Theorem exss 3982

Description: Restricted existence in a class (even if proper) implies restricted existence in a subset. (Contributed by NM, 23-Aug-2003.)

Assertion

Ref	Expression
exss	|- ( E. x e. A ph -> E. y ( y C_ A /\ E. x e. y ph ) )

Distinct variable groups:   x, y, A   ph, y

Allowed substitution hint:   ph( x)

Proof of Theorem exss

Dummy variables z w are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		rabn0m 3272	. . 3 |- ( E. z z e. { x e. A | ph } <-> E. x e. A ph )
2		df-rab 2357	. . . . 5 |- { x e. A | ph } = { x | ( x e. A /\ ph ) }
3	2	eleq2i 2145	. . . 4 |- ( z e. { x e. A | ph } <-> z e. { x | ( x e. A /\ ph ) } )
4	3	exbii 1536	. . 3 |- ( E. z z e. { x e. A | ph } <-> E. z z e. { x | ( x e. A /\ ph ) } )
5	1, 4	bitr3i 184	. 2 |- ( E. x e. A ph <-> E. z z e. { x | ( x e. A /\ ph ) } )
6		vex 2604	. . . . . 6 |- z e. _V
7	6	snss 3516	. . . . 5 |- ( z e. { x | ( x e. A /\ ph ) } <-> { z } C_ { x | ( x e. A /\ ph ) } )
8		ssab2 3078	. . . . . 6 |- { x | ( x e. A /\ ph ) } C_ A
9		sstr2 3006	. . . . . 6 |- ( { z } C_ { x | ( x e. A /\ ph ) } -> ( { x | ( x e. A /\ ph ) } C_ A -> { z } C_ A ) )
10	8, 9	mpi 15	. . . . 5 |- ( { z } C_ { x | ( x e. A /\ ph ) } -> { z } C_ A )
11	7, 10	sylbi 119	. . . 4 |- ( z e. { x | ( x e. A /\ ph ) } -> { z } C_ A )
12		simpr 108	. . . . . . . 8 |- ( ( [ z / x ] x e. A /\ [ z / x ] ph ) -> [ z / x ] ph )
13		equsb1 1708	. . . . . . . . 9 |- [ z / x ] x = z
14		velsn 3415	. . . . . . . . . 10 |- ( x e. { z } <-> x = z )
15	14	sbbii 1688	. . . . . . . . 9 |- ( [ z / x ] x e. { z } <-> [ z / x ] x = z )
16	13, 15	mpbir 144	. . . . . . . 8 |- [ z / x ] x e. { z }
17	12, 16	jctil 305	. . . . . . 7 |- ( ( [ z / x ] x e. A /\ [ z / x ] ph ) -> ( [ z / x ] x e. { z } /\ [ z / x ] ph ) )
18		df-clab 2068	. . . . . . . 8 |- ( z e. { x | ( x e. A /\ ph ) } <-> [ z / x ] ( x e. A /\ ph ) )
19		sban 1870	. . . . . . . 8 |- ( [ z / x ] ( x e. A /\ ph ) <-> ( [ z / x ] x e. A /\ [ z / x ] ph ) )
20	18, 19	bitri 182	. . . . . . 7 |- ( z e. { x | ( x e. A /\ ph ) } <-> ( [ z / x ] x e. A /\ [ z / x ] ph ) )
21		df-rab 2357	. . . . . . . . 9 |- { x e. { z } | ph } = { x | ( x e. { z } /\ ph ) }
22	21	eleq2i 2145	. . . . . . . 8 |- ( z e. { x e. { z } | ph } <-> z e. { x | ( x e. { z } /\ ph ) } )
23		df-clab 2068	. . . . . . . . 9 |- ( z e. { x | ( x e. { z } /\ ph ) } <-> [ z / x ] ( x e. { z } /\ ph ) )
24		sban 1870	. . . . . . . . 9 |- ( [ z / x ] ( x e. { z } /\ ph ) <-> ( [ z / x ] x e. { z } /\ [ z / x ] ph ) )
25	23, 24	bitri 182	. . . . . . . 8 |- ( z e. { x | ( x e. { z } /\ ph ) } <-> ( [ z / x ] x e. { z } /\ [ z / x ] ph ) )
26	22, 25	bitri 182	. . . . . . 7 |- ( z e. { x e. { z } | ph } <-> ( [ z / x ] x e. { z } /\ [ z / x ] ph ) )
27	17, 20, 26	3imtr4i 199	. . . . . 6 |- ( z e. { x | ( x e. A /\ ph ) } -> z e. { x e. { z } | ph } )
28		elex2 2615	. . . . . 6 |- ( z e. { x e. { z } | ph } -> E. w w e. { x e. { z } | ph } )
29	27, 28	syl 14	. . . . 5 |- ( z e. { x | ( x e. A /\ ph ) } -> E. w w e. { x e. { z } | ph } )
30		rabn0m 3272	. . . . 5 |- ( E. w w e. { x e. { z } | ph } <-> E. x e. { z } ph )
31	29, 30	sylib 120	. . . 4 |- ( z e. { x | ( x e. A /\ ph ) } -> E. x e. { z } ph )
32	6	snex 3957	. . . . 5 |- { z } e. _V
33		sseq1 3020	. . . . . 6 |- ( y = { z } -> ( y C_ A <-> { z } C_ A ) )
34		rexeq 2550	. . . . . 6 |- ( y = { z } -> ( E. x e. y ph <-> E. x e. { z } ph ) )
35	33, 34	anbi12d 456	. . . . 5 |- ( y = { z } -> ( ( y C_ A /\ E. x e. y ph ) <-> ( { z } C_ A /\ E. x e. { z } ph ) ) )
36	32, 35	spcev 2692	. . . 4 |- ( ( { z } C_ A /\ E. x e. { z } ph ) -> E. y ( y C_ A /\ E. x e. y ph ) )
37	11, 31, 36	syl2anc 403	. . 3 |- ( z e. { x | ( x e. A /\ ph ) } -> E. y ( y C_ A /\ E. x e. y ph ) )
38	37	exlimiv 1529	. 2 |- ( E. z z e. { x | ( x e. A /\ ph ) } -> E. y ( y C_ A /\ E. x e. y ph ) )
39	5, 38	sylbi 119	1 |- ( E. x e. A ph -> E. y ( y C_ A /\ E. x e. y ph ) )

Syntax hints:   -> wi 4   /\ wa 102   = wceq 1284   E.wex 1421   e. wcel 1433   [wsb 1685   {cab 2067   E.wrex 2349   {crab 2352   C_ wss 2973   {csn 3398

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-sep 3896  ax-pow 3948

This theorem depends on definitions:  df-bi 115  df-tru 1287  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-rex 2354  df-rab 2357  df-v 2603  df-in 2979  df-ss 2986  df-pw 3384  df-sn 3404

This theorem is referenced by: (None)