Theorem moriotass 5766

Description: Restriction of a unique element to a smaller class. (Contributed by NM, 19-Feb-2006.) (Revised by NM, 16-Jun-2017.)

Assertion

Ref	Expression
moriotass	|- ( ( 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 moriotass

Step	Hyp	Ref	Expression
1		ssrexv 3167	. . . . 5 |- ( A C_ B -> ( E. x e. A ph -> E. x e. B ph ) )
2	1	imp 123	. . . 4 |- ( ( A C_ B /\ E. x e. A ph ) -> E. x e. B ph )
3	2	3adant3 1002	. . 3 |- ( ( A C_ B /\ E. x e. A ph /\ E* x e. B ph ) -> E. x e. B ph )
4		simp3 984	. . 3 |- ( ( A C_ B /\ E. x e. A ph /\ E* x e. B ph ) -> E* x e. B ph )
5		reu5 2646	. . 3 |- ( E! x e. B ph <-> ( E. x e. B ph /\ E* x e. B ph ) )
6	3, 4, 5	sylanbrc 414	. 2 |- ( ( A C_ B /\ E. x e. A ph /\ E* x e. B ph ) -> E! x e. B ph )
7		riotass 5765	. 2 |- ( ( A C_ B /\ E. x e. A ph /\ E! x e. B ph ) -> ( iota_ x e. A ph ) = ( iota_ x e. B ph ) )
8	6, 7	syld3an3 1262	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   /\ w3a 963   = wceq 1332   E.wrex 2418   E!wreu 2419   E*wrmo 2420   C_ wss 3076   iota_crio 5737

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-rex 2423  df-reu 2424  df-rmo 2425  df-rab 2426  df-v 2691  df-sbc 2914  df-un 3080  df-in 3082  df-ss 3089  df-sn 3538  df-pr 3539  df-uni 3745  df-iota 5096  df-riota 5738

This theorem is referenced by: (None)