Theorem riotaund 5908

Description: Restricted iota equals the empty set when not meaningful. (Contributed by NM, 16-Jan-2012.) (Revised by Mario Carneiro, 15-Oct-2016.) (Revised by NM, 13-Sep-2018.)

Assertion

Ref	Expression
riotaund	|- ( -. E! x e. A ph -> ( iota_ x e. A ph ) = (/) )

Distinct variable group:   x, A

Allowed substitution hint:   ph( x)

Proof of Theorem riotaund

Step	Hyp	Ref	Expression
1		df-riota 5873	. 2 |- ( iota_ x e. A ph ) = ( iota x ( x e. A /\ ph ) )
2		df-reu 2479	. . 3 |- ( E! x e. A ph <-> E! x ( x e. A /\ ph ) )
3		iotanul 5230	. . 3 |- ( -. E! x ( x e. A /\ ph ) -> ( iota x ( x e. A /\ ph ) ) = (/) )
4	2, 3	sylnbi 679	. 2 |- ( -. E! x e. A ph -> ( iota x ( x e. A /\ ph ) ) = (/) )
5	1, 4	eqtrid 2238	1 |- ( -. E! x e. A ph -> ( iota_ x e. A ph ) = (/) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   = wceq 1364   E!weu 2042   e. wcel 2164   E!wreu 2474   (/)c0 3446   iotacio 5213   iota_crio 5872

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-reu 2479  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-riota 5873

This theorem is referenced by: (None)