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Theorem fnbrafv2b 46691
Description: Equivalence of function value and binary relation, analogous to fnbrfvb 6945. (Contributed by AV, 6-Sep-2022.)
Assertion
Ref Expression
fnbrafv2b ((𝐹 Fn 𝐴𝐵𝐴) → ((𝐹''''𝐵) = 𝐶𝐵𝐹𝐶))

Proof of Theorem fnbrafv2b
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 eqid 2725 . . . 4 (𝐹''''𝐵) = (𝐹''''𝐵)
2 fundmdfat 46572 . . . . . . 7 ((Fun 𝐹𝐵 ∈ dom 𝐹) → 𝐹 defAt 𝐵)
32funfni 6655 . . . . . 6 ((𝐹 Fn 𝐴𝐵𝐴) → 𝐹 defAt 𝐵)
4 dfatafv2ex 46656 . . . . . 6 (𝐹 defAt 𝐵 → (𝐹''''𝐵) ∈ V)
53, 4syl 17 . . . . 5 ((𝐹 Fn 𝐴𝐵𝐴) → (𝐹''''𝐵) ∈ V)
6 eqeq2 2737 . . . . . . 7 (𝑥 = (𝐹''''𝐵) → ((𝐹''''𝐵) = 𝑥 ↔ (𝐹''''𝐵) = (𝐹''''𝐵)))
7 breq2 5147 . . . . . . 7 (𝑥 = (𝐹''''𝐵) → (𝐵𝐹𝑥𝐵𝐹(𝐹''''𝐵)))
86, 7bibi12d 344 . . . . . 6 (𝑥 = (𝐹''''𝐵) → (((𝐹''''𝐵) = 𝑥𝐵𝐹𝑥) ↔ ((𝐹''''𝐵) = (𝐹''''𝐵) ↔ 𝐵𝐹(𝐹''''𝐵))))
98adantl 480 . . . . 5 (((𝐹 Fn 𝐴𝐵𝐴) ∧ 𝑥 = (𝐹''''𝐵)) → (((𝐹''''𝐵) = 𝑥𝐵𝐹𝑥) ↔ ((𝐹''''𝐵) = (𝐹''''𝐵) ↔ 𝐵𝐹(𝐹''''𝐵))))
10 fneu 6659 . . . . . 6 ((𝐹 Fn 𝐴𝐵𝐴) → ∃!𝑥 𝐵𝐹𝑥)
11 tz6.12c-afv2 46685 . . . . . 6 (∃!𝑥 𝐵𝐹𝑥 → ((𝐹''''𝐵) = 𝑥𝐵𝐹𝑥))
1210, 11syl 17 . . . . 5 ((𝐹 Fn 𝐴𝐵𝐴) → ((𝐹''''𝐵) = 𝑥𝐵𝐹𝑥))
135, 9, 12vtocld 3537 . . . 4 ((𝐹 Fn 𝐴𝐵𝐴) → ((𝐹''''𝐵) = (𝐹''''𝐵) ↔ 𝐵𝐹(𝐹''''𝐵)))
141, 13mpbii 232 . . 3 ((𝐹 Fn 𝐴𝐵𝐴) → 𝐵𝐹(𝐹''''𝐵))
15 breq2 5147 . . 3 ((𝐹''''𝐵) = 𝐶 → (𝐵𝐹(𝐹''''𝐵) ↔ 𝐵𝐹𝐶))
1614, 15syl5ibcom 244 . 2 ((𝐹 Fn 𝐴𝐵𝐴) → ((𝐹''''𝐵) = 𝐶𝐵𝐹𝐶))
17 fnfun 6649 . . . 4 (𝐹 Fn 𝐴 → Fun 𝐹)
18 funbrafv2 46690 . . . 4 (Fun 𝐹 → (𝐵𝐹𝐶 → (𝐹''''𝐵) = 𝐶))
1917, 18syl 17 . . 3 (𝐹 Fn 𝐴 → (𝐵𝐹𝐶 → (𝐹''''𝐵) = 𝐶))
2019adantr 479 . 2 ((𝐹 Fn 𝐴𝐵𝐴) → (𝐵𝐹𝐶 → (𝐹''''𝐵) = 𝐶))
2116, 20impbid 211 1 ((𝐹 Fn 𝐴𝐵𝐴) → ((𝐹''''𝐵) = 𝐶𝐵𝐹𝐶))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 394   = wceq 1533  wcel 2098  ∃!weu 2556  Vcvv 3463   class class class wbr 5143  Fun wfun 6537   Fn wfn 6538   defAt wdfat 46559  ''''cafv2 46651
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 2166  ax-ext 2696  ax-sep 5294  ax-nul 5301  ax-pow 5359  ax-pr 5423
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  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 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2931  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3465  df-dif 3942  df-un 3944  df-in 3946  df-ss 3956  df-nul 4319  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-br 5144  df-opab 5206  df-id 5570  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-res 5684  df-iota 6495  df-fun 6545  df-fn 6546  df-dfat 46562  df-afv2 46652
This theorem is referenced by:  fnopafv2b  46692  funbrafv22b  46693
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