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Theorem tz6.12-1 5582
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 5300 . 2  |-  ( F `
 A )  =  ( iota y A F y )
2 iota1 5270 . . . 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 2360 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 1633   E!weu 2176   class class class wbr 4060   iotacio 5254   ` cfv 5292
This theorem is referenced by:  tz6.12  5583  tz6.12c  5585  funbrfv  5599
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1537  ax-5 1548  ax-17 1607  ax-9 1645  ax-8 1666  ax-6 1720  ax-7 1725  ax-11 1732  ax-12 1897  ax-ext 2297
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1533  df-nf 1536  df-sb 1640  df-eu 2180  df-clab 2303  df-cleq 2309  df-clel 2312  df-nfc 2441  df-rex 2583  df-v 2824  df-sbc 3026  df-un 3191  df-sn 3680  df-pr 3681  df-uni 3865  df-iota 5256  df-fv 5300
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