Theorem satffun 35879

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 35841	. . . 4 ⊢ ((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) → Fun ((𝑀 Sat 𝐸)‘∅))
2	1	3adant3 1150	. . 3 ⊢ ((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ∧ 𝑁 ∈ ω) → Fun ((𝑀 Sat 𝐸)‘∅))
3		fveq2 6883	. . . 4 ⊢ (𝑁 = ∅ → ((𝑀 Sat 𝐸)‘𝑁) = ((𝑀 Sat 𝐸)‘∅))
4	3	funeqd 6560	. . 3 ⊢ (𝑁 = ∅ → (Fun ((𝑀 Sat 𝐸)‘𝑁) ↔ Fun ((𝑀 Sat 𝐸)‘∅)))
5	2, 4	imbitrrid 249	. 2 ⊢ (𝑁 = ∅ → ((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ∧ 𝑁 ∈ ω) → Fun ((𝑀 Sat 𝐸)‘𝑁)))
6		df-ne 2959	. . . . . 6 ⊢ (𝑁 ≠ ∅ ↔ ¬ 𝑁 = ∅)
7		nnsuc 7881	. . . . . . . 8 ⊢ ((𝑁 ∈ ω ∧ 𝑁 ≠ ∅) → ∃𝑛 ∈ ω 𝑁 = suc 𝑛)
8		suceq 6431	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = ∅ → suc 𝑥 = suc ∅)
9	8	fveq2d 6887	. . . . . . . . . . . . . 14 ⊢ (𝑥 = ∅ → ((𝑀 Sat 𝐸)‘suc 𝑥) = ((𝑀 Sat 𝐸)‘suc ∅))
10	9	funeqd 6560	. . . . . . . . . . . . 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 6431	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 𝑦 → suc 𝑥 = suc 𝑦)
13	12	fveq2d 6887	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑦 → ((𝑀 Sat 𝐸)‘suc 𝑥) = ((𝑀 Sat 𝐸)‘suc 𝑦))
14	13	funeqd 6560	. . . . . . . . . . . . 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 6431	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = suc 𝑦 → suc 𝑥 = suc suc 𝑦)
17	16	fveq2d 6887	. . . . . . . . . . . . . 14 ⊢ (𝑥 = suc 𝑦 → ((𝑀 Sat 𝐸)‘suc 𝑥) = ((𝑀 Sat 𝐸)‘suc suc 𝑦))
18	17	funeqd 6560	. . . . . . . . . . . . 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 6431	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 𝑛 → suc 𝑥 = suc 𝑛)
21	20	fveq2d 6887	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑛 → ((𝑀 Sat 𝐸)‘suc 𝑥) = ((𝑀 Sat 𝐸)‘suc 𝑛))
22	21	funeqd 6560	. . . . . . . . . . . . 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 35877	. . . . . . . . . . . 12 ⊢ ((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) → Fun ((𝑀 Sat 𝐸)‘suc ∅))
25		pm2.27 43	. . . . . . . . . . . . . 14 ⊢ ((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) → (((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) → Fun ((𝑀 Sat 𝐸)‘suc 𝑦)) → Fun ((𝑀 Sat 𝐸)‘suc 𝑦)))
26		satffunlem2 35878	. . . . . . . . . . . . . . . 16 ⊢ ((𝑦 ∈ ω ∧ (𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊)) → (Fun ((𝑀 Sat 𝐸)‘suc 𝑦) → Fun ((𝑀 Sat 𝐸)‘suc suc 𝑦)))
27	26	expcom 418	. . . . . . . . . . . . . . 15 ⊢ ((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) → (𝑦 ∈ ω → (Fun ((𝑀 Sat 𝐸)‘suc 𝑦) → Fun ((𝑀 Sat 𝐸)‘suc suc 𝑦))))
28	27	com23 87	. . . . . . . . . . . . . 14 ⊢ ((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) → (Fun ((𝑀 Sat 𝐸)‘suc 𝑦) → (𝑦 ∈ ω → Fun ((𝑀 Sat 𝐸)‘suc suc 𝑦))))
29	25, 28	syld 48	. . . . . . . . . . . . 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 89	. . . . . . . . . . . 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 7894	. . . . . . . . . . 11 ⊢ (𝑛 ∈ ω → ((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) → Fun ((𝑀 Sat 𝐸)‘suc 𝑛)))
32	31	adantr 485	. . . . . . . . . 10 ⊢ ((𝑛 ∈ ω ∧ 𝑁 = suc 𝑛) → ((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) → Fun ((𝑀 Sat 𝐸)‘suc 𝑛)))
33		fveq2 6883	. . . . . . . . . . . . 13 ⊢ (𝑁 = suc 𝑛 → ((𝑀 Sat 𝐸)‘𝑁) = ((𝑀 Sat 𝐸)‘suc 𝑛))
34	33	funeqd 6560	. . . . . . . . . . . 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 486	. . . . . . . . . 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 260	. . . . . . . . 9 ⊢ ((𝑛 ∈ ω ∧ 𝑁 = suc 𝑛) → ((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) → Fun ((𝑀 Sat 𝐸)‘𝑁)))
38	37	rexlimiva 3158	. . . . . . . 8 ⊢ (∃𝑛 ∈ ω 𝑁 = suc 𝑛 → ((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) → Fun ((𝑀 Sat 𝐸)‘𝑁)))
39	7, 38	syl 18	. . . . . . 7 ⊢ ((𝑁 ∈ ω ∧ 𝑁 ≠ ∅) → ((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) → Fun ((𝑀 Sat 𝐸)‘𝑁)))
40	39	expcom 418	. . . . . 6 ⊢ (𝑁 ≠ ∅ → (𝑁 ∈ ω → ((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) → Fun ((𝑀 Sat 𝐸)‘𝑁))))
41	6, 40	sylbir 238	. . . . 5 ⊢ (¬ 𝑁 = ∅ → (𝑁 ∈ ω → ((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) → Fun ((𝑀 Sat 𝐸)‘𝑁))))
42	41	com13 89	. . . 4 ⊢ ((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) → (𝑁 ∈ ω → (¬ 𝑁 = ∅ → Fun ((𝑀 Sat 𝐸)‘𝑁))))
43	42	3impia 1135	. . 3 ⊢ ((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ∧ 𝑁 ∈ ω) → (¬ 𝑁 = ∅ → Fun ((𝑀 Sat 𝐸)‘𝑁)))
44	43	com12 33	. 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 209   ∧ wa 400   ∧ w3a 1103   = wceq 1570   ∈ wcel 2143   ≠ wne 2958  ∃wrex 3089  ∅c0 4287  suc csuc 6364  Fun wfun 6532  ‘cfv 6538  (class class class)co 7412  ωcom 7863   Sat csat 35806

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-rep 5239  ax-sep 5258  ax-nul 5270  ax-pow 5338  ax-pr 5406  ax-un 7734  ax-inf2 9611

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-nel 3065  df-ral 3080  df-rex 3090  df-reu 3370  df-rab 3417  df-v 3457  df-sbc 3746  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-pss 3926  df-nul 4288  df-if 4489  df-pw 4565  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4874  df-int 4914  df-iun 4959  df-br 5111  df-opab 5175  df-mpt 5194  df-tr 5220  df-id 5558  df-eprel 5563  df-po 5571  df-so 5572  df-fr 5616  df-we 5618  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-pred 6304  df-ord 6365  df-on 6366  df-lim 6367  df-suc 6368  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-ov 7415  df-oprab 7416  df-mpo 7417  df-om 7864  df-1st 7987  df-2nd 7988  df-frecs 8279  df-wrecs 8310  df-recs 8359  df-rdg 8398  df-1o 8454  df-2o 8455  df-map 8827  df-goel 35810  df-gona 35811  df-goal 35812  df-sat 35813  df-fmla 35815

This theorem is referenced by:  satff  35880  satfv1fvfmla1  35893