Theorem riotaund 5991

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 5954	. 2 |- ( iota_ x e. A ph ) = ( iota x ( x e. A /\ ph ) )
2		df-reu 2515	. . 3 |- ( E! x e. A ph <-> E! x ( x e. A /\ ph ) )
3		iotanul 5294	. . 3 |- ( -. E! x ( x e. A /\ ph ) -> ( iota x ( x e. A /\ ph ) ) = (/) )
4	2, 3	sylnbi 682	. 2 |- ( -. E! x e. A ph -> ( iota x ( x e. A /\ ph ) ) = (/) )
5	1, 4	eqtrid 2274	1 |- ( -. E! x e. A ph -> ( iota_ x e. A ph ) = (/) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   = wceq 1395   E!weu 2077   e. wcel 2200   E!wreu 2510   (/)c0 3491   iotacio 5276   iota_crio 5953

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-reu 2515  df-v 2801  df-dif 3199  df-in 3203  df-ss 3210  df-nul 3492  df-sn 3672  df-uni 3889  df-iota 5278  df-riota 5954

This theorem is referenced by: (None)