Theorem tz6.12-1 5221

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

Step	Hyp	Ref	Expression
1		df-fv 4930	. 2 |- ( F ` A ) = ( iota y A F y )
2		iota1 4901	. . 3 |- ( E! y A F y -> ( A F y <-> ( iota y A F y ) = y ) )
3	2	biimpac 292	. 2 |- ( ( A F y /\ E! y A F y ) -> ( iota y A F y ) = y )
4	1, 3	syl5eq 2125	1 |- ( ( A F y /\ E! y A F y ) -> ( F ` A ) = y )

Syntax hints:   -> wi 4   /\ wa 102   = wceq 1284   E!weu 1941   class class class wbr 3785   iotacio 4885   ` cfv 4922

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 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-tru 1287  df-nf 1390  df-sb 1686  df-eu 1944  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-rex 2354  df-v 2603  df-sbc 2816  df-un 2977  df-sn 3404  df-pr 3405  df-uni 3602  df-iota 4887  df-fv 4930

This theorem is referenced by:  tz6.12  5222  tz6.12c  5224  funbrfv  5233