Theorem riota1 5756

Description: Property of restricted iota. Compare iota1 5110. (Contributed by Mario Carneiro, 15-Oct-2016.)

Assertion

Ref	Expression
riota1	|- ( E! x e. A ph -> ( ( x e. A /\ ph ) <-> ( iota_ x e. A ph ) = x ) )

Distinct variable group:   x, A

Allowed substitution hint:   ph( x)

Proof of Theorem riota1

Step	Hyp	Ref	Expression
1		df-reu 2424	. . 3 |- ( E! x e. A ph <-> E! x ( x e. A /\ ph ) )
2		iota1 5110	. . 3 |- ( E! x ( x e. A /\ ph ) -> ( ( x e. A /\ ph ) <-> ( iota x ( x e. A /\ ph ) ) = x ) )
3	1, 2	sylbi 120	. 2 |- ( E! x e. A ph -> ( ( x e. A /\ ph ) <-> ( iota x ( x e. A /\ ph ) ) = x ) )
4		df-riota 5738	. . 3 |- ( iota_ x e. A ph ) = ( iota x ( x e. A /\ ph ) )
5	4	eqeq1i 2148	. 2 |- ( ( iota_ x e. A ph ) = x <-> ( iota x ( x e. A /\ ph ) ) = x )
6	3, 5	syl6bbr 197	1 |- ( E! x e. A ph -> ( ( x e. A /\ ph ) <-> ( iota_ x e. A ph ) = x ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   = wceq 1332   e. wcel 1481   E!weu 2000   E!wreu 2419   iotacio 5094   iota_crio 5737

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 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122

This theorem depends on definitions:  df-bi 116  df-tru 1335  df-nf 1438  df-sb 1737  df-eu 2003  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-rex 2423  df-reu 2424  df-v 2691  df-sbc 2914  df-un 3080  df-sn 3538  df-pr 3539  df-uni 3745  df-iota 5096  df-riota 5738

This theorem is referenced by:  supelti  6897  oddpwdclemdvds  11884  oddpwdclemndvds  11885