Theorem riotauni 5839

Description: Restricted iota in terms of class union. (Contributed by NM, 11-Oct-2011.)

Assertion

Ref	Expression
riotauni	|- ( E! x e. A ph -> ( iota_ x e. A ph ) = U. { x e. A | ph } )

Proof of Theorem riotauni

Step	Hyp	Ref	Expression
1		df-reu 2462	. . 3 |- ( E! x e. A ph <-> E! x ( x e. A /\ ph ) )
2		iotauni 5192	. . 3 |- ( E! x ( x e. A /\ ph ) -> ( iota x ( x e. A /\ ph ) ) = U. { x | ( x e. A /\ ph ) } )
3	1, 2	sylbi 121	. 2 |- ( E! x e. A ph -> ( iota x ( x e. A /\ ph ) ) = U. { x | ( x e. A /\ ph ) } )
4		df-riota 5833	. 2 |- ( iota_ x e. A ph ) = ( iota x ( x e. A /\ ph ) )
5		df-rab 2464	. . 3 |- { x e. A | ph } = { x | ( x e. A /\ ph ) }
6	5	unieqi 3821	. 2 |- U. { x e. A | ph } = U. { x | ( x e. A /\ ph ) }
7	3, 4, 6	3eqtr4g 2235	1 |- ( E! x e. A ph -> ( iota_ x e. A ph ) = U. { x e. A | ph } )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1353   E!weu 2026   e. wcel 2148   {cab 2163   E!wreu 2457   {crab 2459   U.cuni 3811   iotacio 5178   iota_crio 5832

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-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2741  df-sbc 2965  df-un 3135  df-sn 3600  df-pr 3601  df-uni 3812  df-iota 5180  df-riota 5833

This theorem is referenced by:  supval2ti  6996