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Theorem fnbrafv2b 45470
Description: Equivalence of function value and binary relation, analogous to fnbrfvb 6895. (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 2736 . . . 4 (𝐹''''𝐵) = (𝐹''''𝐵)
2 fundmdfat 45351 . . . . . . 7 ((Fun 𝐹𝐵 ∈ dom 𝐹) → 𝐹 defAt 𝐵)
32funfni 6608 . . . . . 6 ((𝐹 Fn 𝐴𝐵𝐴) → 𝐹 defAt 𝐵)
4 dfatafv2ex 45435 . . . . . 6 (𝐹 defAt 𝐵 → (𝐹''''𝐵) ∈ V)
53, 4syl 17 . . . . 5 ((𝐹 Fn 𝐴𝐵𝐴) → (𝐹''''𝐵) ∈ V)
6 eqeq2 2748 . . . . . . 7 (𝑥 = (𝐹''''𝐵) → ((𝐹''''𝐵) = 𝑥 ↔ (𝐹''''𝐵) = (𝐹''''𝐵)))
7 breq2 5109 . . . . . . 7 (𝑥 = (𝐹''''𝐵) → (𝐵𝐹𝑥𝐵𝐹(𝐹''''𝐵)))
86, 7bibi12d 345 . . . . . 6 (𝑥 = (𝐹''''𝐵) → (((𝐹''''𝐵) = 𝑥𝐵𝐹𝑥) ↔ ((𝐹''''𝐵) = (𝐹''''𝐵) ↔ 𝐵𝐹(𝐹''''𝐵))))
98adantl 482 . . . . 5 (((𝐹 Fn 𝐴𝐵𝐴) ∧ 𝑥 = (𝐹''''𝐵)) → (((𝐹''''𝐵) = 𝑥𝐵𝐹𝑥) ↔ ((𝐹''''𝐵) = (𝐹''''𝐵) ↔ 𝐵𝐹(𝐹''''𝐵))))
10 fneu 6612 . . . . . 6 ((𝐹 Fn 𝐴𝐵𝐴) → ∃!𝑥 𝐵𝐹𝑥)
11 tz6.12c-afv2 45464 . . . . . 6 (∃!𝑥 𝐵𝐹𝑥 → ((𝐹''''𝐵) = 𝑥𝐵𝐹𝑥))
1210, 11syl 17 . . . . 5 ((𝐹 Fn 𝐴𝐵𝐴) → ((𝐹''''𝐵) = 𝑥𝐵𝐹𝑥))
135, 9, 12vtocld 3511 . . . 4 ((𝐹 Fn 𝐴𝐵𝐴) → ((𝐹''''𝐵) = (𝐹''''𝐵) ↔ 𝐵𝐹(𝐹''''𝐵)))
141, 13mpbii 232 . . 3 ((𝐹 Fn 𝐴𝐵𝐴) → 𝐵𝐹(𝐹''''𝐵))
15 breq2 5109 . . 3 ((𝐹''''𝐵) = 𝐶 → (𝐵𝐹(𝐹''''𝐵) ↔ 𝐵𝐹𝐶))
1614, 15syl5ibcom 244 . 2 ((𝐹 Fn 𝐴𝐵𝐴) → ((𝐹''''𝐵) = 𝐶𝐵𝐹𝐶))
17 fnfun 6602 . . . 4 (𝐹 Fn 𝐴 → Fun 𝐹)
18 funbrafv2 45469 . . . 4 (Fun 𝐹 → (𝐵𝐹𝐶 → (𝐹''''𝐵) = 𝐶))
1917, 18syl 17 . . 3 (𝐹 Fn 𝐴 → (𝐵𝐹𝐶 → (𝐹''''𝐵) = 𝐶))
2019adantr 481 . 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 396   = wceq 1541  wcel 2106  ∃!weu 2566  Vcvv 3445   class class class wbr 5105  Fun wfun 6490   Fn wfn 6491   defAt wdfat 45338  ''''cafv2 45430
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-ral 3065  df-rex 3074  df-rab 3408  df-v 3447  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4283  df-if 4487  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-br 5106  df-opab 5168  df-id 5531  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-res 5645  df-iota 6448  df-fun 6498  df-fn 6499  df-dfat 45341  df-afv2 45431
This theorem is referenced by:  fnopafv2b  45471  funbrafv22b  45472
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