Theorem exss 4212

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 3442	. . 3 |- ( E. z z e. { x e. A | ph } <-> E. x e. A ph )
2		df-rab 2457	. . . . 5 |- { x e. A | ph } = { x | ( x e. A /\ ph ) }
3	2	eleq2i 2237	. . . 4 |- ( z e. { x e. A | ph } <-> z e. { x | ( x e. A /\ ph ) } )
4	3	exbii 1598	. . 3 |- ( E. z z e. { x e. A | ph } <-> E. z z e. { x | ( x e. A /\ ph ) } )
5	1, 4	bitr3i 185	. 2 |- ( E. x e. A ph <-> E. z z e. { x | ( x e. A /\ ph ) } )
6		vex 2733	. . . . . 6 |- z e. _V
7	6	snss 3709	. . . . 5 |- ( z e. { x | ( x e. A /\ ph ) } <-> { z } C_ { x | ( x e. A /\ ph ) } )
8		ssab2 3231	. . . . . 6 |- { x | ( x e. A /\ ph ) } C_ A
9		sstr2 3154	. . . . . 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 120	. . . 4 |- ( z e. { x | ( x e. A /\ ph ) } -> { z } C_ A )
12		simpr 109	. . . . . . . 8 |- ( ( [ z / x ] x e. A /\ [ z / x ] ph ) -> [ z / x ] ph )
13		equsb1 1778	. . . . . . . . 9 |- [ z / x ] x = z
14		velsn 3600	. . . . . . . . . 10 |- ( x e. { z } <-> x = z )
15	14	sbbii 1758	. . . . . . . . 9 |- ( [ z / x ] x e. { z } <-> [ z / x ] x = z )
16	13, 15	mpbir 145	. . . . . . . 8 |- [ z / x ] x e. { z }
17	12, 16	jctil 310	. . . . . . 7 |- ( ( [ z / x ] x e. A /\ [ z / x ] ph ) -> ( [ z / x ] x e. { z } /\ [ z / x ] ph ) )
18		df-clab 2157	. . . . . . . 8 |- ( z e. { x | ( x e. A /\ ph ) } <-> [ z / x ] ( x e. A /\ ph ) )
19		sban 1948	. . . . . . . 8 |- ( [ z / x ] ( x e. A /\ ph ) <-> ( [ z / x ] x e. A /\ [ z / x ] ph ) )
20	18, 19	bitri 183	. . . . . . 7 |- ( z e. { x | ( x e. A /\ ph ) } <-> ( [ z / x ] x e. A /\ [ z / x ] ph ) )
21		df-rab 2457	. . . . . . . . 9 |- { x e. { z } | ph } = { x | ( x e. { z } /\ ph ) }
22	21	eleq2i 2237	. . . . . . . 8 |- ( z e. { x e. { z } | ph } <-> z e. { x | ( x e. { z } /\ ph ) } )
23		df-clab 2157	. . . . . . . . 9 |- ( z e. { x | ( x e. { z } /\ ph ) } <-> [ z / x ] ( x e. { z } /\ ph ) )
24		sban 1948	. . . . . . . . 9 |- ( [ z / x ] ( x e. { z } /\ ph ) <-> ( [ z / x ] x e. { z } /\ [ z / x ] ph ) )
25	23, 24	bitri 183	. . . . . . . 8 |- ( z e. { x | ( x e. { z } /\ ph ) } <-> ( [ z / x ] x e. { z } /\ [ z / x ] ph ) )
26	22, 25	bitri 183	. . . . . . 7 |- ( z e. { x e. { z } | ph } <-> ( [ z / x ] x e. { z } /\ [ z / x ] ph ) )
27	17, 20, 26	3imtr4i 200	. . . . . 6 |- ( z e. { x | ( x e. A /\ ph ) } -> z e. { x e. { z } | ph } )
28		elex2 2746	. . . . . 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 3442	. . . . 5 |- ( E. w w e. { x e. { z } | ph } <-> E. x e. { z } ph )
31	29, 30	sylib 121	. . . 4 |- ( z e. { x | ( x e. A /\ ph ) } -> E. x e. { z } ph )
32	6	snex 4171	. . . . 5 |- { z } e. _V
33		sseq1 3170	. . . . . 6 |- ( y = { z } -> ( y C_ A <-> { z } C_ A ) )
34		rexeq 2666	. . . . . 6 |- ( y = { z } -> ( E. x e. y ph <-> E. x e. { z } ph ) )
35	33, 34	anbi12d 470	. . . . 5 |- ( y = { z } -> ( ( y C_ A /\ E. x e. y ph ) <-> ( { z } C_ A /\ E. x e. { z } ph ) ) )
36	32, 35	spcev 2825	. . . 4 |- ( ( { z } C_ A /\ E. x e. { z } ph ) -> E. y ( y C_ A /\ E. x e. y ph ) )
37	11, 31, 36	syl2anc 409	. . 3 |- ( z e. { x | ( x e. A /\ ph ) } -> E. y ( y C_ A /\ E. x e. y ph ) )
38	37	exlimiv 1591	. 2 |- ( E. z z e. { x | ( x e. A /\ ph ) } -> E. y ( y C_ A /\ E. x e. y ph ) )
39	5, 38	sylbi 120	1 |- ( E. x e. A ph -> E. y ( y C_ A /\ E. x e. y ph ) )

Syntax hints:   -> wi 4   /\ wa 103   = wceq 1348   E.wex 1485   [wsb 1755   e. wcel 2141   {cab 2156   E.wrex 2449   {crab 2452   C_ wss 3121   {csn 3583

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160

This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-rex 2454  df-rab 2457  df-v 2732  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589

This theorem is referenced by: (None)