Theorem satffun 35394

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 35356	. . . 4 ⊢ ((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) → Fun ((𝑀 Sat 𝐸)‘∅))
2	1	3adant3 1131	. . 3 ⊢ ((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ∧ 𝑁 ∈ ω) → Fun ((𝑀 Sat 𝐸)‘∅))
3		fveq2 6907	. . . 4 ⊢ (𝑁 = ∅ → ((𝑀 Sat 𝐸)‘𝑁) = ((𝑀 Sat 𝐸)‘∅))
4	3	funeqd 6590	. . 3 ⊢ (𝑁 = ∅ → (Fun ((𝑀 Sat 𝐸)‘𝑁) ↔ Fun ((𝑀 Sat 𝐸)‘∅)))
5	2, 4	imbitrrid 246	. 2 ⊢ (𝑁 = ∅ → ((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ∧ 𝑁 ∈ ω) → Fun ((𝑀 Sat 𝐸)‘𝑁)))
6		df-ne 2939	. . . . . 6 ⊢ (𝑁 ≠ ∅ ↔ ¬ 𝑁 = ∅)
7		nnsuc 7905	. . . . . . . 8 ⊢ ((𝑁 ∈ ω ∧ 𝑁 ≠ ∅) → ∃𝑛 ∈ ω 𝑁 = suc 𝑛)
8		suceq 6452	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = ∅ → suc 𝑥 = suc ∅)
9	8	fveq2d 6911	. . . . . . . . . . . . . 14 ⊢ (𝑥 = ∅ → ((𝑀 Sat 𝐸)‘suc 𝑥) = ((𝑀 Sat 𝐸)‘suc ∅))
10	9	funeqd 6590	. . . . . . . . . . . . 13 ⊢ (𝑥 = ∅ → (Fun ((𝑀 Sat 𝐸)‘suc 𝑥) ↔ Fun ((𝑀 Sat 𝐸)‘suc ∅)))
11	10	imbi2d 340	. . . . . . . . . . . 12 ⊢ (𝑥 = ∅ → (((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) → Fun ((𝑀 Sat 𝐸)‘suc 𝑥)) ↔ ((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) → Fun ((𝑀 Sat 𝐸)‘suc ∅))))
12		suceq 6452	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 𝑦 → suc 𝑥 = suc 𝑦)
13	12	fveq2d 6911	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑦 → ((𝑀 Sat 𝐸)‘suc 𝑥) = ((𝑀 Sat 𝐸)‘suc 𝑦))
14	13	funeqd 6590	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑦 → (Fun ((𝑀 Sat 𝐸)‘suc 𝑥) ↔ Fun ((𝑀 Sat 𝐸)‘suc 𝑦)))
15	14	imbi2d 340	. . . . . . . . . . . 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 6452	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = suc 𝑦 → suc 𝑥 = suc suc 𝑦)
17	16	fveq2d 6911	. . . . . . . . . . . . . 14 ⊢ (𝑥 = suc 𝑦 → ((𝑀 Sat 𝐸)‘suc 𝑥) = ((𝑀 Sat 𝐸)‘suc suc 𝑦))
18	17	funeqd 6590	. . . . . . . . . . . . 13 ⊢ (𝑥 = suc 𝑦 → (Fun ((𝑀 Sat 𝐸)‘suc 𝑥) ↔ Fun ((𝑀 Sat 𝐸)‘suc suc 𝑦)))
19	18	imbi2d 340	. . . . . . . . . . . 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 6452	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 𝑛 → suc 𝑥 = suc 𝑛)
21	20	fveq2d 6911	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑛 → ((𝑀 Sat 𝐸)‘suc 𝑥) = ((𝑀 Sat 𝐸)‘suc 𝑛))
22	21	funeqd 6590	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑛 → (Fun ((𝑀 Sat 𝐸)‘suc 𝑥) ↔ Fun ((𝑀 Sat 𝐸)‘suc 𝑛)))
23	22	imbi2d 340	. . . . . . . . . . . 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 35392	. . . . . . . . . . . 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 35393	. . . . . . . . . . . . . . . 16 ⊢ ((𝑦 ∈ ω ∧ (𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊)) → (Fun ((𝑀 Sat 𝐸)‘suc 𝑦) → Fun ((𝑀 Sat 𝐸)‘suc suc 𝑦)))
27	26	expcom 413	. . . . . . . . . . . . . . 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 7919	. . . . . . . . . . 11 ⊢ (𝑛 ∈ ω → ((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) → Fun ((𝑀 Sat 𝐸)‘suc 𝑛)))
32	31	adantr 480	. . . . . . . . . 10 ⊢ ((𝑛 ∈ ω ∧ 𝑁 = suc 𝑛) → ((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) → Fun ((𝑀 Sat 𝐸)‘suc 𝑛)))
33		fveq2 6907	. . . . . . . . . . . . 13 ⊢ (𝑁 = suc 𝑛 → ((𝑀 Sat 𝐸)‘𝑁) = ((𝑀 Sat 𝐸)‘suc 𝑛))
34	33	funeqd 6590	. . . . . . . . . . . 12 ⊢ (𝑁 = suc 𝑛 → (Fun ((𝑀 Sat 𝐸)‘𝑁) ↔ Fun ((𝑀 Sat 𝐸)‘suc 𝑛)))
35	34	imbi2d 340	. . . . . . . . . . 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 481	. . . . . . . . . 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 257	. . . . . . . . 9 ⊢ ((𝑛 ∈ ω ∧ 𝑁 = suc 𝑛) → ((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) → Fun ((𝑀 Sat 𝐸)‘𝑁)))
38	37	rexlimiva 3145	. . . . . . . 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 413	. . . . . 6 ⊢ (𝑁 ≠ ∅ → (𝑁 ∈ ω → ((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) → Fun ((𝑀 Sat 𝐸)‘𝑁))))
41	6, 40	sylbir 235	. . . . 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 1116	. . 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 182	1 ⊢ ((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ∧ 𝑁 ∈ ω) → Fun ((𝑀 Sat 𝐸)‘𝑁))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1537   ∈ wcel 2106   ≠ wne 2938  ∃wrex 3068  ∅c0 4339  suc csuc 6388  Fun wfun 6557  ‘cfv 6563  (class class class)co 7431  ωcom 7887   Sat csat 35321

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-inf2 9679

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-int 4952  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8013  df-2nd 8014  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-1o 8505  df-2o 8506  df-map 8867  df-goel 35325  df-gona 35326  df-goal 35327  df-sat 35328  df-fmla 35330

This theorem is referenced by:  satff  35395  satfv1fvfmla1  35408