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Theorem tz6.12-2 5516
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 5263 . 2  |-  ( F `
 A )  =  ( iota x A F x )
2 iotanul 5234 . 2  |-  ( -.  E! x  A F x  ->  ( iota x A F x )  =  (/) )
31, 2syl5eq 2327 1  |-  ( -.  E! x  A F x  ->  ( F `  A )  =  (/) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    = wceq 1623   E!weu 2143   (/)c0 3455   class class class wbr 4023   iotacio 5217   ` cfv 5255
This theorem is referenced by:  fvprc  5519  tz6.12i  5548  ndmfv  5552  nfunsn  5558
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-ne 2448  df-ral 2548  df-rex 2549  df-v 2790  df-dif 3155  df-in 3159  df-ss 3166  df-nul 3456  df-sn 3646  df-uni 3828  df-iota 5219  df-fv 5263
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