Theorem moriotass 5906

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 3248	. . . . 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 2714	. . 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 5905	. 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 2476   E!wreu 2477   E*wrmo 2478   C_ wss 3157   iota_crio 5876

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480  df-rex 2481  df-reu 2482  df-rmo 2483  df-rab 2484  df-v 2765  df-sbc 2990  df-un 3161  df-in 3163  df-ss 3170  df-sn 3628  df-pr 3629  df-uni 3840  df-iota 5219  df-riota 5877

This theorem is referenced by: (None)