Theorem riotacl2 5805

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 2449	. . 3 |- ( E! x e. A ph <-> E! x ( x e. A /\ ph ) )
2		iotacl 5170	. . 3 |- ( E! x ( x e. A /\ ph ) -> ( iota x ( x e. A /\ ph ) ) e. { x | ( x e. A /\ ph ) } )
3	1, 2	sylbi 120	. 2 |- ( E! x e. A ph -> ( iota x ( x e. A /\ ph ) ) e. { x | ( x e. A /\ ph ) } )
4		df-riota 5792	. 2 |- ( iota_ x e. A ph ) = ( iota x ( x e. A /\ ph ) )
5		df-rab 2451	. 2 |- { x e. A | ph } = { x | ( x e. A /\ ph ) }
6	3, 4, 5	3eltr4g 2250	1 |- ( E! x e. A ph -> ( iota_ x e. A ph ) e. { x e. A | ph } )

Syntax hints:   -> wi 4   /\ wa 103   E!weu 2013   e. wcel 2135   {cab 2150   E!wreu 2444   {crab 2446   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-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-nf 1448  df-sb 1750  df-eu 2016  df-clab 2151  df-cleq 2157  df-clel 2160  df-nfc 2295  df-rex 2448  df-reu 2449  df-rab 2451  df-v 2723  df-sbc 2947  df-un 3115  df-sn 3576  df-pr 3577  df-uni 3784  df-iota 5147  df-riota 5792

This theorem is referenced by:  riotacl  5806  riotasbc  5807  supubti  6955  suplubti  6956