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Theorem tz6.12-1 5544
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 5263 . 2  |-  ( F `
 A )  =  ( iota y A F y )
2 iota1 5233 . . . 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 2327 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 1623   E!weu 2143   class class class wbr 4023   iotacio 5217   ` cfv 5255
This theorem is referenced by:  tz6.12  5545  tz6.12c  5547  funbrfv  5561  tz6.12-afv  28035
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-rex 2549  df-v 2790  df-sbc 2992  df-un 3157  df-sn 3646  df-pr 3647  df-uni 3828  df-iota 5219  df-fv 5263
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