Theorem tz6.12-2 5545

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 5262	. 2 |- ( F ` A ) = ( iota x A F x )
2		iotanul 5230	. 2 |- ( -. E! x A F x -> ( iota x A F x ) = (/) )
3	1, 2	eqtrid 2238	1 |- ( -. E! x A F x -> ( F ` A ) = (/) )

Syntax hints:   -. wn 3   -> wi 4   = wceq 1364   E!weu 2042   (/)c0 3446   class class class wbr 4029   iotacio 5213   ` cfv 5254

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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2175

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-v 2762  df-dif 3155  df-in 3159  df-ss 3166  df-nul 3447  df-sn 3624  df-uni 3836  df-iota 5215  df-fv 5262

This theorem is referenced by:  fvprc  5548  ndmfvg  5585  nfunsn  5589