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Theorem fnbrafvb 41761
Description: Equivalence of function value and binary relation, analogous to fnbrfvb 6466. (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 6211 . . . . . 6 (𝐹 Fn 𝐴 → dom 𝐹 = 𝐴)
2 eleq2 2885 . . . . . . . 8 (𝐴 = dom 𝐹 → (𝐵𝐴𝐵 ∈ dom 𝐹))
32eqcoms 2825 . . . . . . 7 (dom 𝐹 = 𝐴 → (𝐵𝐴𝐵 ∈ dom 𝐹))
43biimpd 220 . . . . . 6 (dom 𝐹 = 𝐴 → (𝐵𝐴𝐵 ∈ dom 𝐹))
51, 4syl 17 . . . . 5 (𝐹 Fn 𝐴 → (𝐵𝐴𝐵 ∈ dom 𝐹))
65imp 395 . . . 4 ((𝐹 Fn 𝐴𝐵𝐴) → 𝐵 ∈ dom 𝐹)
7 snssi 4540 . . . . . . 7 (𝐵𝐴 → {𝐵} ⊆ 𝐴)
87adantl 469 . . . . . 6 ((𝐹 Fn 𝐴𝐵𝐴) → {𝐵} ⊆ 𝐴)
9 fnssresb 6224 . . . . . . 7 (𝐹 Fn 𝐴 → ((𝐹 ↾ {𝐵}) Fn {𝐵} ↔ {𝐵} ⊆ 𝐴))
109adantr 468 . . . . . 6 ((𝐹 Fn 𝐴𝐵𝐴) → ((𝐹 ↾ {𝐵}) Fn {𝐵} ↔ {𝐵} ⊆ 𝐴))
118, 10mpbird 248 . . . . 5 ((𝐹 Fn 𝐴𝐵𝐴) → (𝐹 ↾ {𝐵}) Fn {𝐵})
12 fnfun 6209 . . . . 5 ((𝐹 ↾ {𝐵}) Fn {𝐵} → Fun (𝐹 ↾ {𝐵}))
1311, 12syl 17 . . . 4 ((𝐹 Fn 𝐴𝐵𝐴) → Fun (𝐹 ↾ {𝐵}))
14 df-dfat 41726 . . . . 5 (𝐹 defAt 𝐵 ↔ (𝐵 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐵})))
15 afvfundmfveq 41745 . . . . 5 (𝐹 defAt 𝐵 → (𝐹'''𝐵) = (𝐹𝐵))
1614, 15sylbir 226 . . . 4 ((𝐵 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐵})) → (𝐹'''𝐵) = (𝐹𝐵))
176, 13, 16syl2anc 575 . . 3 ((𝐹 Fn 𝐴𝐵𝐴) → (𝐹'''𝐵) = (𝐹𝐵))
1817eqeq1d 2819 . 2 ((𝐹 Fn 𝐴𝐵𝐴) → ((𝐹'''𝐵) = 𝐶 ↔ (𝐹𝐵) = 𝐶))
19 fnbrfvb 6466 . 2 ((𝐹 Fn 𝐴𝐵𝐴) → ((𝐹𝐵) = 𝐶𝐵𝐹𝐶))
2018, 19bitrd 270 1 ((𝐹 Fn 𝐴𝐵𝐴) → ((𝐹'''𝐵) = 𝐶𝐵𝐹𝐶))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 197  wa 384   = wceq 1637  wcel 2157  wss 3780  {csn 4381   class class class wbr 4855  dom cdm 5324  cres 5326  Fun wfun 6105   Fn wfn 6106  cfv 6111   defAt wdfat 41723  '''cafv 41724
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2069  ax-7 2105  ax-8 2159  ax-9 2166  ax-10 2186  ax-11 2202  ax-12 2215  ax-13 2422  ax-ext 2795  ax-sep 4988  ax-nul 4996  ax-pow 5048  ax-pr 5109
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-3an 1102  df-tru 1641  df-fal 1651  df-ex 1860  df-nf 1864  df-sb 2062  df-mo 2635  df-eu 2642  df-clab 2804  df-cleq 2810  df-clel 2813  df-nfc 2948  df-ne 2990  df-ral 3112  df-rex 3113  df-rab 3116  df-v 3404  df-sbc 3645  df-csb 3740  df-dif 3783  df-un 3785  df-in 3787  df-ss 3794  df-nul 4128  df-if 4291  df-sn 4382  df-pr 4384  df-op 4388  df-uni 4642  df-int 4681  df-br 4856  df-opab 4918  df-id 5232  df-xp 5330  df-rel 5331  df-cnv 5332  df-co 5333  df-dm 5334  df-res 5336  df-iota 6074  df-fun 6113  df-fn 6114  df-fv 6119  df-aiota 41687  df-dfat 41726  df-afv 41727
This theorem is referenced by:  fnopafvb  41762  funbrafvb  41763  dfafn5a  41767
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