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Theorem funressnbrafv2 43517
Description: The second argument of a binary relation on a function is the function's value, analogous to funbrfv 6713. (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 2899 . . . . . . . 8 (𝑥 = 𝐵 → (𝑥𝑊𝐵𝑊))
32anbi2d 630 . . . . . . 7 (𝑥 = 𝐵 → ((𝐴𝑉𝑥𝑊) ↔ (𝐴𝑉𝐵𝑊)))
43anbi1d 631 . . . . . 6 (𝑥 = 𝐵 → (((𝐴𝑉𝑥𝑊) ∧ Fun (𝐹 ↾ {𝐴})) ↔ ((𝐴𝑉𝐵𝑊) ∧ Fun (𝐹 ↾ {𝐴}))))
5 breq2 5067 . . . . . 6 (𝑥 = 𝐵 → (𝐴𝐹𝑥𝐴𝐹𝐵))
64, 5anbi12d 632 . . . . 5 (𝑥 = 𝐵 → ((((𝐴𝑉𝑥𝑊) ∧ Fun (𝐹 ↾ {𝐴})) ∧ 𝐴𝐹𝑥) ↔ (((𝐴𝑉𝐵𝑊) ∧ Fun (𝐹 ↾ {𝐴})) ∧ 𝐴𝐹𝐵)))
7 eqeq2 2832 . . . . 5 (𝑥 = 𝐵 → ((𝐹''''𝐴) = 𝑥 ↔ (𝐹''''𝐴) = 𝐵))
86, 7imbi12d 347 . . . 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 43356 . . . . . 6 (((𝐴𝑉𝑥𝑊) ∧ Fun (𝐹 ↾ {𝐴}) ∧ 𝐴𝐹𝑥) → ∃!𝑥 𝐴𝐹𝑥)
11103expa 1113 . . . . 5 ((((𝐴𝑉𝑥𝑊) ∧ Fun (𝐹 ↾ {𝐴})) ∧ 𝐴𝐹𝑥) → ∃!𝑥 𝐴𝐹𝑥)
12 tz6.12-1-afv2 43514 . . . . 5 ((𝐴𝐹𝑥 ∧ ∃!𝑥 𝐴𝐹𝑥) → (𝐹''''𝐴) = 𝑥)
139, 11, 12syl2an2 684 . . . 4 ((((𝐴𝑉𝑥𝑊) ∧ Fun (𝐹 ↾ {𝐴})) ∧ 𝐴𝐹𝑥) → (𝐹''''𝐴) = 𝑥)
148, 13vtoclg 3566 . . 3 (𝐵𝑊 → ((((𝐴𝑉𝐵𝑊) ∧ Fun (𝐹 ↾ {𝐴})) ∧ 𝐴𝐹𝐵) → (𝐹''''𝐴) = 𝐵))
151, 14mpcom 38 . 2 ((((𝐴𝑉𝐵𝑊) ∧ Fun (𝐹 ↾ {𝐴})) ∧ 𝐴𝐹𝐵) → (𝐹''''𝐴) = 𝐵)
1615ex 415 1 (((𝐴𝑉𝐵𝑊) ∧ Fun (𝐹 ↾ {𝐴})) → (𝐴𝐹𝐵 → (𝐹''''𝐴) = 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 398   = wceq 1536  wcel 2113  ∃!weu 2652  {csn 4564   class class class wbr 5063  cres 5554  Fun wfun 6346  ''''cafv2 43481
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1969  ax-7 2014  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2160  ax-12 2176  ax-ext 2792  ax-sep 5200  ax-nul 5207  ax-pow 5263  ax-pr 5327
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1084  df-tru 1539  df-ex 1780  df-nf 1784  df-sb 2069  df-mo 2621  df-eu 2653  df-clab 2799  df-cleq 2813  df-clel 2892  df-nfc 2962  df-ral 3142  df-rex 3143  df-rab 3146  df-v 3495  df-sbc 3771  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4465  df-sn 4565  df-pr 4567  df-op 4571  df-uni 4836  df-br 5064  df-opab 5126  df-id 5457  df-xp 5558  df-rel 5559  df-cnv 5560  df-co 5561  df-dm 5562  df-res 5564  df-iota 6311  df-fun 6354  df-fn 6355  df-dfat 43392  df-afv2 43482
This theorem is referenced by:  dfatbrafv2b  43518
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