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Theorem funressnbrafv2 42913
 Description: The second argument of a binary relation on a function is the function's value, analogous to funbrfv 6576. (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 772 . . 3 ((((𝐴𝑉𝐵𝑊) ∧ Fun (𝐹 ↾ {𝐴})) ∧ 𝐴𝐹𝐵) → 𝐵𝑊)
2 eleq1 2868 . . . . . . . 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 4960 . . . . . 6 (𝑥 = 𝐵 → (𝐴𝐹𝑥𝐴𝐹𝐵))
64, 5anbi12d 630 . . . . 5 (𝑥 = 𝐵 → ((((𝐴𝑉𝑥𝑊) ∧ Fun (𝐹 ↾ {𝐴})) ∧ 𝐴𝐹𝑥) ↔ (((𝐴𝑉𝐵𝑊) ∧ Fun (𝐹 ↾ {𝐴})) ∧ 𝐴𝐹𝐵)))
7 eqeq2 2804 . . . . 5 (𝑥 = 𝐵 → ((𝐹''''𝐴) = 𝑥 ↔ (𝐹''''𝐴) = 𝐵))
86, 7imbi12d 346 . . . 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 42752 . . . . . 6 (((𝐴𝑉𝑥𝑊) ∧ Fun (𝐹 ↾ {𝐴}) ∧ 𝐴𝐹𝑥) → ∃!𝑥 𝐴𝐹𝑥)
11103expa 1109 . . . . 5 ((((𝐴𝑉𝑥𝑊) ∧ Fun (𝐹 ↾ {𝐴})) ∧ 𝐴𝐹𝑥) → ∃!𝑥 𝐴𝐹𝑥)
12 tz6.12-1-afv2 42910 . . . . 5 ((𝐴𝐹𝑥 ∧ ∃!𝑥 𝐴𝐹𝑥) → (𝐹''''𝐴) = 𝑥)
139, 11, 12syl2an2 682 . . . 4 ((((𝐴𝑉𝑥𝑊) ∧ Fun (𝐹 ↾ {𝐴})) ∧ 𝐴𝐹𝑥) → (𝐹''''𝐴) = 𝑥)
148, 13vtoclg 3505 . . 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 1520   ∈ wcel 2079  ∃!weu 2609  {csn 4466   class class class wbr 4956   ↾ cres 5437  Fun wfun 6211  ''''cafv2 42877 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1775  ax-4 1789  ax-5 1886  ax-6 1945  ax-7 1990  ax-8 2081  ax-9 2089  ax-10 2110  ax-11 2124  ax-12 2139  ax-13 2342  ax-ext 2767  ax-sep 5088  ax-nul 5095  ax-pow 5150  ax-pr 5214 This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3an 1080  df-tru 1523  df-ex 1760  df-nf 1764  df-sb 2041  df-mo 2574  df-eu 2610  df-clab 2774  df-cleq 2786  df-clel 2861  df-nfc 2933  df-ral 3108  df-rex 3109  df-rab 3112  df-v 3434  df-sbc 3702  df-dif 3857  df-un 3859  df-in 3861  df-ss 3869  df-nul 4207  df-if 4376  df-sn 4467  df-pr 4469  df-op 4473  df-uni 4740  df-br 4957  df-opab 5019  df-id 5340  df-xp 5441  df-rel 5442  df-cnv 5443  df-co 5444  df-dm 5445  df-res 5447  df-iota 6181  df-fun 6219  df-fn 6220  df-dfat 42788  df-afv2 42878 This theorem is referenced by:  dfatbrafv2b  42914
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