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Theorem tz6.12-1 5546
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 5265 . 2  |-  ( F `
 A )  =  ( iota y A F y )
2 iota1 5235 . . . 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 2329 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 1625   E!weu 2145   class class class wbr 4025   iotacio 5219   ` cfv 5257
This theorem is referenced by:  tz6.12  5547  tz6.12c  5549  funbrfv  5563  tz6.12-afv  28046
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1535  ax-5 1546  ax-17 1605  ax-9 1637  ax-8 1645  ax-6 1705  ax-7 1710  ax-11 1717  ax-12 1868  ax-ext 2266
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1531  df-nf 1534  df-sb 1632  df-eu 2149  df-clab 2272  df-cleq 2278  df-clel 2281  df-nfc 2410  df-rex 2551  df-v 2792  df-sbc 2994  df-un 3159  df-sn 3648  df-pr 3649  df-uni 3830  df-iota 5221  df-fv 5265
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