Theorem riota2f 5489

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 2188	. 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 262	. 2 |- ( ( B e. A /\ x = B ) -> ( ph <-> ps ) )
9	2, 3, 5, 6, 8	riota2df 5488	1 |- ( ( B e. A /\ E! x e. A ph ) -> ( ps <-> ( iota_ x e. A ph ) = B ) )

Syntax hints:   -> wi 4   /\ wa 97   <-> wb 98   = wceq 1243   F/wnf 1349   e. wcel 1393   F/_wnfc 2165   E!wreu 2308   iota_crio 5467

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-rex 2312  df-reu 2313  df-v 2559  df-sbc 2765  df-un 2922  df-sn 3381  df-pr 3382  df-uni 3581  df-iota 4867  df-riota 5468

This theorem is referenced by:  riota2  5490  riotaprop  5491  riotass2  5494  riotass  5495