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Theorem fnbrafv2b 43597
Description: Equivalence of function value and binary relation, analogous to fnbrfvb 6691. (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 2821 . . . 4 (𝐹''''𝐵) = (𝐹''''𝐵)
2 fundmdfat 43478 . . . . . . 7 ((Fun 𝐹𝐵 ∈ dom 𝐹) → 𝐹 defAt 𝐵)
32funfni 6430 . . . . . 6 ((𝐹 Fn 𝐴𝐵𝐴) → 𝐹 defAt 𝐵)
4 dfatafv2ex 43562 . . . . . 6 (𝐹 defAt 𝐵 → (𝐹''''𝐵) ∈ V)
53, 4syl 17 . . . . 5 ((𝐹 Fn 𝐴𝐵𝐴) → (𝐹''''𝐵) ∈ V)
6 eqeq2 2833 . . . . . . 7 (𝑥 = (𝐹''''𝐵) → ((𝐹''''𝐵) = 𝑥 ↔ (𝐹''''𝐵) = (𝐹''''𝐵)))
7 breq2 5043 . . . . . . 7 (𝑥 = (𝐹''''𝐵) → (𝐵𝐹𝑥𝐵𝐹(𝐹''''𝐵)))
86, 7bibi12d 349 . . . . . 6 (𝑥 = (𝐹''''𝐵) → (((𝐹''''𝐵) = 𝑥𝐵𝐹𝑥) ↔ ((𝐹''''𝐵) = (𝐹''''𝐵) ↔ 𝐵𝐹(𝐹''''𝐵))))
98adantl 485 . . . . 5 (((𝐹 Fn 𝐴𝐵𝐴) ∧ 𝑥 = (𝐹''''𝐵)) → (((𝐹''''𝐵) = 𝑥𝐵𝐹𝑥) ↔ ((𝐹''''𝐵) = (𝐹''''𝐵) ↔ 𝐵𝐹(𝐹''''𝐵))))
10 fneu 6434 . . . . . 6 ((𝐹 Fn 𝐴𝐵𝐴) → ∃!𝑥 𝐵𝐹𝑥)
11 tz6.12c-afv2 43591 . . . . . 6 (∃!𝑥 𝐵𝐹𝑥 → ((𝐹''''𝐵) = 𝑥𝐵𝐹𝑥))
1210, 11syl 17 . . . . 5 ((𝐹 Fn 𝐴𝐵𝐴) → ((𝐹''''𝐵) = 𝑥𝐵𝐹𝑥))
135, 9, 12vtocld 3533 . . . 4 ((𝐹 Fn 𝐴𝐵𝐴) → ((𝐹''''𝐵) = (𝐹''''𝐵) ↔ 𝐵𝐹(𝐹''''𝐵)))
141, 13mpbii 236 . . 3 ((𝐹 Fn 𝐴𝐵𝐴) → 𝐵𝐹(𝐹''''𝐵))
15 breq2 5043 . . 3 ((𝐹''''𝐵) = 𝐶 → (𝐵𝐹(𝐹''''𝐵) ↔ 𝐵𝐹𝐶))
1614, 15syl5ibcom 248 . 2 ((𝐹 Fn 𝐴𝐵𝐴) → ((𝐹''''𝐵) = 𝐶𝐵𝐹𝐶))
17 fnfun 6426 . . . 4 (𝐹 Fn 𝐴 → Fun 𝐹)
18 funbrafv2 43596 . . . 4 (Fun 𝐹 → (𝐵𝐹𝐶 → (𝐹''''𝐵) = 𝐶))
1917, 18syl 17 . . 3 (𝐹 Fn 𝐴 → (𝐵𝐹𝐶 → (𝐹''''𝐵) = 𝐶))
2019adantr 484 . 2 ((𝐹 Fn 𝐴𝐵𝐴) → (𝐵𝐹𝐶 → (𝐹''''𝐵) = 𝐶))
2116, 20impbid 215 1 ((𝐹 Fn 𝐴𝐵𝐴) → ((𝐹''''𝐵) = 𝐶𝐵𝐹𝐶))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399   = wceq 1538  wcel 2115  ∃!weu 2653  Vcvv 3471   class class class wbr 5039  Fun wfun 6322   Fn wfn 6323   defAt wdfat 43465  ''''cafv2 43557
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 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2178  ax-ext 2793  ax-sep 5176  ax-nul 5183  ax-pow 5239  ax-pr 5303
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2623  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2892  df-nfc 2960  df-ral 3131  df-rex 3132  df-rab 3135  df-v 3473  df-sbc 3750  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4267  df-if 4441  df-sn 4541  df-pr 4543  df-op 4547  df-uni 4812  df-br 5040  df-opab 5102  df-id 5433  df-xp 5534  df-rel 5535  df-cnv 5536  df-co 5537  df-dm 5538  df-res 5540  df-iota 6287  df-fun 6330  df-fn 6331  df-dfat 43468  df-afv2 43558
This theorem is referenced by:  fnopafv2b  43598  funbrafv22b  43599
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