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Theorem tz6.12-2 5839
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 5578 . 2  |-  ( F `
 A )  =  ( iota x A F x )
2 iotanul 5549 . 2  |-  ( -.  E! x  A F x  ->  ( iota x A F x )  =  (/) )
31, 2syl5eq 2507 1  |-  ( -.  E! x  A F x  ->  ( F `  A )  =  (/) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    = wceq 1398   E!weu 2284   (/)c0 3783   class class class wbr 4439   iotacio 5532   ` cfv 5570
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-v 3108  df-dif 3464  df-in 3468  df-ss 3475  df-nul 3784  df-sn 4017  df-uni 4236  df-iota 5534  df-fv 5578
This theorem is referenced by:  fvprc  5842  tz6.12i  5868  ndmfv  5872  nfunsn  5879  funpartfv  29823
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