satffunlem1 - Mathbox for Mario Carneiro

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Theorem satffunlem1 35380

Description: Lemma 1 for satffun 35382: 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 35344	. . 3 ⊢ ((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) → Fun ((𝑀 Sat 𝐸)‘∅))
2		satffunlem1lem1 35375	. . . 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 35376	. . 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 6528	. . 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 7822	. . . 4 ⊢ ∅ ∈ ω
8		eqid 2729	. . . . 5 ⊢ (𝑀 Sat 𝐸) = (𝑀 Sat 𝐸)
9	8	satfvsuc 35334	. . . 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 6504	. 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 3394 ∖ cdif 3900 ∪ cun 3901 ∩ cin 3902 ∅c0 4284 {csn 4577 ⟨cop 4583 {copab 5154 dom cdm 5619 ↾ cres 5621 suc csuc 6309 Fun wfun 6476 ‘cfv 6482 (class class class)co 7349 ωcom 7799 1^st c1st 7922 2^nd c2nd 7923 ↑_m cmap 8753 ⊼_𝑔cgna 35307 ∀_𝑔cgol 35308 Sat csat 35309

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 5218 ax-sep 5235 ax-nul 5245 ax-pow 5304 ax-pr 5371 ax-un 7671 ax-inf2 9537

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 3344 df-rab 3395 df-v 3438 df-sbc 3743 df-csb 3852 df-dif 3906 df-un 3908 df-in 3910 df-ss 3920 df-pss 3923 df-nul 4285 df-if 4477 df-pw 4553 df-sn 4578 df-pr 4580 df-op 4584 df-uni 4859 df-int 4897 df-iun 4943 df-br 5093 df-opab 5155 df-mpt 5174 df-tr 5200 df-id 5514 df-eprel 5519 df-po 5527 df-so 5528 df-fr 5572 df-we 5574 df-xp 5625 df-rel 5626 df-cnv 5627 df-co 5628 df-dm 5629 df-rn 5630 df-res 5631 df-ima 5632 df-pred 6249 df-ord 6310 df-on 6311 df-lim 6312 df-suc 6313 df-iota 6438 df-fun 6484 df-fn 6485 df-f 6486 df-f1 6487 df-fo 6488 df-f1o 6489 df-fv 6490 df-ov 7352 df-oprab 7353 df-mpo 7354 df-om 7800 df-1st 7924 df-2nd 7925 df-frecs 8214 df-wrecs 8245 df-recs 8294 df-rdg 8332 df-1o 8388 df-2o 8389 df-map 8755 df-goel 35313 df-gona 35314 df-goal 35315 df-sat 35316 df-fmla 35318

This theorem is referenced by: satffun 35382

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