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Theorem tz6.12 5808
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 4393 . 2  |-  ( A F y  <->  <. A , 
y >.  e.  F )
21eubii 2285 . 2  |-  ( E! y  A F y  <-> 
E! y <. A , 
y >.  e.  F )
3 tz6.12-1 5807 . 2  |-  ( ( A F y  /\  E! y  A F
y )  ->  ( F `  A )  =  y )
41, 2, 3syl2anbr 480 1  |-  ( (
<. A ,  y >.  e.  F  /\  E! y
<. A ,  y >.  e.  F )  ->  ( F `  A )  =  y )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1370    e. wcel 1758   E!weu 2260   <.cop 3983   class class class wbr 4392   ` cfv 5518
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-rex 2801  df-v 3072  df-sbc 3287  df-un 3433  df-sn 3978  df-pr 3980  df-uni 4192  df-br 4393  df-iota 5481  df-fv 5526
This theorem is referenced by:  tz6.12f  5809  dfac5lem5  8400  tz6.12-afv  30219
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