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Theorem tz6.12-2 5783
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! x  A F x  ->  ( F `  A )  =  (/) )
Distinct variable groups:    x, F    x, A

Proof of Theorem tz6.12-2
StepHypRef Expression
1 df-fv 5527 . 2  |-  ( F `
 A )  =  ( iota x A F x )
2 iotanul 5497 . 2  |-  ( -.  E! x  A F x  ->  ( iota x A F x )  =  (/) )
31, 2syl5eq 2504 1  |-  ( -.  E! x  A F x  ->  ( F `  A )  =  (/) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    = wceq 1370   E!weu 2260   (/)c0 3738   class class class wbr 4393   iotacio 5480   ` cfv 5519
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-ne 2646  df-ral 2800  df-rex 2801  df-v 3073  df-dif 3432  df-in 3436  df-ss 3443  df-nul 3739  df-sn 3979  df-uni 4193  df-iota 5482  df-fv 5527
This theorem is referenced by:  fvprc  5786  tz6.12i  5812  ndmfv  5816  nfunsn  5823  funpartfv  28113
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