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Theorem funbrafv2 45941
Description: The second argument of a binary relation on a function is the function's value, analogous to funbrfv 6939. (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 6562 . . . 4 (Fun 𝐹 → Rel 𝐹)
2 brrelex2 5728 . . . 4 ((Rel 𝐹𝐴𝐹𝐵) → 𝐵 ∈ V)
31, 2sylan 580 . . 3 ((Fun 𝐹𝐴𝐹𝐵) → 𝐵 ∈ V)
4 breq2 5151 . . . . . 6 (𝑥 = 𝐵 → (𝐴𝐹𝑥𝐴𝐹𝐵))
54anbi2d 629 . . . . 5 (𝑥 = 𝐵 → ((Fun 𝐹𝐴𝐹𝑥) ↔ (Fun 𝐹𝐴𝐹𝐵)))
6 eqeq2 2744 . . . . 5 (𝑥 = 𝐵 → ((𝐹''''𝐴) = 𝑥 ↔ (𝐹''''𝐴) = 𝐵))
75, 6imbi12d 344 . . . 4 (𝑥 = 𝐵 → (((Fun 𝐹𝐴𝐹𝑥) → (𝐹''''𝐴) = 𝑥) ↔ ((Fun 𝐹𝐴𝐹𝐵) → (𝐹''''𝐴) = 𝐵)))
8 funeu 6570 . . . . . 6 ((Fun 𝐹𝐴𝐹𝑥) → ∃!𝑥 𝐴𝐹𝑥)
9 tz6.12-1-afv2 45935 . . . . . 6 ((𝐴𝐹𝑥 ∧ ∃!𝑥 𝐴𝐹𝑥) → (𝐹''''𝐴) = 𝑥)
108, 9sylan2 593 . . . . 5 ((𝐴𝐹𝑥 ∧ (Fun 𝐹𝐴𝐹𝑥)) → (𝐹''''𝐴) = 𝑥)
1110anabss7 671 . . . 4 ((Fun 𝐹𝐴𝐹𝑥) → (𝐹''''𝐴) = 𝑥)
127, 11vtoclg 3556 . . 3 (𝐵 ∈ V → ((Fun 𝐹𝐴𝐹𝐵) → (𝐹''''𝐴) = 𝐵))
133, 12mpcom 38 . 2 ((Fun 𝐹𝐴𝐹𝐵) → (𝐹''''𝐴) = 𝐵)
1413ex 413 1 (Fun 𝐹 → (𝐴𝐹𝐵 → (𝐹''''𝐴) = 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1541  wcel 2106  ∃!weu 2562  Vcvv 3474   class class class wbr 5147  Rel wrel 5680  Fun wfun 6534  ''''cafv2 45902
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 2703  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426
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 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-br 5148  df-opab 5210  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-res 5687  df-iota 6492  df-fun 6542  df-fn 6543  df-dfat 45813  df-afv2 45903
This theorem is referenced by:  fnbrafv2b  45942
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