Theorem satffun 32656

Description: The value of the satisfaction predicate as function over wff codes at a natural number is a function. (Contributed by AV, 28-Oct-2023.)

Assertion

Ref	Expression
satffun	⊢ ((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ∧ 𝑁 ∈ ω) → Fun ((𝑀 Sat 𝐸)‘𝑁))

Proof of Theorem satffun

Dummy variables 𝑛 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		satfv0fun 32618	. . . 4 ⊢ ((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) → Fun ((𝑀 Sat 𝐸)‘∅))
2	1	3adant3 1128	. . 3 ⊢ ((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ∧ 𝑁 ∈ ω) → Fun ((𝑀 Sat 𝐸)‘∅))
3		fveq2 6670	. . . 4 ⊢ (𝑁 = ∅ → ((𝑀 Sat 𝐸)‘𝑁) = ((𝑀 Sat 𝐸)‘∅))
4	3	funeqd 6377	. . 3 ⊢ (𝑁 = ∅ → (Fun ((𝑀 Sat 𝐸)‘𝑁) ↔ Fun ((𝑀 Sat 𝐸)‘∅)))
5	2, 4	syl5ibr 248	. 2 ⊢ (𝑁 = ∅ → ((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ∧ 𝑁 ∈ ω) → Fun ((𝑀 Sat 𝐸)‘𝑁)))
6		df-ne 3017	. . . . . 6 ⊢ (𝑁 ≠ ∅ ↔ ¬ 𝑁 = ∅)
7		nnsuc 7597	. . . . . . . 8 ⊢ ((𝑁 ∈ ω ∧ 𝑁 ≠ ∅) → ∃𝑛 ∈ ω 𝑁 = suc 𝑛)
8		suceq 6256	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = ∅ → suc 𝑥 = suc ∅)
9	8	fveq2d 6674	. . . . . . . . . . . . . 14 ⊢ (𝑥 = ∅ → ((𝑀 Sat 𝐸)‘suc 𝑥) = ((𝑀 Sat 𝐸)‘suc ∅))
10	9	funeqd 6377	. . . . . . . . . . . . 13 ⊢ (𝑥 = ∅ → (Fun ((𝑀 Sat 𝐸)‘suc 𝑥) ↔ Fun ((𝑀 Sat 𝐸)‘suc ∅)))
11	10	imbi2d 343	. . . . . . . . . . . 12 ⊢ (𝑥 = ∅ → (((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) → Fun ((𝑀 Sat 𝐸)‘suc 𝑥)) ↔ ((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) → Fun ((𝑀 Sat 𝐸)‘suc ∅))))
12		suceq 6256	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 𝑦 → suc 𝑥 = suc 𝑦)
13	12	fveq2d 6674	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑦 → ((𝑀 Sat 𝐸)‘suc 𝑥) = ((𝑀 Sat 𝐸)‘suc 𝑦))
14	13	funeqd 6377	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑦 → (Fun ((𝑀 Sat 𝐸)‘suc 𝑥) ↔ Fun ((𝑀 Sat 𝐸)‘suc 𝑦)))
15	14	imbi2d 343	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑦 → (((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) → Fun ((𝑀 Sat 𝐸)‘suc 𝑥)) ↔ ((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) → Fun ((𝑀 Sat 𝐸)‘suc 𝑦))))
16		suceq 6256	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = suc 𝑦 → suc 𝑥 = suc suc 𝑦)
17	16	fveq2d 6674	. . . . . . . . . . . . . 14 ⊢ (𝑥 = suc 𝑦 → ((𝑀 Sat 𝐸)‘suc 𝑥) = ((𝑀 Sat 𝐸)‘suc suc 𝑦))
18	17	funeqd 6377	. . . . . . . . . . . . 13 ⊢ (𝑥 = suc 𝑦 → (Fun ((𝑀 Sat 𝐸)‘suc 𝑥) ↔ Fun ((𝑀 Sat 𝐸)‘suc suc 𝑦)))
19	18	imbi2d 343	. . . . . . . . . . . 12 ⊢ (𝑥 = suc 𝑦 → (((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) → Fun ((𝑀 Sat 𝐸)‘suc 𝑥)) ↔ ((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) → Fun ((𝑀 Sat 𝐸)‘suc suc 𝑦))))
20		suceq 6256	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 𝑛 → suc 𝑥 = suc 𝑛)
21	20	fveq2d 6674	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑛 → ((𝑀 Sat 𝐸)‘suc 𝑥) = ((𝑀 Sat 𝐸)‘suc 𝑛))
22	21	funeqd 6377	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑛 → (Fun ((𝑀 Sat 𝐸)‘suc 𝑥) ↔ Fun ((𝑀 Sat 𝐸)‘suc 𝑛)))
23	22	imbi2d 343	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑛 → (((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) → Fun ((𝑀 Sat 𝐸)‘suc 𝑥)) ↔ ((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) → Fun ((𝑀 Sat 𝐸)‘suc 𝑛))))
24		satffunlem1 32654	. . . . . . . . . . . 12 ⊢ ((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) → Fun ((𝑀 Sat 𝐸)‘suc ∅))
25		pm2.27 42	. . . . . . . . . . . . . 14 ⊢ ((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) → (((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) → Fun ((𝑀 Sat 𝐸)‘suc 𝑦)) → Fun ((𝑀 Sat 𝐸)‘suc 𝑦)))
26		satffunlem2 32655	. . . . . . . . . . . . . . . 16 ⊢ ((𝑦 ∈ ω ∧ (𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊)) → (Fun ((𝑀 Sat 𝐸)‘suc 𝑦) → Fun ((𝑀 Sat 𝐸)‘suc suc 𝑦)))
27	26	expcom 416	. . . . . . . . . . . . . . 15 ⊢ ((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) → (𝑦 ∈ ω → (Fun ((𝑀 Sat 𝐸)‘suc 𝑦) → Fun ((𝑀 Sat 𝐸)‘suc suc 𝑦))))
28	27	com23 86	. . . . . . . . . . . . . 14 ⊢ ((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) → (Fun ((𝑀 Sat 𝐸)‘suc 𝑦) → (𝑦 ∈ ω → Fun ((𝑀 Sat 𝐸)‘suc suc 𝑦))))
29	25, 28	syld 47	. . . . . . . . . . . . 13 ⊢ ((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) → (((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) → Fun ((𝑀 Sat 𝐸)‘suc 𝑦)) → (𝑦 ∈ ω → Fun ((𝑀 Sat 𝐸)‘suc suc 𝑦))))
30	29	com13 88	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ ω → (((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) → Fun ((𝑀 Sat 𝐸)‘suc 𝑦)) → ((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) → Fun ((𝑀 Sat 𝐸)‘suc suc 𝑦))))
31	11, 15, 19, 23, 24, 30	finds 7608	. . . . . . . . . . 11 ⊢ (𝑛 ∈ ω → ((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) → Fun ((𝑀 Sat 𝐸)‘suc 𝑛)))
32	31	adantr 483	. . . . . . . . . 10 ⊢ ((𝑛 ∈ ω ∧ 𝑁 = suc 𝑛) → ((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) → Fun ((𝑀 Sat 𝐸)‘suc 𝑛)))
33		fveq2 6670	. . . . . . . . . . . . 13 ⊢ (𝑁 = suc 𝑛 → ((𝑀 Sat 𝐸)‘𝑁) = ((𝑀 Sat 𝐸)‘suc 𝑛))
34	33	funeqd 6377	. . . . . . . . . . . 12 ⊢ (𝑁 = suc 𝑛 → (Fun ((𝑀 Sat 𝐸)‘𝑁) ↔ Fun ((𝑀 Sat 𝐸)‘suc 𝑛)))
35	34	imbi2d 343	. . . . . . . . . . 11 ⊢ (𝑁 = suc 𝑛 → (((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) → Fun ((𝑀 Sat 𝐸)‘𝑁)) ↔ ((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) → Fun ((𝑀 Sat 𝐸)‘suc 𝑛))))
36	35	adantl 484	. . . . . . . . . 10 ⊢ ((𝑛 ∈ ω ∧ 𝑁 = suc 𝑛) → (((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) → Fun ((𝑀 Sat 𝐸)‘𝑁)) ↔ ((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) → Fun ((𝑀 Sat 𝐸)‘suc 𝑛))))
37	32, 36	mpbird 259	. . . . . . . . 9 ⊢ ((𝑛 ∈ ω ∧ 𝑁 = suc 𝑛) → ((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) → Fun ((𝑀 Sat 𝐸)‘𝑁)))
38	37	rexlimiva 3281	. . . . . . . 8 ⊢ (∃𝑛 ∈ ω 𝑁 = suc 𝑛 → ((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) → Fun ((𝑀 Sat 𝐸)‘𝑁)))
39	7, 38	syl 17	. . . . . . 7 ⊢ ((𝑁 ∈ ω ∧ 𝑁 ≠ ∅) → ((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) → Fun ((𝑀 Sat 𝐸)‘𝑁)))
40	39	expcom 416	. . . . . 6 ⊢ (𝑁 ≠ ∅ → (𝑁 ∈ ω → ((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) → Fun ((𝑀 Sat 𝐸)‘𝑁))))
41	6, 40	sylbir 237	. . . . 5 ⊢ (¬ 𝑁 = ∅ → (𝑁 ∈ ω → ((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) → Fun ((𝑀 Sat 𝐸)‘𝑁))))
42	41	com13 88	. . . 4 ⊢ ((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) → (𝑁 ∈ ω → (¬ 𝑁 = ∅ → Fun ((𝑀 Sat 𝐸)‘𝑁))))
43	42	3impia 1113	. . 3 ⊢ ((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ∧ 𝑁 ∈ ω) → (¬ 𝑁 = ∅ → Fun ((𝑀 Sat 𝐸)‘𝑁)))
44	43	com12 32	. 2 ⊢ (¬ 𝑁 = ∅ → ((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ∧ 𝑁 ∈ ω) → Fun ((𝑀 Sat 𝐸)‘𝑁)))
45	5, 44	pm2.61i 184	1 ⊢ ((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ∧ 𝑁 ∈ ω) → Fun ((𝑀 Sat 𝐸)‘𝑁))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 208   ∧ wa 398   ∧ w3a 1083   = wceq 1537   ∈ wcel 2114   ≠ wne 3016  ∃wrex 3139  ∅c0 4291  suc csuc 6193  Fun wfun 6349  ‘cfv 6355  (class class class)co 7156  ωcom 7580   Sat csat 32583

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 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2793  ax-rep 5190  ax-sep 5203  ax-nul 5210  ax-pow 5266  ax-pr 5330  ax-un 7461  ax-inf2 9104

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-nel 3124  df-ral 3143  df-rex 3144  df-reu 3145  df-rab 3147  df-v 3496  df-sbc 3773  df-csb 3884  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-pss 3954  df-nul 4292  df-if 4468  df-pw 4541  df-sn 4568  df-pr 4570  df-tp 4572  df-op 4574  df-uni 4839  df-int 4877  df-iun 4921  df-br 5067  df-opab 5129  df-mpt 5147  df-tr 5173  df-id 5460  df-eprel 5465  df-po 5474  df-so 5475  df-fr 5514  df-we 5516  df-xp 5561  df-rel 5562  df-cnv 5563  df-co 5564  df-dm 5565  df-rn 5566  df-res 5567  df-ima 5568  df-pred 6148  df-ord 6194  df-on 6195  df-lim 6196  df-suc 6197  df-iota 6314  df-fun 6357  df-fn 6358  df-f 6359  df-f1 6360  df-fo 6361  df-f1o 6362  df-fv 6363  df-ov 7159  df-oprab 7160  df-mpo 7161  df-om 7581  df-1st 7689  df-2nd 7690  df-wrecs 7947  df-recs 8008  df-rdg 8046  df-1o 8102  df-2o 8103  df-map 8408  df-goel 32587  df-gona 32588  df-goal 32589  df-sat 32590  df-fmla 32592

This theorem is referenced by:  satff  32657  satfv1fvfmla1  32670