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Theorem funressnbrafv2 45466
Description: The second argument of a binary relation on a function is the function's value, analogous to funbrfv 6893. (Contributed by AV, 7-Sep-2022.)
Assertion
Ref Expression
funressnbrafv2 (((𝐴𝑉𝐵𝑊) ∧ Fun (𝐹 ↾ {𝐴})) → (𝐴𝐹𝐵 → (𝐹''''𝐴) = 𝐵))

Proof of Theorem funressnbrafv2
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 simpllr 774 . . 3 ((((𝐴𝑉𝐵𝑊) ∧ Fun (𝐹 ↾ {𝐴})) ∧ 𝐴𝐹𝐵) → 𝐵𝑊)
2 eleq1 2825 . . . . . . . 8 (𝑥 = 𝐵 → (𝑥𝑊𝐵𝑊))
32anbi2d 629 . . . . . . 7 (𝑥 = 𝐵 → ((𝐴𝑉𝑥𝑊) ↔ (𝐴𝑉𝐵𝑊)))
43anbi1d 630 . . . . . 6 (𝑥 = 𝐵 → (((𝐴𝑉𝑥𝑊) ∧ Fun (𝐹 ↾ {𝐴})) ↔ ((𝐴𝑉𝐵𝑊) ∧ Fun (𝐹 ↾ {𝐴}))))
5 breq2 5109 . . . . . 6 (𝑥 = 𝐵 → (𝐴𝐹𝑥𝐴𝐹𝐵))
64, 5anbi12d 631 . . . . 5 (𝑥 = 𝐵 → ((((𝐴𝑉𝑥𝑊) ∧ Fun (𝐹 ↾ {𝐴})) ∧ 𝐴𝐹𝑥) ↔ (((𝐴𝑉𝐵𝑊) ∧ Fun (𝐹 ↾ {𝐴})) ∧ 𝐴𝐹𝐵)))
7 eqeq2 2748 . . . . 5 (𝑥 = 𝐵 → ((𝐹''''𝐴) = 𝑥 ↔ (𝐹''''𝐴) = 𝐵))
86, 7imbi12d 344 . . . 4 (𝑥 = 𝐵 → (((((𝐴𝑉𝑥𝑊) ∧ Fun (𝐹 ↾ {𝐴})) ∧ 𝐴𝐹𝑥) → (𝐹''''𝐴) = 𝑥) ↔ ((((𝐴𝑉𝐵𝑊) ∧ Fun (𝐹 ↾ {𝐴})) ∧ 𝐴𝐹𝐵) → (𝐹''''𝐴) = 𝐵)))
9 id 22 . . . . 5 (𝐴𝐹𝑥𝐴𝐹𝑥)
10 funressneu 45271 . . . . . 6 (((𝐴𝑉𝑥𝑊) ∧ Fun (𝐹 ↾ {𝐴}) ∧ 𝐴𝐹𝑥) → ∃!𝑥 𝐴𝐹𝑥)
11103expa 1118 . . . . 5 ((((𝐴𝑉𝑥𝑊) ∧ Fun (𝐹 ↾ {𝐴})) ∧ 𝐴𝐹𝑥) → ∃!𝑥 𝐴𝐹𝑥)
12 tz6.12-1-afv2 45463 . . . . 5 ((𝐴𝐹𝑥 ∧ ∃!𝑥 𝐴𝐹𝑥) → (𝐹''''𝐴) = 𝑥)
139, 11, 12syl2an2 684 . . . 4 ((((𝐴𝑉𝑥𝑊) ∧ Fun (𝐹 ↾ {𝐴})) ∧ 𝐴𝐹𝑥) → (𝐹''''𝐴) = 𝑥)
148, 13vtoclg 3525 . . 3 (𝐵𝑊 → ((((𝐴𝑉𝐵𝑊) ∧ Fun (𝐹 ↾ {𝐴})) ∧ 𝐴𝐹𝐵) → (𝐹''''𝐴) = 𝐵))
151, 14mpcom 38 . 2 ((((𝐴𝑉𝐵𝑊) ∧ Fun (𝐹 ↾ {𝐴})) ∧ 𝐴𝐹𝐵) → (𝐹''''𝐴) = 𝐵)
1615ex 413 1 (((𝐴𝑉𝐵𝑊) ∧ Fun (𝐹 ↾ {𝐴})) → (𝐴𝐹𝐵 → (𝐹''''𝐴) = 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1541  wcel 2106  ∃!weu 2566  {csn 4586   class class class wbr 5105  cres 5635  Fun wfun 6490  ''''cafv2 45430
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 2707  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384
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 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ral 3065  df-rex 3074  df-rab 3408  df-v 3447  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4283  df-if 4487  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-br 5106  df-opab 5168  df-id 5531  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-res 5645  df-iota 6448  df-fun 6498  df-fn 6499  df-dfat 45341  df-afv2 45431
This theorem is referenced by:  dfatbrafv2b  45467
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