Theorem moriotass 6001

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 3292	. . . . 5 |- ( A C_ B -> ( E. x e. A ph -> E. x e. B ph ) )
2	1	imp 124	. . . 4 |- ( ( A C_ B /\ E. x e. A ph ) -> E. x e. B ph )
3	2	3adant3 1043	. . 3 |- ( ( A C_ B /\ E. x e. A ph /\ E* x e. B ph ) -> E. x e. B ph )
4		simp3 1025	. . 3 |- ( ( A C_ B /\ E. x e. A ph /\ E* x e. B ph ) -> E* x e. B ph )
5		reu5 2751	. . 3 |- ( E! x e. B ph <-> ( E. x e. B ph /\ E* x e. B ph ) )
6	3, 4, 5	sylanbrc 417	. 2 |- ( ( A C_ B /\ E. x e. A ph /\ E* x e. B ph ) -> E! x e. B ph )
7		riotass 6000	. 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 1318	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 1004   = wceq 1397   E.wrex 2511   E!wreu 2512   E*wrmo 2513   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-rmo 2518  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: (None)