Theorem tz6.12-2 5681

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 5380	. 2 |- ( F ` A ) = ( iota x A F x )
2		iotanul 5348	. 2 |- ( -. E! x A F x -> ( iota x A F x ) = (/) )
3	1, 2	eqtrid 2283	1 |- ( -. E! x A F x -> ( F ` A ) = (/) )

Syntax hints:   -. wn 3   -> wi 4   = wceq 1402   E!weu 2086   (/)c0 3520   class class class wbr 4125   iotacio 5330   ` cfv 5372

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This theorem depends on definitions:  df-bi 117  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-v 2823  df-dif 3222  df-in 3226  df-ss 3233  df-nul 3521  df-sn 3711  df-uni 3931  df-iota 5332  df-fv 5380

This theorem is referenced by:  fvprc  5684  ndmfvg  5721  nfunsn  5727