Theorem exss 3963

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 3245	. . 3 |- ( E. z z e. { x e. A | ph } <-> E. x e. A ph )
2		df-rab 2315	. . . . 5 |- { x e. A | ph } = { x | ( x e. A /\ ph ) }
3	2	eleq2i 2104	. . . 4 |- ( z e. { x e. A | ph } <-> z e. { x | ( x e. A /\ ph ) } )
4	3	exbii 1496	. . 3 |- ( E. z z e. { x e. A | ph } <-> E. z z e. { x | ( x e. A /\ ph ) } )
5	1, 4	bitr3i 175	. 2 |- ( E. x e. A ph <-> E. z z e. { x | ( x e. A /\ ph ) } )
6		vex 2560	. . . . . 6 |- z e. _V
7	6	snss 3494	. . . . 5 |- ( z e. { x | ( x e. A /\ ph ) } <-> { z } C_ { x | ( x e. A /\ ph ) } )
8		ssab2 3024	. . . . . 6 |- { x | ( x e. A /\ ph ) } C_ A
9		sstr2 2952	. . . . . 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 114	. . . 4 |- ( z e. { x | ( x e. A /\ ph ) } -> { z } C_ A )
12		simpr 103	. . . . . . . 8 |- ( ( [ z / x ] x e. A /\ [ z / x ] ph ) -> [ z / x ] ph )
13		equsb1 1668	. . . . . . . . 9 |- [ z / x ] x = z
14		velsn 3392	. . . . . . . . . 10 |- ( x e. { z } <-> x = z )
15	14	sbbii 1648	. . . . . . . . 9 |- ( [ z / x ] x e. { z } <-> [ z / x ] x = z )
16	13, 15	mpbir 134	. . . . . . . 8 |- [ z / x ] x e. { z }
17	12, 16	jctil 295	. . . . . . 7 |- ( ( [ z / x ] x e. A /\ [ z / x ] ph ) -> ( [ z / x ] x e. { z } /\ [ z / x ] ph ) )
18		df-clab 2027	. . . . . . . 8 |- ( z e. { x | ( x e. A /\ ph ) } <-> [ z / x ] ( x e. A /\ ph ) )
19		sban 1829	. . . . . . . 8 |- ( [ z / x ] ( x e. A /\ ph ) <-> ( [ z / x ] x e. A /\ [ z / x ] ph ) )
20	18, 19	bitri 173	. . . . . . 7 |- ( z e. { x | ( x e. A /\ ph ) } <-> ( [ z / x ] x e. A /\ [ z / x ] ph ) )
21		df-rab 2315	. . . . . . . . 9 |- { x e. { z } | ph } = { x | ( x e. { z } /\ ph ) }
22	21	eleq2i 2104	. . . . . . . 8 |- ( z e. { x e. { z } | ph } <-> z e. { x | ( x e. { z } /\ ph ) } )
23		df-clab 2027	. . . . . . . . 9 |- ( z e. { x | ( x e. { z } /\ ph ) } <-> [ z / x ] ( x e. { z } /\ ph ) )
24		sban 1829	. . . . . . . . 9 |- ( [ z / x ] ( x e. { z } /\ ph ) <-> ( [ z / x ] x e. { z } /\ [ z / x ] ph ) )
25	23, 24	bitri 173	. . . . . . . 8 |- ( z e. { x | ( x e. { z } /\ ph ) } <-> ( [ z / x ] x e. { z } /\ [ z / x ] ph ) )
26	22, 25	bitri 173	. . . . . . 7 |- ( z e. { x e. { z } | ph } <-> ( [ z / x ] x e. { z } /\ [ z / x ] ph ) )
27	17, 20, 26	3imtr4i 190	. . . . . 6 |- ( z e. { x | ( x e. A /\ ph ) } -> z e. { x e. { z } | ph } )
28		elex2 2570	. . . . . 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 3245	. . . . 5 |- ( E. w w e. { x e. { z } | ph } <-> E. x e. { z } ph )
31	29, 30	sylib 127	. . . 4 |- ( z e. { x | ( x e. A /\ ph ) } -> E. x e. { z } ph )
32		snexgOLD 3935	. . . . . 6 |- ( z e. _V -> { z } e. _V )
33	6, 32	ax-mp 7	. . . . 5 |- { z } e. _V
34		sseq1 2966	. . . . . 6 |- ( y = { z } -> ( y C_ A <-> { z } C_ A ) )
35		rexeq 2506	. . . . . 6 |- ( y = { z } -> ( E. x e. y ph <-> E. x e. { z } ph ) )
36	34, 35	anbi12d 442	. . . . 5 |- ( y = { z } -> ( ( y C_ A /\ E. x e. y ph ) <-> ( { z } C_ A /\ E. x e. { z } ph ) ) )
37	33, 36	spcev 2647	. . . 4 |- ( ( { z } C_ A /\ E. x e. { z } ph ) -> E. y ( y C_ A /\ E. x e. y ph ) )
38	11, 31, 37	syl2anc 391	. . 3 |- ( z e. { x | ( x e. A /\ ph ) } -> E. y ( y C_ A /\ E. x e. y ph ) )
39	38	exlimiv 1489	. 2 |- ( E. z z e. { x | ( x e. A /\ ph ) } -> E. y ( y C_ A /\ E. x e. y ph ) )
40	5, 39	sylbi 114	1 |- ( E. x e. A ph -> E. y ( y C_ A /\ E. x e. y ph ) )

Syntax hints:   -> wi 4   /\ wa 97   = wceq 1243   E.wex 1381   e. wcel 1393   [wsb 1645   {cab 2026   E.wrex 2307   {crab 2310   _Vcvv 2557   C_ wss 2917   {csn 3375

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927

This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-rex 2312  df-rab 2315  df-v 2559  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381

This theorem is referenced by: (None)