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Theorem funbrafv2 45955
Description: The second argument of a binary relation on a function is the function's value, analogous to funbrfv 6943. (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 6566 . . . 4 (Fun 𝐹 → Rel 𝐹)
2 brrelex2 5731 . . . 4 ((Rel 𝐹𝐴𝐹𝐵) → 𝐵 ∈ V)
31, 2sylan 581 . . 3 ((Fun 𝐹𝐴𝐹𝐵) → 𝐵 ∈ V)
4 breq2 5153 . . . . . 6 (𝑥 = 𝐵 → (𝐴𝐹𝑥𝐴𝐹𝐵))
54anbi2d 630 . . . . 5 (𝑥 = 𝐵 → ((Fun 𝐹𝐴𝐹𝑥) ↔ (Fun 𝐹𝐴𝐹𝐵)))
6 eqeq2 2745 . . . . 5 (𝑥 = 𝐵 → ((𝐹''''𝐴) = 𝑥 ↔ (𝐹''''𝐴) = 𝐵))
75, 6imbi12d 345 . . . 4 (𝑥 = 𝐵 → (((Fun 𝐹𝐴𝐹𝑥) → (𝐹''''𝐴) = 𝑥) ↔ ((Fun 𝐹𝐴𝐹𝐵) → (𝐹''''𝐴) = 𝐵)))
8 funeu 6574 . . . . . 6 ((Fun 𝐹𝐴𝐹𝑥) → ∃!𝑥 𝐴𝐹𝑥)
9 tz6.12-1-afv2 45949 . . . . . 6 ((𝐴𝐹𝑥 ∧ ∃!𝑥 𝐴𝐹𝑥) → (𝐹''''𝐴) = 𝑥)
108, 9sylan2 594 . . . . 5 ((𝐴𝐹𝑥 ∧ (Fun 𝐹𝐴𝐹𝑥)) → (𝐹''''𝐴) = 𝑥)
1110anabss7 672 . . . 4 ((Fun 𝐹𝐴𝐹𝑥) → (𝐹''''𝐴) = 𝑥)
127, 11vtoclg 3557 . . 3 (𝐵 ∈ V → ((Fun 𝐹𝐴𝐹𝐵) → (𝐹''''𝐴) = 𝐵))
133, 12mpcom 38 . 2 ((Fun 𝐹𝐴𝐹𝐵) → (𝐹''''𝐴) = 𝐵)
1413ex 414 1 (Fun 𝐹 → (𝐴𝐹𝐵 → (𝐹''''𝐴) = 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397   = wceq 1542  wcel 2107  ∃!weu 2563  Vcvv 3475   class class class wbr 5149  Rel wrel 5682  Fun wfun 6538  ''''cafv2 45916
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-opab 5212  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-res 5689  df-iota 6496  df-fun 6546  df-fn 6547  df-dfat 45827  df-afv2 45917
This theorem is referenced by:  fnbrafv2b  45956
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