Theorem riotass 5515

Description: Restriction of a unique element to a smaller class. (Contributed by NM, 19-Oct-2005.) (Revised by Mario Carneiro, 24-Dec-2016.)

Assertion

Ref	Expression
riotass	|- ( ( A C_ B /\ E. x e. A ph /\ E! x e. B ph ) -> ( iota_ x e. A ph ) = ( iota_ x e. B ph ) )

Distinct variable groups:   x, A   x, B

Allowed substitution hint:   ph( x)

Proof of Theorem riotass

Step	Hyp	Ref	Expression
1		reuss 3245	. . . 4 |- ( ( A C_ B /\ E. x e. A ph /\ E! x e. B ph ) -> E! x e. A ph )
2		riotasbc 5503	. . . 4 |- ( E! x e. A ph -> [. ( iota_ x e. A ph ) / x ]. ph )
3	1, 2	syl 14	. . 3 |- ( ( A C_ B /\ E. x e. A ph /\ E! x e. B ph ) -> [. ( iota_ x e. A ph ) / x ]. ph )
4		simp1 938	. . . . 5 |- ( ( A C_ B /\ E. x e. A ph /\ E! x e. B ph ) -> A C_ B )
5		riotacl 5502	. . . . . 6 |- ( E! x e. A ph -> ( iota_ x e. A ph ) e. A )
6	1, 5	syl 14	. . . . 5 |- ( ( A C_ B /\ E. x e. A ph /\ E! x e. B ph ) -> ( iota_ x e. A ph ) e. A )
7	4, 6	sseldd 3000	. . . 4 |- ( ( A C_ B /\ E. x e. A ph /\ E! x e. B ph ) -> ( iota_ x e. A ph ) e. B )
8		simp3 940	. . . 4 |- ( ( A C_ B /\ E. x e. A ph /\ E! x e. B ph ) -> E! x e. B ph )
9		nfriota1 5495	. . . . 5 |- F/_ x ( iota_ x e. A ph )
10	9	nfsbc1 2832	. . . . 5 |- F/ x [. ( iota_ x e. A ph ) / x ]. ph
11		sbceq1a 2824	. . . . 5 |- ( x = ( iota_ x e. A ph ) -> ( ph <-> [. ( iota_ x e. A ph ) / x ]. ph ) )
12	9, 10, 11	riota2f 5509	. . . 4 |- ( ( ( iota_ x e. A ph ) e. B /\ E! x e. B ph ) -> ( [. ( iota_ x e. A ph ) / x ]. ph <-> ( iota_ x e. B ph ) = ( iota_ x e. A ph ) ) )
13	7, 8, 12	syl2anc 403	. . 3 |- ( ( A C_ B /\ E. x e. A ph /\ E! x e. B ph ) -> ( [. ( iota_ x e. A ph ) / x ]. ph <-> ( iota_ x e. B ph ) = ( iota_ x e. A ph ) ) )
14	3, 13	mpbid 145	. 2 |- ( ( A C_ B /\ E. x e. A ph /\ E! x e. B ph ) -> ( iota_ x e. B ph ) = ( iota_ x e. A ph ) )
15	14	eqcomd 2086	1 |- ( ( A C_ B /\ E. x e. A ph /\ E! x e. B ph ) -> ( iota_ x e. A ph ) = ( iota_ x e. B ph ) )

Syntax hints:   -> wi 4   <-> wb 103   /\ w3a 919   = wceq 1284   e. wcel 1433   E.wrex 2349   E!wreu 2350   [.wsbc 2815   C_ wss 2973   iota_crio 5487

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-3an 921  df-tru 1287  df-nf 1390  df-sb 1686  df-eu 1944  df-mo 1945  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ral 2353  df-rex 2354  df-reu 2355  df-rab 2357  df-v 2603  df-sbc 2816  df-un 2977  df-in 2979  df-ss 2986  df-sn 3404  df-pr 3405  df-uni 3602  df-iota 4887  df-riota 5488

This theorem is referenced by:  moriotass  5516