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Theorem tz6.12-1 5651
Description: Function value. Theorem 6.12(1) of [TakeutiZaring] p. 27. (Contributed by NM, 30-Apr-2004.)
Assertion
Ref Expression
tz6.12-1  |-  ( ( A F y  /\  E! y  A F
y )  ->  ( F `  A )  =  y )
Distinct variable groups:    y, F    y, A

Proof of Theorem tz6.12-1
StepHypRef Expression
1 df-fv 5366 . 2  |-  ( F `
 A )  =  ( iota y A F y )
2 iota1 5336 . . . 4  |-  ( E! y  A F y  ->  ( A F y  <->  ( iota y A F y )  =  y ) )
32biimpd 198 . . 3  |-  ( E! y  A F y  ->  ( A F y  ->  ( iota y A F y )  =  y ) )
43impcom 419 . 2  |-  ( ( A F y  /\  E! y  A F
y )  ->  ( iota y A F y )  =  y )
51, 4syl5eq 2410 1  |-  ( ( A F y  /\  E! y  A F
y )  ->  ( F `  A )  =  y )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1647   E!weu 2217   class class class wbr 4125   iotacio 5320   ` cfv 5358
This theorem is referenced by:  tz6.12  5652  tz6.12c  5654  funbrfv  5668
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1551  ax-5 1562  ax-17 1621  ax-9 1659  ax-8 1680  ax-6 1734  ax-7 1739  ax-11 1751  ax-12 1937  ax-ext 2347
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1324  df-ex 1547  df-nf 1550  df-sb 1654  df-eu 2221  df-clab 2353  df-cleq 2359  df-clel 2362  df-nfc 2491  df-rex 2634  df-v 2875  df-sbc 3078  df-un 3243  df-sn 3735  df-pr 3736  df-uni 3930  df-iota 5322  df-fv 5366
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