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Theorem tz6.12 5506
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 4025 . 2  |-  ( A F y  <->  <. A , 
y >.  e.  F )
21eubii 2153 . 2  |-  ( E! y  A F y  <-> 
E! y <. A , 
y >.  e.  F )
3 tz6.12-1 5505 . 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 1685   E!weu 2144   <.cop 3644   class class class wbr 4024   ` cfv 5221
This theorem is referenced by:  tz6.12f  5507  dfac5lem5  7750
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 1636  ax-8 1644  ax-14 1689  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1868  ax-ext 2265  ax-sep 4142  ax-nul 4150  ax-pr 4213
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1631  df-eu 2148  df-mo 2149  df-clab 2271  df-cleq 2277  df-clel 2280  df-nfc 2409  df-ne 2449  df-ral 2549  df-rex 2550  df-rab 2553  df-v 2791  df-sbc 2993  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-nul 3457  df-if 3567  df-sn 3647  df-pr 3648  df-op 3650  df-uni 3829  df-br 4025  df-opab 4079  df-xp 4694  df-cnv 4696  df-dm 4698  df-rn 4699  df-res 4700  df-ima 4701  df-fv 5229
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