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Theorem fnbrafvb 44079
 Description: Equivalence of function value and binary relation, analogous to fnbrfvb 6707. (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 6437 . . . . . 6 (𝐹 Fn 𝐴 → dom 𝐹 = 𝐴)
2 eleq2 2841 . . . . . . . 8 (𝐴 = dom 𝐹 → (𝐵𝐴𝐵 ∈ dom 𝐹))
32eqcoms 2767 . . . . . . 7 (dom 𝐹 = 𝐴 → (𝐵𝐴𝐵 ∈ dom 𝐹))
43biimpd 232 . . . . . 6 (dom 𝐹 = 𝐴 → (𝐵𝐴𝐵 ∈ dom 𝐹))
51, 4syl 17 . . . . 5 (𝐹 Fn 𝐴 → (𝐵𝐴𝐵 ∈ dom 𝐹))
65imp 411 . . . 4 ((𝐹 Fn 𝐴𝐵𝐴) → 𝐵 ∈ dom 𝐹)
7 snssi 4699 . . . . . . 7 (𝐵𝐴 → {𝐵} ⊆ 𝐴)
87adantl 486 . . . . . 6 ((𝐹 Fn 𝐴𝐵𝐴) → {𝐵} ⊆ 𝐴)
9 fnssresb 6453 . . . . . . 7 (𝐹 Fn 𝐴 → ((𝐹 ↾ {𝐵}) Fn {𝐵} ↔ {𝐵} ⊆ 𝐴))
109adantr 485 . . . . . 6 ((𝐹 Fn 𝐴𝐵𝐴) → ((𝐹 ↾ {𝐵}) Fn {𝐵} ↔ {𝐵} ⊆ 𝐴))
118, 10mpbird 260 . . . . 5 ((𝐹 Fn 𝐴𝐵𝐴) → (𝐹 ↾ {𝐵}) Fn {𝐵})
12 fnfun 6435 . . . . 5 ((𝐹 ↾ {𝐵}) Fn {𝐵} → Fun (𝐹 ↾ {𝐵}))
1311, 12syl 17 . . . 4 ((𝐹 Fn 𝐴𝐵𝐴) → Fun (𝐹 ↾ {𝐵}))
14 df-dfat 44044 . . . . 5 (𝐹 defAt 𝐵 ↔ (𝐵 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐵})))
15 afvfundmfveq 44063 . . . . 5 (𝐹 defAt 𝐵 → (𝐹'''𝐵) = (𝐹𝐵))
1614, 15sylbir 238 . . . 4 ((𝐵 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐵})) → (𝐹'''𝐵) = (𝐹𝐵))
176, 13, 16syl2anc 588 . . 3 ((𝐹 Fn 𝐴𝐵𝐴) → (𝐹'''𝐵) = (𝐹𝐵))
1817eqeq1d 2761 . 2 ((𝐹 Fn 𝐴𝐵𝐴) → ((𝐹'''𝐵) = 𝐶 ↔ (𝐹𝐵) = 𝐶))
19 fnbrfvb 6707 . 2 ((𝐹 Fn 𝐴𝐵𝐴) → ((𝐹𝐵) = 𝐶𝐵𝐹𝐶))
2018, 19bitrd 282 1 ((𝐹 Fn 𝐴𝐵𝐴) → ((𝐹'''𝐵) = 𝐶𝐵𝐹𝐶))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   = wceq 1539   ∈ wcel 2112   ⊆ wss 3859  {csn 4523   class class class wbr 5033  dom cdm 5525   ↾ cres 5527  Fun wfun 6330   Fn wfn 6331  ‘cfv 6336   defAt wdfat 44041  '''cafv 44042 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2730  ax-sep 5170  ax-nul 5177  ax-pow 5235  ax-pr 5299 This theorem depends on definitions:  df-bi 210  df-an 401  df-or 846  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2071  df-mo 2558  df-eu 2589  df-clab 2737  df-cleq 2751  df-clel 2831  df-nfc 2902  df-ne 2953  df-ral 3076  df-rex 3077  df-rab 3080  df-v 3412  df-sbc 3698  df-csb 3807  df-dif 3862  df-un 3864  df-in 3866  df-ss 3876  df-nul 4227  df-if 4422  df-sn 4524  df-pr 4526  df-op 4530  df-uni 4800  df-int 4840  df-br 5034  df-opab 5096  df-id 5431  df-xp 5531  df-rel 5532  df-cnv 5533  df-co 5534  df-dm 5535  df-res 5537  df-iota 6295  df-fun 6338  df-fn 6339  df-fv 6344  df-aiota 44009  df-dfat 44044  df-afv 44045 This theorem is referenced by:  fnopafvb  44080  funbrafvb  44081  dfafn5a  44085
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