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Theorem funressnbrafv2 46437
Description: The second argument of a binary relation on a function is the function's value, analogous to funbrfv 6932. (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 773 . . 3 ((((𝐴𝑉𝐵𝑊) ∧ Fun (𝐹 ↾ {𝐴})) ∧ 𝐴𝐹𝐵) → 𝐵𝑊)
2 eleq1 2813 . . . . . . . 8 (𝑥 = 𝐵 → (𝑥𝑊𝐵𝑊))
32anbi2d 628 . . . . . . 7 (𝑥 = 𝐵 → ((𝐴𝑉𝑥𝑊) ↔ (𝐴𝑉𝐵𝑊)))
43anbi1d 629 . . . . . 6 (𝑥 = 𝐵 → (((𝐴𝑉𝑥𝑊) ∧ Fun (𝐹 ↾ {𝐴})) ↔ ((𝐴𝑉𝐵𝑊) ∧ Fun (𝐹 ↾ {𝐴}))))
5 breq2 5142 . . . . . 6 (𝑥 = 𝐵 → (𝐴𝐹𝑥𝐴𝐹𝐵))
64, 5anbi12d 630 . . . . 5 (𝑥 = 𝐵 → ((((𝐴𝑉𝑥𝑊) ∧ Fun (𝐹 ↾ {𝐴})) ∧ 𝐴𝐹𝑥) ↔ (((𝐴𝑉𝐵𝑊) ∧ Fun (𝐹 ↾ {𝐴})) ∧ 𝐴𝐹𝐵)))
7 eqeq2 2736 . . . . 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 46242 . . . . . 6 (((𝐴𝑉𝑥𝑊) ∧ Fun (𝐹 ↾ {𝐴}) ∧ 𝐴𝐹𝑥) → ∃!𝑥 𝐴𝐹𝑥)
11103expa 1115 . . . . 5 ((((𝐴𝑉𝑥𝑊) ∧ Fun (𝐹 ↾ {𝐴})) ∧ 𝐴𝐹𝑥) → ∃!𝑥 𝐴𝐹𝑥)
12 tz6.12-1-afv2 46434 . . . . 5 ((𝐴𝐹𝑥 ∧ ∃!𝑥 𝐴𝐹𝑥) → (𝐹''''𝐴) = 𝑥)
139, 11, 12syl2an2 683 . . . 4 ((((𝐴𝑉𝑥𝑊) ∧ Fun (𝐹 ↾ {𝐴})) ∧ 𝐴𝐹𝑥) → (𝐹''''𝐴) = 𝑥)
148, 13vtoclg 3535 . . 3 (𝐵𝑊 → ((((𝐴𝑉𝐵𝑊) ∧ Fun (𝐹 ↾ {𝐴})) ∧ 𝐴𝐹𝐵) → (𝐹''''𝐴) = 𝐵))
151, 14mpcom 38 . 2 ((((𝐴𝑉𝐵𝑊) ∧ Fun (𝐹 ↾ {𝐴})) ∧ 𝐴𝐹𝐵) → (𝐹''''𝐴) = 𝐵)
1615ex 412 1 (((𝐴𝑉𝐵𝑊) ∧ Fun (𝐹 ↾ {𝐴})) → (𝐴𝐹𝐵 → (𝐹''''𝐴) = 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1533  wcel 2098  ∃!weu 2554  {csn 4620   class class class wbr 5138  cres 5668  Fun wfun 6527  ''''cafv2 46401
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 2695  ax-sep 5289  ax-nul 5296  ax-pow 5353  ax-pr 5417
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 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ral 3054  df-rex 3063  df-rab 3425  df-v 3468  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-nul 4315  df-if 4521  df-sn 4621  df-pr 4623  df-op 4627  df-uni 4900  df-br 5139  df-opab 5201  df-id 5564  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-res 5678  df-iota 6485  df-fun 6535  df-fn 6536  df-dfat 46312  df-afv2 46402
This theorem is referenced by:  dfatbrafv2b  46438
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