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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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