Theorem tz6.12-2 5666

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 5365	. 2 |- ( F ` A ) = ( iota x A F x )
2		iotanul 5333	. 2 |- ( -. E! x A F x -> ( iota x A F x ) = (/) )
3	1, 2	eqtrid 2279	1 |- ( -. E! x A F x -> ( F ` A ) = (/) )

Syntax hints:   -. wn 3   -> wi 4   = wceq 1398   E!weu 2082   (/)c0 3512   class class class wbr 4114   iotacio 5315   ` cfv 5357

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2216

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2085  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ral 2527  df-rex 2528  df-v 2817  df-dif 3216  df-in 3220  df-ss 3227  df-nul 3513  df-sn 3700  df-uni 3920  df-iota 5317  df-fv 5365

This theorem is referenced by:  fvprc  5669  ndmfvg  5706  nfunsn  5712