Theorem exss 1881

Description: Restricted existence in a class (even if proper) implies restricted existence in a subset.

Assertion

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

Distinct variable group(s):   x,y,A   ph,y

Proof of Theorem exss

Step	Hyp	Ref	Expression
1		df-rab 1208	. . . . 5 |- {x e. A | ph} = {x | (x e. A /\ ph)}
2	1	cleq1i 1108	. . . 4 |- ({x e. A | ph} = (/) <-> {x | (x e. A /\ ph)} = (/))
3	2	negbii 162	. . 3 |- (-. {x e. A | ph} = (/) <-> -. {x | (x e. A /\ ph)} = (/))
4		rabn0 1716	. . 3 |- (-. {x e. A | ph} = (/) <-> E.x e. A ph)
5		n0 1714	. . 3 |- (-. {x | (x e. A /\ ph)} = (/) <-> E.z z e. {x | (x e. A /\ ph)})
6	3, 4, 5	3bitr3 156	. 2 |- (E.x e. A ph <-> E.z z e. {x | (x e. A /\ ph)})
7		snex 1859	. . . . 5 |- {z} e. V
8		sseq1 1521	. . . . . 6 |- (y = {z} -> (y (_ A <-> {z} (_ A))
9		rexeq 1325	. . . . . 6 |- (y = {z} -> (E.x e. y ph <-> E.x e. {z}ph))
10	8, 9	anbi12d 476	. . . . 5 |- (y = {z} -> ((y (_ A /\ E.x e. y ph) <-> ({z} (_ A /\ E.x e. {z}ph)))
11	7, 10	cla4ev 1401	. . . 4 |- (({z} (_ A /\ E.x e. {z}ph) -> E.y(y (_ A /\ E.x e. y ph))
12		visset 1350	. . . . . 6 |- z e. V
13	12	snss 1849	. . . . 5 |- (z e. {x | (x e. A /\ ph)} <-> {z} (_ {x | (x e. A /\ ph)})
14		ssab 1555	. . . . . 6 |- {x | (x e. A /\ ph)} (_ A
15		sstr2 1510	. . . . . 6 |- ({z} (_ {x | (x e. A /\ ph)} -> ({x | (x e. A /\ ph)} (_ A -> {z} (_ A))
16	14, 15	mpi 44	. . . . 5 |- ({z} (_ {x | (x e. A /\ ph)} -> {z} (_ A)
17	13, 16	sylbi 174	. . . 4 |- (z e. {x | (x e. A /\ ph)} -> {z} (_ A)
18		pm3.27 260	. . . . . . . 8 |- (([z / x]x e. A /\ [z / x]ph) -> [z / x]ph)
19		sbeq1 900	. . . . . . . . 9 |- [z / x]x = z
20		elsn 1820	. . . . . . . . . 10 |- (x e. {z} <-> x = z)
21	20	bisb 855	. . . . . . . . 9 |- ([z / x]x e. {z} <-> [z / x]x = z)
22	19, 21	mpbir 165	. . . . . . . 8 |- [z / x]x e. {z}
23	18, 22	jctil 240	. . . . . . 7 |- (([z / x]x e. A /\ [z / x]ph) -> ([z / x]x e. {z} /\ [z / x]ph))
24		df-clab 1093	. . . . . . . 8 |- (z e. {x | (x e. A /\ ph)} <-> [z / x](x e. A /\ ph))
25		sban 889	. . . . . . . 8 |- ([z / x](x e. A /\ ph) <-> ([z / x]x e. A /\ [z / x]ph))
26	24, 25	bitr 151	. . . . . . 7 |- (z e. {x | (x e. A /\ ph)} <-> ([z / x]x e. A /\ [z / x]ph))
27		df-rab 1208	. . . . . . . . 9 |- {x e. {z} | ph} = {x | (x e. {z} /\ ph)}
28	27	eleq2i 1153	. . . . . . . 8 |- (z e. {x e. {z} | ph} <-> z e. {x | (x e. {z} /\ ph)})
29		df-clab 1093	. . . . . . . 8 |- (z e. {x | (x e. {z} /\ ph)} <-> [z / x](x e. {z} /\ ph))
30		sban 889	. . . . . . . 8 |- ([z / x](x e. {z} /\ ph) <-> ([z / x]x e. {z} /\ [z / x]ph))
31	28, 29, 30	3bitr 155	. . . . . . 7 |- (z e. {x e. {z} | ph} <-> ([z / x]x e. {z} /\ [z / x]ph))
32	23, 26, 31	3imtr4 192	. . . . . 6 |- (z e. {x | (x e. A /\ ph)} -> z e. {x e. {z} | ph})
33		n0i 1712	. . . . . 6 |- (z e. {x e. {z} | ph} -> -. {x e. {z} | ph} = (/))
34	32, 33	syl 12	. . . . 5 |- (z e. {x | (x e. A /\ ph)} -> -. {x e. {z} | ph} = (/))
35		rabn0 1716	. . . . 5 |- (-. {x e. {z} | ph} = (/) <-> E.x e. {z}ph)
36	34, 35	sylib 173	. . . 4 |- (z e. {x | (x e. A /\ ph)} -> E.x e. {z}ph)
37	11, 17, 36	sylanc 361	. . 3 |- (z e. {x | (x e. A /\ ph)} -> E.y(y (_ A /\ E.x e. y ph))
38	37	19.23aiv 952	. 2 |- (E.z z e. {x | (x e. A /\ ph)} -> E.y(y (_ A /\ E.x e. y ph))
39	6, 38	sylbi 174	1 |- (E.x e. A ph -> E.y(y (_ A /\ E.x e. y ph))

Syntax hints:  -. wn 1   -> wi 2   /\ wa 196  E.wex 678   = weq 797  [wsb 852  {cab 1090   = wceq 1091   e. wcel 1092  E.wrex 1202  {crab 1204   (_ wss 1487  (/)c0 1707  {csn 1808

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-rex 1206  df-rab 1208  df-v 1349  df-dif 1489  df-in 1491  df-ss 1492  df-nul 1708  df-pw 1799  df-sn 1811

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