Theorem riotaund 5934

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 5899	. 2 |- ( iota_ x e. A ph ) = ( iota x ( x e. A /\ ph ) )
2		df-reu 2491	. . 3 |- ( E! x e. A ph <-> E! x ( x e. A /\ ph ) )
3		iotanul 5247	. . 3 |- ( -. E! x ( x e. A /\ ph ) -> ( iota x ( x e. A /\ ph ) ) = (/) )
4	2, 3	sylnbi 680	. 2 |- ( -. E! x e. A ph -> ( iota x ( x e. A /\ ph ) ) = (/) )
5	1, 4	eqtrid 2250	1 |- ( -. E! x e. A ph -> ( iota_ x e. A ph ) = (/) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   = wceq 1373   E!weu 2054   e. wcel 2176   E!wreu 2486   (/)c0 3460   iotacio 5230   iota_crio 5898

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-ext 2187

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-fal 1379  df-nf 1484  df-sb 1786  df-eu 2057  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ral 2489  df-rex 2490  df-reu 2491  df-v 2774  df-dif 3168  df-in 3172  df-ss 3179  df-nul 3461  df-sn 3639  df-uni 3851  df-iota 5232  df-riota 5899

This theorem is referenced by: (None)