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Theorem funbrafv2 43800
Description: The second argument of a binary relation on a function is the function's value, analogous to funbrfv 6695. (Contributed by AV, 6-Sep-2022.)
Assertion
Ref Expression
funbrafv2 (Fun 𝐹 → (𝐴𝐹𝐵 → (𝐹''''𝐴) = 𝐵))

Proof of Theorem funbrafv2
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 funrel 6345 . . . 4 (Fun 𝐹 → Rel 𝐹)
2 brrelex2 5574 . . . 4 ((Rel 𝐹𝐴𝐹𝐵) → 𝐵 ∈ V)
31, 2sylan 583 . . 3 ((Fun 𝐹𝐴𝐹𝐵) → 𝐵 ∈ V)
4 breq2 5037 . . . . . 6 (𝑥 = 𝐵 → (𝐴𝐹𝑥𝐴𝐹𝐵))
54anbi2d 631 . . . . 5 (𝑥 = 𝐵 → ((Fun 𝐹𝐴𝐹𝑥) ↔ (Fun 𝐹𝐴𝐹𝐵)))
6 eqeq2 2813 . . . . 5 (𝑥 = 𝐵 → ((𝐹''''𝐴) = 𝑥 ↔ (𝐹''''𝐴) = 𝐵))
75, 6imbi12d 348 . . . 4 (𝑥 = 𝐵 → (((Fun 𝐹𝐴𝐹𝑥) → (𝐹''''𝐴) = 𝑥) ↔ ((Fun 𝐹𝐴𝐹𝐵) → (𝐹''''𝐴) = 𝐵)))
8 funeu 6353 . . . . . 6 ((Fun 𝐹𝐴𝐹𝑥) → ∃!𝑥 𝐴𝐹𝑥)
9 tz6.12-1-afv2 43794 . . . . . 6 ((𝐴𝐹𝑥 ∧ ∃!𝑥 𝐴𝐹𝑥) → (𝐹''''𝐴) = 𝑥)
108, 9sylan2 595 . . . . 5 ((𝐴𝐹𝑥 ∧ (Fun 𝐹𝐴𝐹𝑥)) → (𝐹''''𝐴) = 𝑥)
1110anabss7 672 . . . 4 ((Fun 𝐹𝐴𝐹𝑥) → (𝐹''''𝐴) = 𝑥)
127, 11vtoclg 3518 . . 3 (𝐵 ∈ V → ((Fun 𝐹𝐴𝐹𝐵) → (𝐹''''𝐴) = 𝐵))
133, 12mpcom 38 . 2 ((Fun 𝐹𝐴𝐹𝐵) → (𝐹''''𝐴) = 𝐵)
1413ex 416 1 (Fun 𝐹 → (𝐴𝐹𝐵 → (𝐹''''𝐴) = 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399   = wceq 1538  wcel 2112  ∃!weu 2631  Vcvv 3444   class class class wbr 5033  Rel wrel 5528  Fun wfun 6322  ''''cafv2 43761
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 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298
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 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ral 3114  df-rex 3115  df-rab 3118  df-v 3446  df-sbc 3724  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4247  df-if 4429  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4804  df-br 5034  df-opab 5096  df-id 5428  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-res 5535  df-iota 6287  df-fun 6330  df-fn 6331  df-dfat 43672  df-afv2 43762
This theorem is referenced by:  fnbrafv2b  43801
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