Theorem riotass 6011

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 3490	. . . 4 |- ( ( A C_ B /\ E. x e. A ph /\ E! x e. B ph ) -> E! x e. A ph )
2		riotasbc 5998	. . . 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 1024	. . . . 5 |- ( ( A C_ B /\ E. x e. A ph /\ E! x e. B ph ) -> A C_ B )
5		riotacl 5997	. . . . . 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 3229	. . . 4 |- ( ( A C_ B /\ E. x e. A ph /\ E! x e. B ph ) -> ( iota_ x e. A ph ) e. B )
8		simp3 1026	. . . 4 |- ( ( A C_ B /\ E. x e. A ph /\ E! x e. B ph ) -> E! x e. B ph )
9		nfriota1 5989	. . . . 5 |- F/_ x ( iota_ x e. A ph )
10	9	nfsbc1 3050	. . . . 5 |- F/ x [. ( iota_ x e. A ph ) / x ]. ph
11		sbceq1a 3042	. . . . 5 |- ( x = ( iota_ x e. A ph ) -> ( ph <-> [. ( iota_ x e. A ph ) / x ]. ph ) )
12	9, 10, 11	riota2f 6004	. . . 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 411	. . 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 147	. 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 2237	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 105   /\ w3a 1005   = wceq 1398   e. wcel 2202   E.wrex 2512   E!wreu 2513   [.wsbc 3032   C_ wss 3201   iota_crio 5980

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ral 2516  df-rex 2517  df-reu 2518  df-rab 2520  df-v 2805  df-sbc 3033  df-un 3205  df-in 3207  df-ss 3214  df-sn 3679  df-pr 3680  df-uni 3899  df-iota 5293  df-riota 5981

This theorem is referenced by:  moriotass  6012