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

Step	Hyp	Ref	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 ) )
3	2	biimpac 292	. 2 |- ( ( A F y /\ E! y A F y ) -> ( iota y A F y ) = y )
4	1, 3	syl5eq 2126	1 |- ( ( A F y /\ E! y A F y ) -> ( F ` A ) = y )

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