MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  tz6.12c Structured version   Visualization version   GIF version

Theorem tz6.12c 6923
Description: Corollary of Theorem 6.12(1) of [TakeutiZaring] p. 27. (Contributed by NM, 30-Apr-2004.) (Proof shortened by SN, 23-Dec-2024.)
Assertion
Ref Expression
tz6.12c (∃!𝑦 𝐴𝐹𝑦 → ((𝐹𝐴) = 𝑦𝐴𝐹𝑦))
Distinct variable groups:   𝑦,𝐹   𝑦,𝐴

Proof of Theorem tz6.12c
StepHypRef Expression
1 df-fv 6562 . . 3 (𝐹𝐴) = (℩𝑦𝐴𝐹𝑦)
21eqeq1i 2731 . 2 ((𝐹𝐴) = 𝑦 ↔ (℩𝑦𝐴𝐹𝑦) = 𝑦)
3 iota1 6531 . 2 (∃!𝑦 𝐴𝐹𝑦 → (𝐴𝐹𝑦 ↔ (℩𝑦𝐴𝐹𝑦) = 𝑦))
42, 3bitr4id 289 1 (∃!𝑦 𝐴𝐹𝑦 → ((𝐹𝐴) = 𝑦𝐴𝐹𝑦))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205   = wceq 1534  ∃!weu 2557   class class class wbr 5153  cio 6504  cfv 6554
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-12 2167  ax-ext 2697
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-tru 1537  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2704  df-cleq 2718  df-clel 2803  df-v 3464  df-un 3952  df-ss 3964  df-sn 4634  df-pr 4636  df-uni 4914  df-iota 6506  df-fv 6562
This theorem is referenced by:  tz6.12-1  6924  tz6.12i  6929  fnbrfvb  6954
  Copyright terms: Public domain W3C validator