  df-fo 6566  df-f1o 6567  df-fv 6568  df-ov 7435  df-oprab 7436  df-mpo 7437  df-om 7889  df-1st 8015  df-2nd 8016  df-frecs 8307  df-wrecs 8338  df-recs 8412  df-rdg 8451  df-1o 8507  df-2o 8508  df-map 8869  df-goel 35346  df-gona 35347  df-goal 35348  df-sat 35349  df-fmla 35351
This theorem is referenced by:  satffun  35415
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