Theorem tz6.12-1 5675

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 5341	. 2 |- ( F ` A ) = ( iota y A F y )
2		iota1 5308	. . 3 |- ( E! y A F y -> ( A F y <-> ( iota y A F y ) = y ) )
3	2	biimpac 298	. 2 |- ( ( A F y /\ E! y A F y ) -> ( iota y A F y ) = y )
4	1, 3	eqtrid 2276	1 |- ( ( A F y /\ E! y A F y ) -> ( F ` A ) = y )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1398   E!weu 2079   class class class wbr 4093   iotacio 5291   ` cfv 5333

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1811  df-eu 2082  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-rex 2517  df-v 2805  df-sbc 3033  df-un 3205  df-sn 3679  df-pr 3680  df-uni 3899  df-iota 5293  df-fv 5341

This theorem is referenced by:  tz6.12  5676  tz6.12c  5678  funbrfv  5691