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Theorem tz6.12-2 5719
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 5462 . 2  |-  ( F `
 A )  =  ( iota x A F x )
2 iotanul 5433 . 2  |-  ( -.  E! x  A F x  ->  ( iota x A F x )  =  (/) )
31, 2syl5eq 2480 1  |-  ( -.  E! x  A F x  ->  ( F `  A )  =  (/) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    = wceq 1652   E!weu 2281   (/)c0 3628   class class class wbr 4212   iotacio 5416   ` cfv 5454
This theorem is referenced by:  fvprc  5722  tz6.12i  5751  ndmfv  5755  nfunsn  5761  funpartfv  25790
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-v 2958  df-dif 3323  df-in 3327  df-ss 3334  df-nul 3629  df-sn 3820  df-uni 4016  df-iota 5418  df-fv 5462
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