Theorem riota2f 5977

Description: This theorem shows a condition that allows us to represent a descriptor with a class expression B. (Contributed by NM, 23-Aug-2011.) (Revised by Mario Carneiro, 15-Oct-2016.)

Hypotheses

Ref	Expression
riota2f.1	|- F/_ x B
riota2f.2	|- F/ x ps
riota2f.3	|- ( x = B -> ( ph <-> ps ) )

Assertion

Ref	Expression
riota2f	|- ( ( B e. A /\ E! x e. A ph ) -> ( ps <-> ( iota_ x e. A ph ) = B ) )

Distinct variable group:   x, A

Allowed substitution hints:   ph( x)   ps( x)   B( x)

Proof of Theorem riota2f

Step	Hyp	Ref	Expression
1		riota2f.1	. . 3 |- F/_ x B
2	1	nfel1 2383	. 2 |- F/ x B e. A
3	1	a1i 9	. 2 |- ( B e. A -> F/_ x B )
4		riota2f.2	. . 3 |- F/ x ps
5	4	a1i 9	. 2 |- ( B e. A -> F/ x ps )
6		id 19	. 2 |- ( B e. A -> B e. A )
7		riota2f.3	. . 3 |- ( x = B -> ( ph <-> ps ) )
8	7	adantl 277	. 2 |- ( ( B e. A /\ x = B ) -> ( ph <-> ps ) )
9	2, 3, 5, 6, 8	riota2df 5976	1 |- ( ( B e. A /\ E! x e. A ph ) -> ( ps <-> ( iota_ x e. A ph ) = B ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1395   F/wnf 1506   e. wcel 2200   F/_wnfc 2359   E!wreu 2510   iota_crio 5953

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-eu 2080  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-rex 2514  df-reu 2515  df-v 2801  df-sbc 3029  df-un 3201  df-sn 3672  df-pr 3673  df-uni 3889  df-iota 5278  df-riota 5954

This theorem is referenced by:  riota2  5978  riotaprop  5980  riotass2  5983  riotass  5984