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Theorem satffun 34003
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 ((𝑀𝑉𝐸𝑊𝑁 ∈ ω) → Fun ((𝑀 Sat 𝐸)‘𝑁))

Proof of Theorem satffun
Dummy variables 𝑛 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 satfv0fun 33965 . . . 4 ((𝑀𝑉𝐸𝑊) → Fun ((𝑀 Sat 𝐸)‘∅))
213adant3 1132 . . 3 ((𝑀𝑉𝐸𝑊𝑁 ∈ ω) → Fun ((𝑀 Sat 𝐸)‘∅))
3 fveq2 6842 . . . 4 (𝑁 = ∅ → ((𝑀 Sat 𝐸)‘𝑁) = ((𝑀 Sat 𝐸)‘∅))
43funeqd 6523 . . 3 (𝑁 = ∅ → (Fun ((𝑀 Sat 𝐸)‘𝑁) ↔ Fun ((𝑀 Sat 𝐸)‘∅)))
52, 4syl5ibr 245 . 2 (𝑁 = ∅ → ((𝑀𝑉𝐸𝑊𝑁 ∈ ω) → Fun ((𝑀 Sat 𝐸)‘𝑁)))
6 df-ne 2944 . . . . . 6 (𝑁 ≠ ∅ ↔ ¬ 𝑁 = ∅)
7 nnsuc 7820 . . . . . . . 8 ((𝑁 ∈ ω ∧ 𝑁 ≠ ∅) → ∃𝑛 ∈ ω 𝑁 = suc 𝑛)
8 suceq 6383 . . . . . . . . . . . . . . 15 (𝑥 = ∅ → suc 𝑥 = suc ∅)
98fveq2d 6846 . . . . . . . . . . . . . 14 (𝑥 = ∅ → ((𝑀 Sat 𝐸)‘suc 𝑥) = ((𝑀 Sat 𝐸)‘suc ∅))
109funeqd 6523 . . . . . . . . . . . . 13 (𝑥 = ∅ → (Fun ((𝑀 Sat 𝐸)‘suc 𝑥) ↔ Fun ((𝑀 Sat 𝐸)‘suc ∅)))
1110imbi2d 340 . . . . . . . . . . . 12 (𝑥 = ∅ → (((𝑀𝑉𝐸𝑊) → Fun ((𝑀 Sat 𝐸)‘suc 𝑥)) ↔ ((𝑀𝑉𝐸𝑊) → Fun ((𝑀 Sat 𝐸)‘suc ∅))))
12 suceq 6383 . . . . . . . . . . . . . . 15 (𝑥 = 𝑦 → suc 𝑥 = suc 𝑦)
1312fveq2d 6846 . . . . . . . . . . . . . 14 (𝑥 = 𝑦 → ((𝑀 Sat 𝐸)‘suc 𝑥) = ((𝑀 Sat 𝐸)‘suc 𝑦))
1413funeqd 6523 . . . . . . . . . . . . 13 (𝑥 = 𝑦 → (Fun ((𝑀 Sat 𝐸)‘suc 𝑥) ↔ Fun ((𝑀 Sat 𝐸)‘suc 𝑦)))
1514imbi2d 340 . . . . . . . . . . . 12 (𝑥 = 𝑦 → (((𝑀𝑉𝐸𝑊) → Fun ((𝑀 Sat 𝐸)‘suc 𝑥)) ↔ ((𝑀𝑉𝐸𝑊) → Fun ((𝑀 Sat 𝐸)‘suc 𝑦))))
16 suceq 6383 . . . . . . . . . . . . . . 15 (𝑥 = suc 𝑦 → suc 𝑥 = suc suc 𝑦)
1716fveq2d 6846 . . . . . . . . . . . . . 14 (𝑥 = suc 𝑦 → ((𝑀 Sat 𝐸)‘suc 𝑥) = ((𝑀 Sat 𝐸)‘suc suc 𝑦))
1817funeqd 6523 . . . . . . . . . . . . 13 (𝑥 = suc 𝑦 → (Fun ((𝑀 Sat 𝐸)‘suc 𝑥) ↔ Fun ((𝑀 Sat 𝐸)‘suc suc 𝑦)))
1918imbi2d 340 . . . . . . . . . . . 12 (𝑥 = suc 𝑦 → (((𝑀𝑉𝐸𝑊) → Fun ((𝑀 Sat 𝐸)‘suc 𝑥)) ↔ ((𝑀𝑉𝐸𝑊) → Fun ((𝑀 Sat 𝐸)‘suc suc 𝑦))))
20 suceq 6383 . . . . . . . . . . . . . . 15 (𝑥 = 𝑛 → suc 𝑥 = suc 𝑛)
2120fveq2d 6846 . . . . . . . . . . . . . 14 (𝑥 = 𝑛 → ((𝑀 Sat 𝐸)‘suc 𝑥) = ((𝑀 Sat 𝐸)‘suc 𝑛))
2221funeqd 6523 . . . . . . . . . . . . 13 (𝑥 = 𝑛 → (Fun ((𝑀 Sat 𝐸)‘suc 𝑥) ↔ Fun ((𝑀 Sat 𝐸)‘suc 𝑛)))
2322imbi2d 340 . . . . . . . . . . . 12 (𝑥 = 𝑛 → (((𝑀𝑉𝐸𝑊) → Fun ((𝑀 Sat 𝐸)‘suc 𝑥)) ↔ ((𝑀𝑉𝐸𝑊) → Fun ((𝑀 Sat 𝐸)‘suc 𝑛))))
24 satffunlem1 34001 . . . . . . . . . . . 12 ((𝑀𝑉𝐸𝑊) → Fun ((𝑀 Sat 𝐸)‘suc ∅))
25 pm2.27 42 . . . . . . . . . . . . . 14 ((𝑀𝑉𝐸𝑊) → (((𝑀𝑉𝐸𝑊) → Fun ((𝑀 Sat 𝐸)‘suc 𝑦)) → Fun ((𝑀 Sat 𝐸)‘suc 𝑦)))
26 satffunlem2 34002 . . . . . . . . . . . . . . . 16 ((𝑦 ∈ ω ∧ (𝑀𝑉𝐸𝑊)) → (Fun ((𝑀 Sat 𝐸)‘suc 𝑦) → Fun ((𝑀 Sat 𝐸)‘suc suc 𝑦)))
2726expcom 414 . . . . . . . . . . . . . . 15 ((𝑀𝑉𝐸𝑊) → (𝑦 ∈ ω → (Fun ((𝑀 Sat 𝐸)‘suc 𝑦) → Fun ((𝑀 Sat 𝐸)‘suc suc 𝑦))))
2827com23 86 . . . . . . . . . . . . . 14 ((𝑀𝑉𝐸𝑊) → (Fun ((𝑀 Sat 𝐸)‘suc 𝑦) → (𝑦 ∈ ω → Fun ((𝑀 Sat 𝐸)‘suc suc 𝑦))))
2925, 28syld 47 . . . . . . . . . . . . 13 ((𝑀𝑉𝐸𝑊) → (((𝑀𝑉𝐸𝑊) → Fun ((𝑀 Sat 𝐸)‘suc 𝑦)) → (𝑦 ∈ ω → Fun ((𝑀 Sat 𝐸)‘suc suc 𝑦))))
3029com13 88 . . . . . . . . . . . 12 (𝑦 ∈ ω → (((𝑀𝑉𝐸𝑊) → Fun ((𝑀 Sat 𝐸)‘suc 𝑦)) → ((𝑀𝑉𝐸𝑊) → Fun ((𝑀 Sat 𝐸)‘suc suc 𝑦))))
3111, 15, 19, 23, 24, 30finds 7835 . . . . . . . . . . 11 (𝑛 ∈ ω → ((𝑀𝑉𝐸𝑊) → Fun ((𝑀 Sat 𝐸)‘suc 𝑛)))
3231adantr 481 . . . . . . . . . 10 ((𝑛 ∈ ω ∧ 𝑁 = suc 𝑛) → ((𝑀𝑉𝐸𝑊) → Fun ((𝑀 Sat 𝐸)‘suc 𝑛)))
33 fveq2 6842 . . . . . . . . . . . . 13 (𝑁 = suc 𝑛 → ((𝑀 Sat 𝐸)‘𝑁) = ((𝑀 Sat 𝐸)‘suc 𝑛))
3433funeqd 6523 . . . . . . . . . . . 12 (𝑁 = suc 𝑛 → (Fun ((𝑀 Sat 𝐸)‘𝑁) ↔ Fun ((𝑀 Sat 𝐸)‘suc 𝑛)))
3534imbi2d 340 . . . . . . . . . . 11 (𝑁 = suc 𝑛 → (((𝑀𝑉𝐸𝑊) → Fun ((𝑀 Sat 𝐸)‘𝑁)) ↔ ((𝑀𝑉𝐸𝑊) → Fun ((𝑀 Sat 𝐸)‘suc 𝑛))))
3635adantl 482 . . . . . . . . . 10 ((𝑛 ∈ ω ∧ 𝑁 = suc 𝑛) → (((𝑀𝑉𝐸𝑊) → Fun ((𝑀 Sat 𝐸)‘𝑁)) ↔ ((𝑀𝑉𝐸𝑊) → Fun ((𝑀 Sat 𝐸)‘suc 𝑛))))
3732, 36mpbird 256 . . . . . . . . 9 ((𝑛 ∈ ω ∧ 𝑁 = suc 𝑛) → ((𝑀𝑉𝐸𝑊) → Fun ((𝑀 Sat 𝐸)‘𝑁)))
3837rexlimiva 3144 . . . . . . . 8 (∃𝑛 ∈ ω 𝑁 = suc 𝑛 → ((𝑀𝑉𝐸𝑊) → Fun ((𝑀 Sat 𝐸)‘𝑁)))
397, 38syl 17 . . . . . . 7 ((𝑁 ∈ ω ∧ 𝑁 ≠ ∅) → ((𝑀𝑉𝐸𝑊) → Fun ((𝑀 Sat 𝐸)‘𝑁)))
4039expcom 414 . . . . . 6 (𝑁 ≠ ∅ → (𝑁 ∈ ω → ((𝑀𝑉𝐸𝑊) → Fun ((𝑀 Sat 𝐸)‘𝑁))))
416, 40sylbir 234 . . . . 5 𝑁 = ∅ → (𝑁 ∈ ω → ((𝑀𝑉𝐸𝑊) → Fun ((𝑀 Sat 𝐸)‘𝑁))))
4241com13 88 . . . 4 ((𝑀𝑉𝐸𝑊) → (𝑁 ∈ ω → (¬ 𝑁 = ∅ → Fun ((𝑀 Sat 𝐸)‘𝑁))))
43423impia 1117 . . 3 ((𝑀𝑉𝐸𝑊𝑁 ∈ ω) → (¬ 𝑁 = ∅ → Fun ((𝑀 Sat 𝐸)‘𝑁)))
4443com12 32 . 2 𝑁 = ∅ → ((𝑀𝑉𝐸𝑊𝑁 ∈ ω) → Fun ((𝑀 Sat 𝐸)‘𝑁)))
455, 44pm2.61i 182 1 ((𝑀𝑉𝐸𝑊𝑁 ∈ ω) → Fun ((𝑀 Sat 𝐸)‘𝑁))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 396  w3a 1087   = wceq 1541  wcel 2106  wne 2943  wrex 3073  c0 4282  suc csuc 6319  Fun wfun 6490  cfv 6496  (class class class)co 7357  ωcom 7802   Sat csat 33930
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-inf2 9577
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3065  df-rex 3074  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-int 4908  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-ov 7360  df-oprab 7361  df-mpo 7362  df-om 7803  df-1st 7921  df-2nd 7922  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-1o 8412  df-2o 8413  df-map 8767  df-goel 33934  df-gona 33935  df-goal 33936  df-sat 33937  df-fmla 33939
This theorem is referenced by:  satff  34004  satfv1fvfmla1  34017
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