Theorem riota2f 5865

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 2340	. 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 5864	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 1363   F/wnf 1470   e. wcel 2158   F/_wnfc 2316   E!wreu 2467   iota_crio 5843

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 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-ext 2169

This theorem depends on definitions:  df-bi 117  df-3an 981  df-tru 1366  df-nf 1471  df-sb 1773  df-eu 2039  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-rex 2471  df-reu 2472  df-v 2751  df-sbc 2975  df-un 3145  df-sn 3610  df-pr 3611  df-uni 3822  df-iota 5190  df-riota 5844

This theorem is referenced by:  riota2  5866  riotaprop  5867  riotass2  5870  riotass  5871