Theorem riotaund 5522

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 5488	. 2 |- ( iota_ x e. A ph ) = ( iota x ( x e. A /\ ph ) )
2		df-reu 2355	. . 3 |- ( E! x e. A ph <-> E! x ( x e. A /\ ph ) )
3		iotanul 4902	. . 3 |- ( -. E! x ( x e. A /\ ph ) -> ( iota x ( x e. A /\ ph ) ) = (/) )
4	2, 3	sylnbi 635	. 2 |- ( -. E! x e. A ph -> ( iota x ( x e. A /\ ph ) ) = (/) )
5	1, 4	syl5eq 2125	1 |- ( -. E! x e. A ph -> ( iota_ x e. A ph ) = (/) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 102   = wceq 1284   e. wcel 1433   E!weu 1941   E!wreu 2350   (/)c0 3251   iotacio 4885   iota_crio 5487

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 576  ax-in2 577  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-tru 1287  df-fal 1290  df-nf 1390  df-sb 1686  df-eu 1944  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ral 2353  df-rex 2354  df-reu 2355  df-v 2603  df-dif 2975  df-in 2979  df-ss 2986  df-nul 3252  df-sn 3404  df-uni 3602  df-iota 4887  df-riota 5488

This theorem is referenced by: (None)