Theorem riotacl2 5985

Description: Membership law for "the unique element in A such that ph."

(Contributed by NM, 21-Aug-2011.) (Revised by Mario Carneiro, 23-Dec-2016.)

Assertion

Ref	Expression
riotacl2	|- ( E! x e. A ph -> ( iota_ x e. A ph ) e. { x e. A | ph } )

Proof of Theorem riotacl2

Step	Hyp	Ref	Expression
1		df-reu 2517	. . 3 |- ( E! x e. A ph <-> E! x ( x e. A /\ ph ) )
2		iotacl 5311	. . 3 |- ( E! x ( x e. A /\ ph ) -> ( iota x ( x e. A /\ ph ) ) e. { x | ( x e. A /\ ph ) } )
3	1, 2	sylbi 121	. 2 |- ( E! x e. A ph -> ( iota x ( x e. A /\ ph ) ) e. { x | ( x e. A /\ ph ) } )
4		df-riota 5970	. 2 |- ( iota_ x e. A ph ) = ( iota x ( x e. A /\ ph ) )
5		df-rab 2519	. 2 |- { x e. A | ph } = { x | ( x e. A /\ ph ) }
6	3, 4, 5	3eltr4g 2317	1 |- ( E! x e. A ph -> ( iota_ x e. A ph ) e. { x e. A | ph } )

Syntax hints:   -> wi 4   /\ wa 104   E!weu 2079   e. wcel 2202   {cab 2217   E!wreu 2512   {crab 2514   iotacio 5284   iota_crio 5969

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-tru 1400  df-nf 1509  df-sb 1811  df-eu 2082  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-rex 2516  df-reu 2517  df-rab 2519  df-v 2804  df-sbc 3032  df-un 3204  df-sn 3675  df-pr 3676  df-uni 3894  df-iota 5286  df-riota 5970

This theorem is referenced by:  riotacl  5986  riotasbc  5987  supubti  7197  suplubti  7198  grplinv  13632