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Theorem satffun 32770
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 32732 . . . 4 ((𝑀𝑉𝐸𝑊) → Fun ((𝑀 Sat 𝐸)‘∅))
213adant3 1129 . . 3 ((𝑀𝑉𝐸𝑊𝑁 ∈ ω) → Fun ((𝑀 Sat 𝐸)‘∅))
3 fveq2 6649 . . . 4 (𝑁 = ∅ → ((𝑀 Sat 𝐸)‘𝑁) = ((𝑀 Sat 𝐸)‘∅))
43funeqd 6350 . . 3 (𝑁 = ∅ → (Fun ((𝑀 Sat 𝐸)‘𝑁) ↔ Fun ((𝑀 Sat 𝐸)‘∅)))
52, 4syl5ibr 249 . 2 (𝑁 = ∅ → ((𝑀𝑉𝐸𝑊𝑁 ∈ ω) → Fun ((𝑀 Sat 𝐸)‘𝑁)))
6 df-ne 2991 . . . . . 6 (𝑁 ≠ ∅ ↔ ¬ 𝑁 = ∅)
7 nnsuc 7581 . . . . . . . 8 ((𝑁 ∈ ω ∧ 𝑁 ≠ ∅) → ∃𝑛 ∈ ω 𝑁 = suc 𝑛)
8 suceq 6228 . . . . . . . . . . . . . . 15 (𝑥 = ∅ → suc 𝑥 = suc ∅)
98fveq2d 6653 . . . . . . . . . . . . . 14 (𝑥 = ∅ → ((𝑀 Sat 𝐸)‘suc 𝑥) = ((𝑀 Sat 𝐸)‘suc ∅))
109funeqd 6350 . . . . . . . . . . . . 13 (𝑥 = ∅ → (Fun ((𝑀 Sat 𝐸)‘suc 𝑥) ↔ Fun ((𝑀 Sat 𝐸)‘suc ∅)))
1110imbi2d 344 . . . . . . . . . . . 12 (𝑥 = ∅ → (((𝑀𝑉𝐸𝑊) → Fun ((𝑀 Sat 𝐸)‘suc 𝑥)) ↔ ((𝑀𝑉𝐸𝑊) → Fun ((𝑀 Sat 𝐸)‘suc ∅))))
12 suceq 6228 . . . . . . . . . . . . . . 15 (𝑥 = 𝑦 → suc 𝑥 = suc 𝑦)
1312fveq2d 6653 . . . . . . . . . . . . . 14 (𝑥 = 𝑦 → ((𝑀 Sat 𝐸)‘suc 𝑥) = ((𝑀 Sat 𝐸)‘suc 𝑦))
1413funeqd 6350 . . . . . . . . . . . . 13 (𝑥 = 𝑦 → (Fun ((𝑀 Sat 𝐸)‘suc 𝑥) ↔ Fun ((𝑀 Sat 𝐸)‘suc 𝑦)))
1514imbi2d 344 . . . . . . . . . . . 12 (𝑥 = 𝑦 → (((𝑀𝑉𝐸𝑊) → Fun ((𝑀 Sat 𝐸)‘suc 𝑥)) ↔ ((𝑀𝑉𝐸𝑊) → Fun ((𝑀 Sat 𝐸)‘suc 𝑦))))
16 suceq 6228 . . . . . . . . . . . . . . 15 (𝑥 = suc 𝑦 → suc 𝑥 = suc suc 𝑦)
1716fveq2d 6653 . . . . . . . . . . . . . 14 (𝑥 = suc 𝑦 → ((𝑀 Sat 𝐸)‘suc 𝑥) = ((𝑀 Sat 𝐸)‘suc suc 𝑦))
1817funeqd 6350 . . . . . . . . . . . . 13 (𝑥 = suc 𝑦 → (Fun ((𝑀 Sat 𝐸)‘suc 𝑥) ↔ Fun ((𝑀 Sat 𝐸)‘suc suc 𝑦)))
1918imbi2d 344 . . . . . . . . . . . 12 (𝑥 = suc 𝑦 → (((𝑀𝑉𝐸𝑊) → Fun ((𝑀 Sat 𝐸)‘suc 𝑥)) ↔ ((𝑀𝑉𝐸𝑊) → Fun ((𝑀 Sat 𝐸)‘suc suc 𝑦))))
20 suceq 6228 . . . . . . . . . . . . . . 15 (𝑥 = 𝑛 → suc 𝑥 = suc 𝑛)
2120fveq2d 6653 . . . . . . . . . . . . . 14 (𝑥 = 𝑛 → ((𝑀 Sat 𝐸)‘suc 𝑥) = ((𝑀 Sat 𝐸)‘suc 𝑛))
2221funeqd 6350 . . . . . . . . . . . . 13 (𝑥 = 𝑛 → (Fun ((𝑀 Sat 𝐸)‘suc 𝑥) ↔ Fun ((𝑀 Sat 𝐸)‘suc 𝑛)))
2322imbi2d 344 . . . . . . . . . . . 12 (𝑥 = 𝑛 → (((𝑀𝑉𝐸𝑊) → Fun ((𝑀 Sat 𝐸)‘suc 𝑥)) ↔ ((𝑀𝑉𝐸𝑊) → Fun ((𝑀 Sat 𝐸)‘suc 𝑛))))
24 satffunlem1 32768 . . . . . . . . . . . 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 32769 . . . . . . . . . . . . . . . 16 ((𝑦 ∈ ω ∧ (𝑀𝑉𝐸𝑊)) → (Fun ((𝑀 Sat 𝐸)‘suc 𝑦) → Fun ((𝑀 Sat 𝐸)‘suc suc 𝑦)))
2726expcom 417 . . . . . . . . . . . . . . 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 7593 . . . . . . . . . . 11 (𝑛 ∈ ω → ((𝑀𝑉𝐸𝑊) → Fun ((𝑀 Sat 𝐸)‘suc 𝑛)))
3231adantr 484 . . . . . . . . . 10 ((𝑛 ∈ ω ∧ 𝑁 = suc 𝑛) → ((𝑀𝑉𝐸𝑊) → Fun ((𝑀 Sat 𝐸)‘suc 𝑛)))
33 fveq2 6649 . . . . . . . . . . . . 13 (𝑁 = suc 𝑛 → ((𝑀 Sat 𝐸)‘𝑁) = ((𝑀 Sat 𝐸)‘suc 𝑛))
3433funeqd 6350 . . . . . . . . . . . 12 (𝑁 = suc 𝑛 → (Fun ((𝑀 Sat 𝐸)‘𝑁) ↔ Fun ((𝑀 Sat 𝐸)‘suc 𝑛)))
3534imbi2d 344 . . . . . . . . . . 11 (𝑁 = suc 𝑛 → (((𝑀𝑉𝐸𝑊) → Fun ((𝑀 Sat 𝐸)‘𝑁)) ↔ ((𝑀𝑉𝐸𝑊) → Fun ((𝑀 Sat 𝐸)‘suc 𝑛))))
3635adantl 485 . . . . . . . . . 10 ((𝑛 ∈ ω ∧ 𝑁 = suc 𝑛) → (((𝑀𝑉𝐸𝑊) → Fun ((𝑀 Sat 𝐸)‘𝑁)) ↔ ((𝑀𝑉𝐸𝑊) → Fun ((𝑀 Sat 𝐸)‘suc 𝑛))))
3732, 36mpbird 260 . . . . . . . . 9 ((𝑛 ∈ ω ∧ 𝑁 = suc 𝑛) → ((𝑀𝑉𝐸𝑊) → Fun ((𝑀 Sat 𝐸)‘𝑁)))
3837rexlimiva 3243 . . . . . . . 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 417 . . . . . 6 (𝑁 ≠ ∅ → (𝑁 ∈ ω → ((𝑀𝑉𝐸𝑊) → Fun ((𝑀 Sat 𝐸)‘𝑁))))
416, 40sylbir 238 . . . . 5 𝑁 = ∅ → (𝑁 ∈ ω → ((𝑀𝑉𝐸𝑊) → Fun ((𝑀 Sat 𝐸)‘𝑁))))
4241com13 88 . . . 4 ((𝑀𝑉𝐸𝑊) → (𝑁 ∈ ω → (¬ 𝑁 = ∅ → Fun ((𝑀 Sat 𝐸)‘𝑁))))
43423impia 1114 . . 3 ((𝑀𝑉𝐸𝑊𝑁 ∈ ω) → (¬ 𝑁 = ∅ → Fun ((𝑀 Sat 𝐸)‘𝑁)))
4443com12 32 . 2 𝑁 = ∅ → ((𝑀𝑉𝐸𝑊𝑁 ∈ ω) → Fun ((𝑀 Sat 𝐸)‘𝑁)))
455, 44pm2.61i 185 1 ((𝑀𝑉𝐸𝑊𝑁 ∈ ω) → Fun ((𝑀 Sat 𝐸)‘𝑁))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 209  wa 399  w3a 1084   = wceq 1538  wcel 2112  wne 2990  wrex 3110  c0 4246  suc csuc 6165  Fun wfun 6322  cfv 6328  (class class class)co 7139  ωcom 7564   Sat csat 32697
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 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-rep 5157  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298  ax-un 7445  ax-inf2 9092
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-nel 3095  df-ral 3114  df-rex 3115  df-reu 3116  df-rab 3118  df-v 3446  df-sbc 3724  df-csb 3832  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-pss 3903  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-tp 4533  df-op 4535  df-uni 4804  df-int 4842  df-iun 4886  df-br 5034  df-opab 5096  df-mpt 5114  df-tr 5140  df-id 5428  df-eprel 5433  df-po 5442  df-so 5443  df-fr 5482  df-we 5484  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-pred 6120  df-ord 6166  df-on 6167  df-lim 6168  df-suc 6169  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336  df-ov 7142  df-oprab 7143  df-mpo 7144  df-om 7565  df-1st 7675  df-2nd 7676  df-wrecs 7934  df-recs 7995  df-rdg 8033  df-1o 8089  df-2o 8090  df-map 8395  df-goel 32701  df-gona 32702  df-goal 32703  df-sat 32704  df-fmla 32706
This theorem is referenced by:  satff  32771  satfv1fvfmla1  32784
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