Theorem tz6.12-1 5513

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 5196	. 2 |- ( F ` A ) = ( iota y A F y )
2		iota1 5167	. . 3 |- ( E! y A F y -> ( A F y <-> ( iota y A F y ) = y ) )
3	2	biimpac 296	. 2 |- ( ( A F y /\ E! y A F y ) -> ( iota y A F y ) = y )
4	1, 3	syl5eq 2211	1 |- ( ( A F y /\ E! y A F y ) -> ( F ` A ) = y )

Syntax hints:   -> wi 4   /\ wa 103   = wceq 1343   E!weu 2014   class class class wbr 3982   iotacio 5151   ` cfv 5188

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-tru 1346  df-nf 1449  df-sb 1751  df-eu 2017  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-rex 2450  df-v 2728  df-sbc 2952  df-un 3120  df-sn 3582  df-pr 3583  df-uni 3790  df-iota 5153  df-fv 5196

This theorem is referenced by:  tz6.12  5514  tz6.12c  5516  funbrfv  5525