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Theorem tz6.12-1 6567
Description: Function value. Theorem 6.12(1) of [TakeutiZaring] p. 27. (Contributed by NM, 30-Apr-2004.)
Assertion
Ref Expression
tz6.12-1 ((𝐴𝐹𝑦 ∧ ∃!𝑦 𝐴𝐹𝑦) → (𝐹𝐴) = 𝑦)
Distinct variable groups:   𝑦,𝐹   𝑦,𝐴

Proof of Theorem tz6.12-1
StepHypRef Expression
1 df-fv 6240 . 2 (𝐹𝐴) = (℩𝑦𝐴𝐹𝑦)
2 iota1 6210 . . 3 (∃!𝑦 𝐴𝐹𝑦 → (𝐴𝐹𝑦 ↔ (℩𝑦𝐴𝐹𝑦) = 𝑦))
32biimpac 479 . 2 ((𝐴𝐹𝑦 ∧ ∃!𝑦 𝐴𝐹𝑦) → (℩𝑦𝐴𝐹𝑦) = 𝑦)
41, 3syl5eq 2845 1 ((𝐴𝐹𝑦 ∧ ∃!𝑦 𝐴𝐹𝑦) → (𝐹𝐴) = 𝑦)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1525  ∃!weu 2613   class class class wbr 4968  cio 6194  cfv 6232
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1781  ax-4 1795  ax-5 1892  ax-6 1951  ax-7 1996  ax-8 2085  ax-9 2093  ax-10 2114  ax-11 2128  ax-12 2143  ax-ext 2771
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-tru 1528  df-ex 1766  df-nf 1770  df-sb 2045  df-mo 2578  df-eu 2614  df-clab 2778  df-cleq 2790  df-clel 2865  df-nfc 2937  df-rex 3113  df-v 3442  df-sbc 3712  df-un 3870  df-sn 4479  df-pr 4481  df-uni 4752  df-iota 6196  df-fv 6240
This theorem is referenced by:  tz6.12  6568  tz6.12c  6570  funbrfv  6591  setrec2lem2  44299
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