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Theorem fnbrafvb 46431
Description: Equivalence of function value and binary relation, analogous to fnbrfvb 6938. (Contributed by Alexander van der Vekens, 25-May-2017.)
Assertion
Ref Expression
fnbrafvb ((𝐹 Fn 𝐴𝐵𝐴) → ((𝐹'''𝐵) = 𝐶𝐵𝐹𝐶))

Proof of Theorem fnbrafvb
StepHypRef Expression
1 fndm 6646 . . . . . 6 (𝐹 Fn 𝐴 → dom 𝐹 = 𝐴)
2 eleq2 2816 . . . . . . . 8 (𝐴 = dom 𝐹 → (𝐵𝐴𝐵 ∈ dom 𝐹))
32eqcoms 2734 . . . . . . 7 (dom 𝐹 = 𝐴 → (𝐵𝐴𝐵 ∈ dom 𝐹))
43biimpd 228 . . . . . 6 (dom 𝐹 = 𝐴 → (𝐵𝐴𝐵 ∈ dom 𝐹))
51, 4syl 17 . . . . 5 (𝐹 Fn 𝐴 → (𝐵𝐴𝐵 ∈ dom 𝐹))
65imp 406 . . . 4 ((𝐹 Fn 𝐴𝐵𝐴) → 𝐵 ∈ dom 𝐹)
7 snssi 4806 . . . . . . 7 (𝐵𝐴 → {𝐵} ⊆ 𝐴)
87adantl 481 . . . . . 6 ((𝐹 Fn 𝐴𝐵𝐴) → {𝐵} ⊆ 𝐴)
9 fnssresb 6666 . . . . . . 7 (𝐹 Fn 𝐴 → ((𝐹 ↾ {𝐵}) Fn {𝐵} ↔ {𝐵} ⊆ 𝐴))
109adantr 480 . . . . . 6 ((𝐹 Fn 𝐴𝐵𝐴) → ((𝐹 ↾ {𝐵}) Fn {𝐵} ↔ {𝐵} ⊆ 𝐴))
118, 10mpbird 257 . . . . 5 ((𝐹 Fn 𝐴𝐵𝐴) → (𝐹 ↾ {𝐵}) Fn {𝐵})
12 fnfun 6643 . . . . 5 ((𝐹 ↾ {𝐵}) Fn {𝐵} → Fun (𝐹 ↾ {𝐵}))
1311, 12syl 17 . . . 4 ((𝐹 Fn 𝐴𝐵𝐴) → Fun (𝐹 ↾ {𝐵}))
14 df-dfat 46396 . . . . 5 (𝐹 defAt 𝐵 ↔ (𝐵 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐵})))
15 afvfundmfveq 46415 . . . . 5 (𝐹 defAt 𝐵 → (𝐹'''𝐵) = (𝐹𝐵))
1614, 15sylbir 234 . . . 4 ((𝐵 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐵})) → (𝐹'''𝐵) = (𝐹𝐵))
176, 13, 16syl2anc 583 . . 3 ((𝐹 Fn 𝐴𝐵𝐴) → (𝐹'''𝐵) = (𝐹𝐵))
1817eqeq1d 2728 . 2 ((𝐹 Fn 𝐴𝐵𝐴) → ((𝐹'''𝐵) = 𝐶 ↔ (𝐹𝐵) = 𝐶))
19 fnbrfvb 6938 . 2 ((𝐹 Fn 𝐴𝐵𝐴) → ((𝐹𝐵) = 𝐶𝐵𝐹𝐶))
2018, 19bitrd 279 1 ((𝐹 Fn 𝐴𝐵𝐴) → ((𝐹'''𝐵) = 𝐶𝐵𝐹𝐶))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 395   = wceq 1533  wcel 2098  wss 3943  {csn 4623   class class class wbr 5141  dom cdm 5669  cres 5671  Fun wfun 6531   Fn wfn 6532  cfv 6537   defAt wdfat 46393  '''cafv 46394
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pr 5420
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-int 4944  df-br 5142  df-opab 5204  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-res 5681  df-iota 6489  df-fun 6539  df-fn 6540  df-fv 6545  df-aiota 46362  df-dfat 46396  df-afv 46397
This theorem is referenced by:  fnopafvb  46432  funbrafvb  46433  dfafn5a  46437
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