Theorem riotass 5854

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 3416	. . . 4 |- ( ( A C_ B /\ E. x e. A ph /\ E! x e. B ph ) -> E! x e. A ph )
2		riotasbc 5842	. . . 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 997	. . . . 5 |- ( ( A C_ B /\ E. x e. A ph /\ E! x e. B ph ) -> A C_ B )
5		riotacl 5841	. . . . . 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 3156	. . . 4 |- ( ( A C_ B /\ E. x e. A ph /\ E! x e. B ph ) -> ( iota_ x e. A ph ) e. B )
8		simp3 999	. . . 4 |- ( ( A C_ B /\ E. x e. A ph /\ E! x e. B ph ) -> E! x e. B ph )
9		nfriota1 5834	. . . . 5 |- F/_ x ( iota_ x e. A ph )
10	9	nfsbc1 2980	. . . . 5 |- F/ x [. ( iota_ x e. A ph ) / x ]. ph
11		sbceq1a 2972	. . . . 5 |- ( x = ( iota_ x e. A ph ) -> ( ph <-> [. ( iota_ x e. A ph ) / x ]. ph ) )
12	9, 10, 11	riota2f 5848	. . . 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 2183	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 978   = wceq 1353   e. wcel 2148   E.wrex 2456   E!wreu 2457   [.wsbc 2962   C_ wss 3129   iota_crio 5826

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2739  df-sbc 2963  df-un 3133  df-in 3135  df-ss 3142  df-sn 3598  df-pr 3599  df-uni 3810  df-iota 5176  df-riota 5827

This theorem is referenced by:  moriotass  5855