Theorem moriotass 5902

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 3244	. . . . 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 1019	. . 3 |- ( ( A C_ B /\ E. x e. A ph /\ E* x e. B ph ) -> E. x e. B ph )
4		simp3 1001	. . 3 |- ( ( A C_ B /\ E. x e. A ph /\ E* x e. B ph ) -> E* x e. B ph )
5		reu5 2711	. . 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 5901	. 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 1294	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 980   = wceq 1364   E.wrex 2473   E!wreu 2474   E*wrmo 2475   C_ wss 3153   iota_crio 5872

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2175

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-reu 2479  df-rmo 2480  df-rab 2481  df-v 2762  df-sbc 2986  df-un 3157  df-in 3159  df-ss 3166  df-sn 3624  df-pr 3625  df-uni 3836  df-iota 5215  df-riota 5873

This theorem is referenced by: (None)