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Theorem tz6.12-2 5653
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 5396 . 2  |-  ( F `
 A )  =  ( iota x A F x )
2 iotanul 5367 . 2  |-  ( -.  E! x  A F x  ->  ( iota x A F x )  =  (/) )
31, 2syl5eq 2425 1  |-  ( -.  E! x  A F x  ->  ( F `  A )  =  (/) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    = wceq 1649   E!weu 2232   (/)c0 3565   class class class wbr 4147   iotacio 5350   ` cfv 5388
This theorem is referenced by:  fvprc  5656  tz6.12i  5685  ndmfv  5689  nfunsn  5695  funpartfv  25502
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2362
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2236  df-clab 2368  df-cleq 2374  df-clel 2377  df-nfc 2506  df-ne 2546  df-ral 2648  df-rex 2649  df-v 2895  df-dif 3260  df-in 3264  df-ss 3271  df-nul 3566  df-sn 3757  df-uni 3952  df-iota 5352  df-fv 5396
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