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Theorem tz6.12 5545
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 4024 . 2  |-  ( A F y  <->  <. A , 
y >.  e.  F )
21eubii 2152 . 2  |-  ( E! y  A F y  <-> 
E! y <. A , 
y >.  e.  F )
3 tz6.12-1 5544 . 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 1623    e. wcel 1684   E!weu 2143   <.cop 3643   class class class wbr 4023   ` cfv 5255
This theorem is referenced by:  tz6.12f  5546  dfac5lem5  7754
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-rex 2549  df-v 2790  df-sbc 2992  df-un 3157  df-sn 3646  df-pr 3647  df-uni 3828  df-br 4024  df-iota 5219  df-fv 5263
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