Theorem riotass 6000

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 3488	. . . 4 |- ( ( A C_ B /\ E. x e. A ph /\ E! x e. B ph ) -> E! x e. A ph )
2		riotasbc 5987	. . . 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 1023	. . . . 5 |- ( ( A C_ B /\ E. x e. A ph /\ E! x e. B ph ) -> A C_ B )
5		riotacl 5986	. . . . . 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 3228	. . . 4 |- ( ( A C_ B /\ E. x e. A ph /\ E! x e. B ph ) -> ( iota_ x e. A ph ) e. B )
8		simp3 1025	. . . 4 |- ( ( A C_ B /\ E. x e. A ph /\ E! x e. B ph ) -> E! x e. B ph )
9		nfriota1 5978	. . . . 5 |- F/_ x ( iota_ x e. A ph )
10	9	nfsbc1 3049	. . . . 5 |- F/ x [. ( iota_ x e. A ph ) / x ]. ph
11		sbceq1a 3041	. . . . 5 |- ( x = ( iota_ x e. A ph ) -> ( ph <-> [. ( iota_ x e. A ph ) / x ]. ph ) )
12	9, 10, 11	riota2f 5993	. . . 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 1004   = wceq 1397   e. wcel 2202   E.wrex 2511   E!wreu 2512   [.wsbc 3031   C_ wss 3200   iota_crio 5969

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ral 2515  df-rex 2516  df-reu 2517  df-rab 2519  df-v 2804  df-sbc 3032  df-un 3204  df-in 3206  df-ss 3213  df-sn 3675  df-pr 3676  df-uni 3894  df-iota 5286  df-riota 5970

This theorem is referenced by:  moriotass  6001