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Theorem satffun 33412
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 33374 . . . 4 ((𝑀𝑉𝐸𝑊) → Fun ((𝑀 Sat 𝐸)‘∅))
213adant3 1132 . . 3 ((𝑀𝑉𝐸𝑊𝑁 ∈ ω) → Fun ((𝑀 Sat 𝐸)‘∅))
3 fveq2 6800 . . . 4 (𝑁 = ∅ → ((𝑀 Sat 𝐸)‘𝑁) = ((𝑀 Sat 𝐸)‘∅))
43funeqd 6481 . . 3 (𝑁 = ∅ → (Fun ((𝑀 Sat 𝐸)‘𝑁) ↔ Fun ((𝑀 Sat 𝐸)‘∅)))
52, 4syl5ibr 247 . 2 (𝑁 = ∅ → ((𝑀𝑉𝐸𝑊𝑁 ∈ ω) → Fun ((𝑀 Sat 𝐸)‘𝑁)))
6 df-ne 2942 . . . . . 6 (𝑁 ≠ ∅ ↔ ¬ 𝑁 = ∅)
7 nnsuc 7758 . . . . . . . 8 ((𝑁 ∈ ω ∧ 𝑁 ≠ ∅) → ∃𝑛 ∈ ω 𝑁 = suc 𝑛)
8 suceq 6342 . . . . . . . . . . . . . . 15 (𝑥 = ∅ → suc 𝑥 = suc ∅)
98fveq2d 6804 . . . . . . . . . . . . . 14 (𝑥 = ∅ → ((𝑀 Sat 𝐸)‘suc 𝑥) = ((𝑀 Sat 𝐸)‘suc ∅))
109funeqd 6481 . . . . . . . . . . . . 13 (𝑥 = ∅ → (Fun ((𝑀 Sat 𝐸)‘suc 𝑥) ↔ Fun ((𝑀 Sat 𝐸)‘suc ∅)))
1110imbi2d 342 . . . . . . . . . . . 12 (𝑥 = ∅ → (((𝑀𝑉𝐸𝑊) → Fun ((𝑀 Sat 𝐸)‘suc 𝑥)) ↔ ((𝑀𝑉𝐸𝑊) → Fun ((𝑀 Sat 𝐸)‘suc ∅))))
12 suceq 6342 . . . . . . . . . . . . . . 15 (𝑥 = 𝑦 → suc 𝑥 = suc 𝑦)
1312fveq2d 6804 . . . . . . . . . . . . . 14 (𝑥 = 𝑦 → ((𝑀 Sat 𝐸)‘suc 𝑥) = ((𝑀 Sat 𝐸)‘suc 𝑦))
1413funeqd 6481 . . . . . . . . . . . . 13 (𝑥 = 𝑦 → (Fun ((𝑀 Sat 𝐸)‘suc 𝑥) ↔ Fun ((𝑀 Sat 𝐸)‘suc 𝑦)))
1514imbi2d 342 . . . . . . . . . . . 12 (𝑥 = 𝑦 → (((𝑀𝑉𝐸𝑊) → Fun ((𝑀 Sat 𝐸)‘suc 𝑥)) ↔ ((𝑀𝑉𝐸𝑊) → Fun ((𝑀 Sat 𝐸)‘suc 𝑦))))
16 suceq 6342 . . . . . . . . . . . . . . 15 (𝑥 = suc 𝑦 → suc 𝑥 = suc suc 𝑦)
1716fveq2d 6804 . . . . . . . . . . . . . 14 (𝑥 = suc 𝑦 → ((𝑀 Sat 𝐸)‘suc 𝑥) = ((𝑀 Sat 𝐸)‘suc suc 𝑦))
1817funeqd 6481 . . . . . . . . . . . . 13 (𝑥 = suc 𝑦 → (Fun ((𝑀 Sat 𝐸)‘suc 𝑥) ↔ Fun ((𝑀 Sat 𝐸)‘suc suc 𝑦)))
1918imbi2d 342 . . . . . . . . . . . 12 (𝑥 = suc 𝑦 → (((𝑀𝑉𝐸𝑊) → Fun ((𝑀 Sat 𝐸)‘suc 𝑥)) ↔ ((𝑀𝑉𝐸𝑊) → Fun ((𝑀 Sat 𝐸)‘suc suc 𝑦))))
20 suceq 6342 . . . . . . . . . . . . . . 15 (𝑥 = 𝑛 → suc 𝑥 = suc 𝑛)
2120fveq2d 6804 . . . . . . . . . . . . . 14 (𝑥 = 𝑛 → ((𝑀 Sat 𝐸)‘suc 𝑥) = ((𝑀 Sat 𝐸)‘suc 𝑛))
2221funeqd 6481 . . . . . . . . . . . . 13 (𝑥 = 𝑛 → (Fun ((𝑀 Sat 𝐸)‘suc 𝑥) ↔ Fun ((𝑀 Sat 𝐸)‘suc 𝑛)))
2322imbi2d 342 . . . . . . . . . . . 12 (𝑥 = 𝑛 → (((𝑀𝑉𝐸𝑊) → Fun ((𝑀 Sat 𝐸)‘suc 𝑥)) ↔ ((𝑀𝑉𝐸𝑊) → Fun ((𝑀 Sat 𝐸)‘suc 𝑛))))
24 satffunlem1 33410 . . . . . . . . . . . 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 33411 . . . . . . . . . . . . . . . 16 ((𝑦 ∈ ω ∧ (𝑀𝑉𝐸𝑊)) → (Fun ((𝑀 Sat 𝐸)‘suc 𝑦) → Fun ((𝑀 Sat 𝐸)‘suc suc 𝑦)))
2726expcom 415 . . . . . . . . . . . . . . 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 7773 . . . . . . . . . . 11 (𝑛 ∈ ω → ((𝑀𝑉𝐸𝑊) → Fun ((𝑀 Sat 𝐸)‘suc 𝑛)))
3231adantr 482 . . . . . . . . . 10 ((𝑛 ∈ ω ∧ 𝑁 = suc 𝑛) → ((𝑀𝑉𝐸𝑊) → Fun ((𝑀 Sat 𝐸)‘suc 𝑛)))
33 fveq2 6800 . . . . . . . . . . . . 13 (𝑁 = suc 𝑛 → ((𝑀 Sat 𝐸)‘𝑁) = ((𝑀 Sat 𝐸)‘suc 𝑛))
3433funeqd 6481 . . . . . . . . . . . 12 (𝑁 = suc 𝑛 → (Fun ((𝑀 Sat 𝐸)‘𝑁) ↔ Fun ((𝑀 Sat 𝐸)‘suc 𝑛)))
3534imbi2d 342 . . . . . . . . . . 11 (𝑁 = suc 𝑛 → (((𝑀𝑉𝐸𝑊) → Fun ((𝑀 Sat 𝐸)‘𝑁)) ↔ ((𝑀𝑉𝐸𝑊) → Fun ((𝑀 Sat 𝐸)‘suc 𝑛))))
3635adantl 483 . . . . . . . . . 10 ((𝑛 ∈ ω ∧ 𝑁 = suc 𝑛) → (((𝑀𝑉𝐸𝑊) → Fun ((𝑀 Sat 𝐸)‘𝑁)) ↔ ((𝑀𝑉𝐸𝑊) → Fun ((𝑀 Sat 𝐸)‘suc 𝑛))))
3732, 36mpbird 258 . . . . . . . . 9 ((𝑛 ∈ ω ∧ 𝑁 = suc 𝑛) → ((𝑀𝑉𝐸𝑊) → Fun ((𝑀 Sat 𝐸)‘𝑁)))
3837rexlimiva 3141 . . . . . . . 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 415 . . . . . 6 (𝑁 ≠ ∅ → (𝑁 ∈ ω → ((𝑀𝑉𝐸𝑊) → Fun ((𝑀 Sat 𝐸)‘𝑁))))
416, 40sylbir 235 . . . . 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 397  w3a 1087   = wceq 1539  wcel 2104  wne 2941  wrex 3071  c0 4262  suc csuc 6279  Fun wfun 6448  cfv 6454  (class class class)co 7303  ωcom 7740   Sat csat 33339
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 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2707  ax-rep 5218  ax-sep 5232  ax-nul 5239  ax-pow 5297  ax-pr 5361  ax-un 7616  ax-inf2 9439
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 846  df-3or 1088  df-3an 1089  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2887  df-ne 2942  df-nel 3048  df-ral 3063  df-rex 3072  df-reu 3286  df-rab 3287  df-v 3439  df-sbc 3722  df-csb 3838  df-dif 3895  df-un 3897  df-in 3899  df-ss 3909  df-pss 3911  df-nul 4263  df-if 4466  df-pw 4541  df-sn 4566  df-pr 4568  df-op 4572  df-uni 4845  df-int 4887  df-iun 4933  df-br 5082  df-opab 5144  df-mpt 5165  df-tr 5199  df-id 5496  df-eprel 5502  df-po 5510  df-so 5511  df-fr 5551  df-we 5553  df-xp 5602  df-rel 5603  df-cnv 5604  df-co 5605  df-dm 5606  df-rn 5607  df-res 5608  df-ima 5609  df-pred 6213  df-ord 6280  df-on 6281  df-lim 6282  df-suc 6283  df-iota 6406  df-fun 6456  df-fn 6457  df-f 6458  df-f1 6459  df-fo 6460  df-f1o 6461  df-fv 6462  df-ov 7306  df-oprab 7307  df-mpo 7308  df-om 7741  df-1st 7859  df-2nd 7860  df-frecs 8124  df-wrecs 8155  df-recs 8229  df-rdg 8268  df-1o 8324  df-2o 8325  df-map 8644  df-goel 33343  df-gona 33344  df-goal 33345  df-sat 33346  df-fmla 33348
This theorem is referenced by:  satff  33413  satfv1fvfmla1  33426
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