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Theorem tz6.12-1 5232
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 4940 . 2  |-  ( F `
 A )  =  ( iota y A F y )
2 iota1 4911 . . 3  |-  ( E! y  A F y  ->  ( A F y  <->  ( iota y A F y )  =  y ) )
32biimpac 292 . 2  |-  ( ( A F y  /\  E! y  A F
y )  ->  ( iota y A F y )  =  y )
41, 3syl5eq 2126 1  |-  ( ( A F y  /\  E! y  A F
y )  ->  ( F `  A )  =  y )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    = wceq 1285   E!weu 1942   class class class wbr 3793   iotacio 4895   ` cfv 4932
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064
This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1687  df-eu 1945  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-rex 2355  df-v 2604  df-sbc 2817  df-un 2978  df-sn 3412  df-pr 3413  df-uni 3610  df-iota 4897  df-fv 4940
This theorem is referenced by:  tz6.12  5233  tz6.12c  5235  funbrfv  5244
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