Theorem moriotass 5859

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 3221	. . . . 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 1017	. . 3 |- ( ( A C_ B /\ E. x e. A ph /\ E* x e. B ph ) -> E. x e. B ph )
4		simp3 999	. . 3 |- ( ( A C_ B /\ E. x e. A ph /\ E* x e. B ph ) -> E* x e. B ph )
5		reu5 2690	. . 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 5858	. 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 1283	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 978   = wceq 1353   E.wrex 2456   E!wreu 2457   E*wrmo 2458   C_ wss 3130   iota_crio 5830

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-rmo 2463  df-rab 2464  df-v 2740  df-sbc 2964  df-un 3134  df-in 3136  df-ss 3143  df-sn 3599  df-pr 3600  df-uni 3811  df-iota 5179  df-riota 5831

This theorem is referenced by: (None)