Theorem tz6.12-2 5552

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 5267	. 2 |- ( F ` A ) = ( iota x A F x )
2		iotanul 5235	. 2 |- ( -. E! x A F x -> ( iota x A F x ) = (/) )
3	1, 2	eqtrid 2241	1 |- ( -. E! x A F x -> ( F ` A ) = (/) )

Syntax hints:   -. wn 3   -> wi 4   = wceq 1364   E!weu 2045   (/)c0 3451   class class class wbr 4034   iotacio 5218   ` cfv 5259

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 615  ax-in2 616  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480  df-rex 2481  df-v 2765  df-dif 3159  df-in 3163  df-ss 3170  df-nul 3452  df-sn 3629  df-uni 3841  df-iota 5220  df-fv 5267

This theorem is referenced by:  fvprc  5555  ndmfvg  5592  nfunsn  5596