Theorem tz6.12-1 5554

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 5236	. 2 |- ( F ` A ) = ( iota y A F y )
2		iota1 5204	. . 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 2232	1 |- ( ( A F y /\ E! y A F y ) -> ( F ` A ) = y )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1363   E!weu 2036   class class class wbr 4015   iotacio 5188   ` cfv 5228

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 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-ext 2169

This theorem depends on definitions:  df-bi 117  df-tru 1366  df-nf 1471  df-sb 1773  df-eu 2039  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-rex 2471  df-v 2751  df-sbc 2975  df-un 3145  df-sn 3610  df-pr 3611  df-uni 3822  df-iota 5190  df-fv 5236

This theorem is referenced by:  tz6.12  5555  tz6.12c  5557  funbrfv  5567