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Theorem fnbrafvb 47103
Description: Equivalence of function value and binary relation, analogous to fnbrfvb 6959. (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 6671 . . . . . 6 (𝐹 Fn 𝐴 → dom 𝐹 = 𝐴)
2 eleq2 2827 . . . . . . . 8 (𝐴 = dom 𝐹 → (𝐵𝐴𝐵 ∈ dom 𝐹))
32eqcoms 2742 . . . . . . 7 (dom 𝐹 = 𝐴 → (𝐵𝐴𝐵 ∈ dom 𝐹))
43biimpd 229 . . . . . 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 4812 . . . . . . 7 (𝐵𝐴 → {𝐵} ⊆ 𝐴)
87adantl 481 . . . . . 6 ((𝐹 Fn 𝐴𝐵𝐴) → {𝐵} ⊆ 𝐴)
9 fnssresb 6690 . . . . . . 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 6668 . . . . 5 ((𝐹 ↾ {𝐵}) Fn {𝐵} → Fun (𝐹 ↾ {𝐵}))
1311, 12syl 17 . . . 4 ((𝐹 Fn 𝐴𝐵𝐴) → Fun (𝐹 ↾ {𝐵}))
14 df-dfat 47068 . . . . 5 (𝐹 defAt 𝐵 ↔ (𝐵 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐵})))
15 afvfundmfveq 47087 . . . . 5 (𝐹 defAt 𝐵 → (𝐹'''𝐵) = (𝐹𝐵))
1614, 15sylbir 235 . . . 4 ((𝐵 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐵})) → (𝐹'''𝐵) = (𝐹𝐵))
176, 13, 16syl2anc 584 . . 3 ((𝐹 Fn 𝐴𝐵𝐴) → (𝐹'''𝐵) = (𝐹𝐵))
1817eqeq1d 2736 . 2 ((𝐹 Fn 𝐴𝐵𝐴) → ((𝐹'''𝐵) = 𝐶 ↔ (𝐹𝐵) = 𝐶))
19 fnbrfvb 6959 . 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 206  wa 395   = wceq 1536  wcel 2105  wss 3962  {csn 4630   class class class wbr 5147  dom cdm 5688  cres 5690  Fun wfun 6556   Fn wfn 6557  cfv 6562   defAt wdfat 47065  '''cafv 47066
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-sep 5301  ax-nul 5311  ax-pr 5437
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3433  df-v 3479  df-sbc 3791  df-csb 3908  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-nul 4339  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-int 4951  df-br 5148  df-opab 5210  df-id 5582  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-res 5700  df-iota 6515  df-fun 6564  df-fn 6565  df-fv 6570  df-aiota 47034  df-dfat 47068  df-afv 47069
This theorem is referenced by:  fnopafvb  47104  funbrafvb  47105  dfafn5a  47109
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