satffunlem1 - Mathbox for Mario Carneiro

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Theorem satffunlem1 35394

Description: Lemma 1 for satffun 35396: induction basis. (Contributed by AV, 28-Oct-2023.)

**Assertion**
Ref	Expression
satffunlem1	⊢ ((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) → Fun ((𝑀 Sat 𝐸)‘suc ∅))

Proof of Theorem satffunlem1 Dummy variables 𝑓 𝑖 𝑗 𝑢 𝑣 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		satfv0fun 35358	. . 3 ⊢ ((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) → Fun ((𝑀 Sat 𝐸)‘∅))
2		satffunlem1lem1 35389	. . . 4 ⊢ (Fun ((𝑀 Sat 𝐸)‘∅) → Fun {⟨𝑥, 𝑦⟩ ∣ ∃𝑢 ∈ ((𝑀 Sat 𝐸)‘∅)(∃𝑣 ∈ ((𝑀 Sat 𝐸)‘∅)(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)}))})
3	1, 2	syl 17	. . 3 ⊢ ((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) → Fun {⟨𝑥, 𝑦⟩ ∣ ∃𝑢 ∈ ((𝑀 Sat 𝐸)‘∅)(∃𝑣 ∈ ((𝑀 Sat 𝐸)‘∅)(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)}))})
4		satffunlem1lem2 35390	. . 3 ⊢ ((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) → (dom ((𝑀 Sat 𝐸)‘∅) ∩ dom {⟨𝑥, 𝑦⟩ ∣ ∃𝑢 ∈ ((𝑀 Sat 𝐸)‘∅)(∃𝑣 ∈ ((𝑀 Sat 𝐸)‘∅)(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)}))}) = ∅)
5		funun 6562	. . 3 ⊢ (((Fun ((𝑀 Sat 𝐸)‘∅) ∧ Fun {⟨𝑥, 𝑦⟩ ∣ ∃𝑢 ∈ ((𝑀 Sat 𝐸)‘∅)(∃𝑣 ∈ ((𝑀 Sat 𝐸)‘∅)(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)}))}) ∧ (dom ((𝑀 Sat 𝐸)‘∅) ∩ dom {⟨𝑥, 𝑦⟩ ∣ ∃𝑢 ∈ ((𝑀 Sat 𝐸)‘∅)(∃𝑣 ∈ ((𝑀 Sat 𝐸)‘∅)(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)}))}) = ∅) → Fun (((𝑀 Sat 𝐸)‘∅) ∪ {⟨𝑥, 𝑦⟩ ∣ ∃𝑢 ∈ ((𝑀 Sat 𝐸)‘∅)(∃𝑣 ∈ ((𝑀 Sat 𝐸)‘∅)(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)}))}))
6	1, 3, 4, 5	syl21anc 837	. 2 ⊢ ((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) → Fun (((𝑀 Sat 𝐸)‘∅) ∪ {⟨𝑥, 𝑦⟩ ∣ ∃𝑢 ∈ ((𝑀 Sat 𝐸)‘∅)(∃𝑣 ∈ ((𝑀 Sat 𝐸)‘∅)(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)}))}))
7		peano1 7865	. . . 4 ⊢ ∅ ∈ ω
8		eqid 2729	. . . . 5 ⊢ (𝑀 Sat 𝐸) = (𝑀 Sat 𝐸)
9	8	satfvsuc 35348	. . . 4 ⊢ ((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ∧ ∅ ∈ ω) → ((𝑀 Sat 𝐸)‘suc ∅) = (((𝑀 Sat 𝐸)‘∅) ∪ {⟨𝑥, 𝑦⟩ ∣ ∃𝑢 ∈ ((𝑀 Sat 𝐸)‘∅)(∃𝑣 ∈ ((𝑀 Sat 𝐸)‘∅)(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)}))}))
10	7, 9	mp3an3 1452	. . 3 ⊢ ((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) → ((𝑀 Sat 𝐸)‘suc ∅) = (((𝑀 Sat 𝐸)‘∅) ∪ {⟨𝑥, 𝑦⟩ ∣ ∃𝑢 ∈ ((𝑀 Sat 𝐸)‘∅)(∃𝑣 ∈ ((𝑀 Sat 𝐸)‘∅)(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)}))}))
11	10	funeqd 6538	. 2 ⊢ ((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) → (Fun ((𝑀 Sat 𝐸)‘suc ∅) ↔ Fun (((𝑀 Sat 𝐸)‘∅) ∪ {⟨𝑥, 𝑦⟩ ∣ ∃𝑢 ∈ ((𝑀 Sat 𝐸)‘∅)(∃𝑣 ∈ ((𝑀 Sat 𝐸)‘∅)(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)}))})))
12	6, 11	mpbird 257	1 ⊢ ((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) → Fun ((𝑀 Sat 𝐸)‘suc ∅))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 ∨ wo 847 = wceq 1540 ∈ wcel 2109 ∀wral 3044 ∃wrex 3053 {crab 3405 ∖ cdif 3911 ∪ cun 3912 ∩ cin 3913 ∅c0 4296 {csn 4589 ⟨cop 4595 {copab 5169 dom cdm 5638 ↾ cres 5640 suc csuc 6334 Fun wfun 6505 ‘cfv 6511 (class class class)co 7387 ωcom 7842 1^st c1st 7966 2^nd c2nd 7967 ↑_m cmap 8799 ⊼_𝑔cgna 35321 ∀_𝑔cgol 35322 Sat csat 35323

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-rep 5234 ax-sep 5251 ax-nul 5261 ax-pow 5320 ax-pr 5387 ax-un 7711 ax-inf2 9594

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-ral 3045 df-rex 3054 df-reu 3355 df-rab 3406 df-v 3449 df-sbc 3754 df-csb 3863 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-pss 3934 df-nul 4297 df-if 4489 df-pw 4565 df-sn 4590 df-pr 4592 df-op 4596 df-uni 4872 df-int 4911 df-iun 4957 df-br 5108 df-opab 5170 df-mpt 5189 df-tr 5215 df-id 5533 df-eprel 5538 df-po 5546 df-so 5547 df-fr 5591 df-we 5593 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-pred 6274 df-ord 6335 df-on 6336 df-lim 6337 df-suc 6338 df-iota 6464 df-fun 6513 df-fn 6514 df-f 6515 df-f1 6516 df-fo 6517 df-f1o 6518 df-fv 6519 df-ov 7390 df-oprab 7391 df-mpo 7392 df-om 7843 df-1st 7968 df-2nd 7969 df-frecs 8260 df-wrecs 8291 df-recs 8340 df-rdg 8378 df-1o 8434 df-2o 8435 df-map 8801 df-goel 35327 df-gona 35328 df-goal 35329 df-sat 35330 df-fmla 35332

This theorem is referenced by: satffun 35396

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