Theorem riota1 5920

Description: Property of restricted iota. Compare iota1 5247. (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 2491	. . 3 |- ( E! x e. A ph <-> E! x ( x e. A /\ ph ) )
2		iota1 5247	. . 3 |- ( E! x ( x e. A /\ ph ) -> ( ( x e. A /\ ph ) <-> ( iota x ( x e. A /\ ph ) ) = x ) )
3	1, 2	sylbi 121	. 2 |- ( E! x e. A ph -> ( ( x e. A /\ ph ) <-> ( iota x ( x e. A /\ ph ) ) = x ) )
4		df-riota 5901	. . 3 |- ( iota_ x e. A ph ) = ( iota x ( x e. A /\ ph ) )
5	4	eqeq1i 2213	. 2 |- ( ( iota_ x e. A ph ) = x <-> ( iota x ( x e. A /\ ph ) ) = x )
6	3, 5	bitr4di 198	1 |- ( E! x e. A ph -> ( ( x e. A /\ ph ) <-> ( iota_ x e. A ph ) = x ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1373   E!weu 2054   e. wcel 2176   E!wreu 2486   iotacio 5231   iota_crio 5900

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 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-ext 2187

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1484  df-sb 1786  df-eu 2057  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-rex 2490  df-reu 2491  df-v 2774  df-sbc 2999  df-un 3170  df-sn 3639  df-pr 3640  df-uni 3851  df-iota 5233  df-riota 5901

This theorem is referenced by:  supelti  7106  oddpwdclemdvds  12525  oddpwdclemndvds  12526