Theorem ssrel 4845

Description: A subclass relationship depends only on a relation's ordered pairs. Theorem 3.2(i) of [Monk1] p. 33. (The proof was shortened by Andrew Salmon, 27-Aug-2011.) (Contributed by NM, 2-Aug-1994.) (Revised by set.mm contributors, 27-Aug-2011.)

Assertion

Ref	Expression
ssrel	|- (A C_ B <-> A.xA.y(<.x, y>. e. A -> <.x, y>. e. B))

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

Proof of Theorem ssrel

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ssel 3268	. . 3 |- (A C_ B -> (<.x, y>. e. A -> <.x, y>. e. B))
2	1	alrimivv 1632	. 2 |- (A C_ B -> A.xA.y(<.x, y>. e. A -> <.x, y>. e. B))
3		vex 2863	. . . . . . 7 |- z e. _V
4	3	proj1ex 4594	. . . . . 6 |- Proj1 z e. _V
5		opeq1 4579	. . . . . . . . 9 |- (x = Proj1 z -> <.x, y>. = <. Proj1 z, y>.)
6	5	eleq1d 2419	. . . . . . . 8 |- (x = Proj1 z -> (<.x, y>. e. A <-> <. Proj1 z, y>. e. A))
7	5	eleq1d 2419	. . . . . . . 8 |- (x = Proj1 z -> (<.x, y>. e. B <-> <. Proj1 z, y>. e. B))
8	6, 7	imbi12d 311	. . . . . . 7 |- (x = Proj1 z -> ((<.x, y>. e. A -> <.x, y>. e. B) <-> (<. Proj1 z, y>. e. A -> <. Proj1 z, y>. e. B)))
9	8	albidv 1625	. . . . . 6 |- (x = Proj1 z -> (A.y(<.x, y>. e. A -> <.x, y>. e. B) <-> A.y(<. Proj1 z, y>. e. A -> <. Proj1 z, y>. e. B)))
10	4, 9	spcv 2946	. . . . 5 |- (A.xA.y(<.x, y>. e. A -> <.x, y>. e. B) -> A.y(<. Proj1 z, y>. e. A -> <. Proj1 z, y>. e. B))
11	3	proj2ex 4595	. . . . . 6 |- Proj2 z e. _V
12		opeq2 4580	. . . . . . . 8 |- (y = Proj2 z -> <. Proj1 z, y>. = <. Proj1 z, Proj2 z>.)
13	12	eleq1d 2419	. . . . . . 7 |- (y = Proj2 z -> (<. Proj1 z, y>. e. A <-> <. Proj1 z, Proj2 z>. e. A))
14	12	eleq1d 2419	. . . . . . 7 |- (y = Proj2 z -> (<. Proj1 z, y>. e. B <-> <. Proj1 z, Proj2 z>. e. B))
15	13, 14	imbi12d 311	. . . . . 6 |- (y = Proj2 z -> ((<. Proj1 z, y>. e. A -> <. Proj1 z, y>. e. B) <-> (<. Proj1 z, Proj2 z>. e. A -> <. Proj1 z, Proj2 z>. e. B)))
16	11, 15	spcv 2946	. . . . 5 |- (A.y(<. Proj1 z, y>. e. A -> <. Proj1 z, y>. e. B) -> (<. Proj1 z, Proj2 z>. e. A -> <. Proj1 z, Proj2 z>. e. B))
17	10, 16	syl 15	. . . 4 |- (A.xA.y(<.x, y>. e. A -> <.x, y>. e. B) -> (<. Proj1 z, Proj2 z>. e. A -> <. Proj1 z, Proj2 z>. e. B))
18		opeq 4620	. . . . 5 |- z = <. Proj1 z, Proj2 z>.
19	18	eleq1i 2416	. . . 4 |- (z e. A <-> <. Proj1 z, Proj2 z>. e. A)
20	18	eleq1i 2416	. . . 4 |- (z e. B <-> <. Proj1 z, Proj2 z>. e. B)
21	17, 19, 20	3imtr4g 261	. . 3 |- (A.xA.y(<.x, y>. e. A -> <.x, y>. e. B) -> (z e. A -> z e. B))
22	21	ssrdv 3279	. 2 |- (A.xA.y(<.x, y>. e. A -> <.x, y>. e. B) -> A C_ B)
23	2, 22	impbii 180	1 |- (A C_ B <-> A.xA.y(<.x, y>. e. A -> <.x, y>. e. B))

Syntax hints:   -> wi 4   <-> wb 176  A.wal 1540   = wceq 1642   e. wcel 1710   C_ wss 3258  <.cop 4562   Proj1 cproj1 4564   Proj2 cproj2 4565

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334  ax-nin 4079  ax-xp 4080  ax-cnv 4081  ax-1c 4082  ax-sset 4083  ax-si 4084  ax-ins2 4085  ax-ins3 4086  ax-typlower 4087  ax-sn 4088

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-ne 2519  df-ral 2620  df-rex 2621  df-rab 2624  df-v 2862  df-sbc 3048  df-nin 3212  df-compl 3213  df-in 3214  df-un 3215  df-dif 3216  df-symdif 3217  df-ss 3260  df-nul 3552  df-if 3664  df-pw 3725  df-sn 3742  df-pr 3743  df-uni 3893  df-int 3928  df-opk 4059  df-1c 4137  df-pw1 4138  df-uni1 4139  df-xpk 4186  df-cnvk 4187  df-ins2k 4188  df-ins3k 4189  df-imak 4190  df-cok 4191  df-p6 4192  df-sik 4193  df-ssetk 4194  df-imagek 4195  df-idk 4196  df-0c 4378  df-addc 4379  df-nnc 4380  df-phi 4566  df-op 4567  df-proj1 4568  df-proj2 4569

This theorem is referenced by:  eqrel  4846  ssopr  4847  relssi  4849  relssdv  4850  cotr  5027  cnvsym  5028  intasym  5029  intirr  5030  ssdmrn  5100  dffun2  5120  fvfullfunlem2  5863