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Theorem tz6.12-1 6208
 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 5894 . 2 (𝐹𝐴) = (℩𝑦𝐴𝐹𝑦)
2 iota1 5863 . . 3 (∃!𝑦 𝐴𝐹𝑦 → (𝐴𝐹𝑦 ↔ (℩𝑦𝐴𝐹𝑦) = 𝑦))
32biimpac 503 . 2 ((𝐴𝐹𝑦 ∧ ∃!𝑦 𝐴𝐹𝑦) → (℩𝑦𝐴𝐹𝑦) = 𝑦)
41, 3syl5eq 2667 1 ((𝐴𝐹𝑦 ∧ ∃!𝑦 𝐴𝐹𝑦) → (𝐹𝐴) = 𝑦)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 384   = wceq 1482  ∃!weu 2469   class class class wbr 4651  ℩cio 5847  ‘cfv 5886 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1721  ax-4 1736  ax-5 1838  ax-6 1887  ax-7 1934  ax-9 1998  ax-10 2018  ax-11 2033  ax-12 2046  ax-13 2245  ax-ext 2601 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1485  df-ex 1704  df-nf 1709  df-sb 1880  df-eu 2473  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2752  df-rex 2917  df-v 3200  df-sbc 3434  df-un 3577  df-sn 4176  df-pr 4178  df-uni 4435  df-iota 5849  df-fv 5894 This theorem is referenced by:  tz6.12  6209  tz6.12c  6211  funbrfv  6232  setrec2lem2  42212
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