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Theorem ssrel 4844
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 Bxy(x, y Ax, y B))
Distinct variable groups:   x,y,A   x,B,y

Proof of Theorem ssrel
Dummy variable z is distinct from all other variables.
StepHypRef Expression
1 ssel 3267 . . 3 (A B → (x, y Ax, y B))
21alrimivv 1632 . 2 (A Bxy(x, y Ax, y B))
3 vex 2862 . . . . . . 7 z V
43proj1ex 4593 . . . . . 6 Proj1 z V
5 opeq1 4578 . . . . . . . . 9 (x = Proj1 zx, y = Proj1 z, y)
65eleq1d 2419 . . . . . . . 8 (x = Proj1 z → (x, y A Proj1 z, y A))
75eleq1d 2419 . . . . . . . 8 (x = Proj1 z → (x, y B Proj1 z, y B))
86, 7imbi12d 311 . . . . . . 7 (x = Proj1 z → ((x, y Ax, y B) ↔ ( Proj1 z, y A Proj1 z, y B)))
98albidv 1625 . . . . . 6 (x = Proj1 z → (y(x, y Ax, y B) ↔ y( Proj1 z, y A Proj1 z, y B)))
104, 9spcv 2945 . . . . 5 (xy(x, y Ax, y B) → y( Proj1 z, y A Proj1 z, y B))
113proj2ex 4594 . . . . . 6 Proj2 z V
12 opeq2 4579 . . . . . . . 8 (y = Proj2 z Proj1 z, y = Proj1 z, Proj2 z)
1312eleq1d 2419 . . . . . . 7 (y = Proj2 z → ( Proj1 z, y A Proj1 z, Proj2 z A))
1412eleq1d 2419 . . . . . . 7 (y = Proj2 z → ( Proj1 z, y B Proj1 z, Proj2 z B))
1513, 14imbi12d 311 . . . . . 6 (y = Proj2 z → (( Proj1 z, y A Proj1 z, y B) ↔ ( Proj1 z, Proj2 z A Proj1 z, Proj2 z B)))
1611, 15spcv 2945 . . . . 5 (y( Proj1 z, y A Proj1 z, y B) → ( Proj1 z, Proj2 z A Proj1 z, Proj2 z B))
1710, 16syl 15 . . . 4 (xy(x, y Ax, y B) → ( Proj1 z, Proj2 z A Proj1 z, Proj2 z B))
18 opeq 4619 . . . . 5 z = Proj1 z, Proj2 z
1918eleq1i 2416 . . . 4 (z A Proj1 z, Proj2 z A)
2018eleq1i 2416 . . . 4 (z B Proj1 z, Proj2 z B)
2117, 19, 203imtr4g 261 . . 3 (xy(x, y Ax, y B) → (z Az B))
2221ssrdv 3278 . 2 (xy(x, y Ax, y B) → A B)
232, 22impbii 180 1 (A Bxy(x, y Ax, y B))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 176  wal 1540   = wceq 1642   wcel 1710   wss 3257  cop 4561   Proj1 cproj1 4563   Proj2 cproj2 4564
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 4078  ax-xp 4079  ax-cnv 4080  ax-1c 4081  ax-sset 4082  ax-si 4083  ax-ins2 4084  ax-ins3 4085  ax-typlower 4086  ax-sn 4087
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 2478  df-ne 2518  df-ral 2619  df-rex 2620  df-rab 2623  df-v 2861  df-sbc 3047  df-nin 3211  df-compl 3212  df-in 3213  df-un 3214  df-dif 3215  df-symdif 3216  df-ss 3259  df-nul 3551  df-if 3663  df-pw 3724  df-sn 3741  df-pr 3742  df-uni 3892  df-int 3927  df-opk 4058  df-1c 4136  df-pw1 4137  df-uni1 4138  df-xpk 4185  df-cnvk 4186  df-ins2k 4187  df-ins3k 4188  df-imak 4189  df-cok 4190  df-p6 4191  df-sik 4192  df-ssetk 4193  df-imagek 4194  df-idk 4195  df-0c 4377  df-addc 4378  df-nnc 4379  df-phi 4565  df-op 4566  df-proj1 4567  df-proj2 4568
This theorem is referenced by:  eqrel  4845  ssopr  4846  relssi  4848  relssdv  4849  cotr  5026  cnvsym  5027  intasym  5028  intirr  5029  ssdmrn  5099  dffun2  5119  fvfullfunlem2  5862
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