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Theorem tz6.12 5547
Description: Function value. Theorem 6.12(1) of [TakeutiZaring] p. 27. (Contributed by NM, 10-Jul-1994.)
Assertion
Ref Expression
tz6.12  |-  ( (
<. A ,  y >.  e.  F  /\  E! y
<. A ,  y >.  e.  F )  ->  ( F `  A )  =  y )
Distinct variable groups:    y, F    y, A

Proof of Theorem tz6.12
StepHypRef Expression
1 df-br 4026 . 2  |-  ( A F y  <->  <. A , 
y >.  e.  F )
21eubii 2154 . 2  |-  ( E! y  A F y  <-> 
E! y <. A , 
y >.  e.  F )
3 tz6.12-1 5546 . 2  |-  ( ( A F y  /\  E! y  A F
y )  ->  ( F `  A )  =  y )
41, 2, 3syl2anbr 466 1  |-  ( (
<. A ,  y >.  e.  F  /\  E! y
<. A ,  y >.  e.  F )  ->  ( F `  A )  =  y )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1625    e. wcel 1686   E!weu 2145   <.cop 3645   class class class wbr 4025   ` cfv 5257
This theorem is referenced by:  tz6.12f  5548  dfac5lem5  7756
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1535  ax-5 1546  ax-17 1605  ax-9 1637  ax-8 1645  ax-6 1705  ax-7 1710  ax-11 1717  ax-12 1868  ax-ext 2266
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1531  df-nf 1534  df-sb 1632  df-eu 2149  df-clab 2272  df-cleq 2278  df-clel 2281  df-nfc 2410  df-rex 2551  df-v 2792  df-sbc 2994  df-un 3159  df-sn 3648  df-pr 3649  df-uni 3830  df-br 4026  df-iota 5221  df-fv 5265
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