Theorem riotaund 5826

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 5792	. 2 |- ( iota_ x e. A ph ) = ( iota x ( x e. A /\ ph ) )
2		df-reu 2449	. . 3 |- ( E! x e. A ph <-> E! x ( x e. A /\ ph ) )
3		iotanul 5162	. . 3 |- ( -. E! x ( x e. A /\ ph ) -> ( iota x ( x e. A /\ ph ) ) = (/) )
4	2, 3	sylnbi 668	. 2 |- ( -. E! x e. A ph -> ( iota x ( x e. A /\ ph ) ) = (/) )
5	1, 4	syl5eq 2209	1 |- ( -. E! x e. A ph -> ( iota_ x e. A ph ) = (/) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 103   = wceq 1342   E!weu 2013   e. wcel 2135   E!wreu 2444   (/)c0 3404   iotacio 5145   iota_crio 5791

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1434  ax-7 1435  ax-gen 1436  ax-ie1 1480  ax-ie2 1481  ax-8 1491  ax-10 1492  ax-11 1493  ax-i12 1494  ax-bndl 1496  ax-4 1497  ax-17 1513  ax-i9 1517  ax-ial 1521  ax-i5r 1522  ax-ext 2146

This theorem depends on definitions:  df-bi 116  df-tru 1345  df-fal 1348  df-nf 1448  df-sb 1750  df-eu 2016  df-clab 2151  df-cleq 2157  df-clel 2160  df-nfc 2295  df-ral 2447  df-rex 2448  df-reu 2449  df-v 2723  df-dif 3113  df-in 3117  df-ss 3124  df-nul 3405  df-sn 3576  df-uni 3784  df-iota 5147  df-riota 5792

This theorem is referenced by: (None)