Theorem riotacl2 5931

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 2492	. . 3 |- ( E! x e. A ph <-> E! x ( x e. A /\ ph ) )
2		iotacl 5270	. . 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 5917	. 2 |- ( iota_ x e. A ph ) = ( iota x ( x e. A /\ ph ) )
5		df-rab 2494	. 2 |- { x e. A | ph } = { x | ( x e. A /\ ph ) }
6	3, 4, 5	3eltr4g 2292	1 |- ( E! x e. A ph -> ( iota_ x e. A ph ) e. { x e. A | ph } )

Syntax hints:   -> wi 4   /\ wa 104   E!weu 2055   e. wcel 2177   {cab 2192   E!wreu 2487   {crab 2489   iotacio 5244   iota_crio 5916

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2188

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1485  df-sb 1787  df-eu 2058  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-rex 2491  df-reu 2492  df-rab 2494  df-v 2775  df-sbc 3003  df-un 3174  df-sn 3644  df-pr 3645  df-uni 3860  df-iota 5246  df-riota 5917

This theorem is referenced by:  riotacl  5932  riotasbc  5933  supubti  7122  suplubti  7123  grplinv  13467