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Theorem funbrafv2 46527
Description: The second argument of a binary relation on a function is the function's value, analogous to funbrfv 6936. (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 6559 . . . 4 (Fun 𝐹 → Rel 𝐹)
2 brrelex2 5723 . . . 4 ((Rel 𝐹𝐴𝐹𝐵) → 𝐵 ∈ V)
31, 2sylan 579 . . 3 ((Fun 𝐹𝐴𝐹𝐵) → 𝐵 ∈ V)
4 breq2 5145 . . . . . 6 (𝑥 = 𝐵 → (𝐴𝐹𝑥𝐴𝐹𝐵))
54anbi2d 628 . . . . 5 (𝑥 = 𝐵 → ((Fun 𝐹𝐴𝐹𝑥) ↔ (Fun 𝐹𝐴𝐹𝐵)))
6 eqeq2 2738 . . . . 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 6567 . . . . . 6 ((Fun 𝐹𝐴𝐹𝑥) → ∃!𝑥 𝐴𝐹𝑥)
9 tz6.12-1-afv2 46521 . . . . . 6 ((𝐴𝐹𝑥 ∧ ∃!𝑥 𝐴𝐹𝑥) → (𝐹''''𝐴) = 𝑥)
108, 9sylan2 592 . . . . 5 ((𝐴𝐹𝑥 ∧ (Fun 𝐹𝐴𝐹𝑥)) → (𝐹''''𝐴) = 𝑥)
1110anabss7 670 . . . 4 ((Fun 𝐹𝐴𝐹𝑥) → (𝐹''''𝐴) = 𝑥)
127, 11vtoclg 3537 . . 3 (𝐵 ∈ V → ((Fun 𝐹𝐴𝐹𝐵) → (𝐹''''𝐴) = 𝐵))
133, 12mpcom 38 . 2 ((Fun 𝐹𝐴𝐹𝐵) → (𝐹''''𝐴) = 𝐵)
1413ex 412 1 (Fun 𝐹 → (𝐴𝐹𝐵 → (𝐹''''𝐴) = 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1533  wcel 2098  ∃!weu 2556  Vcvv 3468   class class class wbr 5141  Rel wrel 5674  Fun wfun 6531  ''''cafv2 46488
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-br 5142  df-opab 5204  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-res 5681  df-iota 6489  df-fun 6539  df-fn 6540  df-dfat 46399  df-afv2 46489
This theorem is referenced by:  fnbrafv2b  46528
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