Theorem tz6.12-2 6182

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

Step	Hyp	Ref	Expression
1		df-fv 5896	. 2 |- ( F ` A ) = ( iota x A F x )
2		iotanul 5866	. 2 |- ( -. E! x A F x -> ( iota x A F x ) = (/) )
3	1, 2	syl5eq 2668	1 |- ( -. E! x A F x -> ( F ` A ) = (/) )

Syntax hints:   -. wn 3   -> wi 4   = wceq 1483   E!weu 2470   (/)c0 3915   class class class wbr 4653   iotacio 5849   ` cfv 5888

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-v 3202  df-dif 3577  df-in 3581  df-ss 3588  df-nul 3916  df-sn 4178  df-uni 4437  df-iota 5851  df-fv 5896

This theorem is referenced by:  fvprc  6185  tz6.12i  6214  ndmfv  6218  nfunsn  6225  funpartfv  32052  setrec2lem1  42440