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Theorem tz6.12-2 5508
Description: Function value when  F is not a function. Theorem 6.12(2) of [TakeutiZaring] p. 27. (Contributed by NM, 30-Apr-2004.) (Proof shortened by Mario Carneiro, 31-Aug-2015.)
Assertion
Ref Expression
tz6.12-2  |-  ( -.  E! y  A F y  ->  ( F `  A )  =  (/) )
Distinct variable groups:    y, A    y, F

Proof of Theorem tz6.12-2
Dummy variables  x  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simpr 447 . . . . 5  |-  ( ( x  e.  z  /\  A. y ( A F y  <->  y  =  z ) )  ->  A. y
( A F y  <-> 
y  =  z ) )
21eximi 1563 . . . 4  |-  ( E. z ( x  e.  z  /\  A. y
( A F y  <-> 
y  =  z ) )  ->  E. z A. y ( A F y  <->  y  =  z ) )
3 elfv 5484 . . . 4  |-  ( x  e.  ( F `  A )  <->  E. z
( x  e.  z  /\  A. y ( A F y  <->  y  =  z ) ) )
4 df-eu 2148 . . . 4  |-  ( E! y  A F y  <->  E. z A. y ( A F y  <->  y  =  z ) )
52, 3, 43imtr4i 257 . . 3  |-  ( x  e.  ( F `  A )  ->  E! y  A F y )
65con3i 127 . 2  |-  ( -.  E! y  A F y  ->  -.  x  e.  ( F `  A
) )
76eq0rdv 3490 1  |-  ( -.  E! y  A F y  ->  ( F `  A )  =  (/) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    /\ wa 358   A.wal 1527   E.wex 1528    = wceq 1623    e. wcel 1685   E!weu 2144   (/)c0 3456   class class class wbr 4024   ` cfv 5221
This theorem is referenced by:  tz6.12i  5510  ndmfv  5514  nfunsn  5520
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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