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Theorem ssrel2 5098
Description: A subclass relationship depends only on a relation's ordered pairs. This version of ssrel 5096 is restricted to the relation's domain. (Contributed by Thierry Arnoux, 25-Jan-2018.)
Assertion
Ref Expression
ssrel2
Distinct variable groups:   , ,   , ,   , ,   ,S,

Proof of Theorem ssrel2
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 ssel 3497 . . . 4
21a1d 25 . . 3
32ralrimivv 2877 . 2
4 eleq1 2529 . . . . . . . . . . . 12
5 eleq1 2529 . . . . . . . . . . . 12
64, 5imbi12d 320 . . . . . . . . . . 11
76biimprcd 225 . . . . . . . . . 10
87ralimi 2850 . . . . . . . . 9
98ralimi 2850 . . . . . . . 8
10 r19.23v 2937 . . . . . . . . . 10
1110ralbii 2888 . . . . . . . . 9
12 r19.23v 2937 . . . . . . . . 9
1311, 12bitri 249 . . . . . . . 8
149, 13sylib 196 . . . . . . 7
1514com23 78 . . . . . 6
1615a2d 26 . . . . 5
1716alimdv 1709 . . . 4
18 dfss2 3492 . . . . 5
19 elxp2 5022 . . . . . . 7
2019imbi2i 312 . . . . . 6
2120albii 1640 . . . . 5
2218, 21bitri 249 . . . 4
23 dfss2 3492 . . . 4
2417, 22, 233imtr4g 270 . . 3
2524com12 31 . 2
263, 25impbid2 204 1
Colors of variables: wff setvar class
Syntax hints:  ->wi 4  <->wb 184  /\wa 369  A.wal 1393  =wceq 1395  e.wcel 1818  A.wral 2807  E.wrex 2808  C_wss 3475  <.cop 4035  X.cxp 5002
This theorem is referenced by:  metuel2  21082  isarchi  27726
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1618  ax-4 1631  ax-5 1704  ax-6 1747  ax-7 1790  ax-9 1822  ax-10 1837  ax-11 1842  ax-12 1854  ax-13 1999  ax-ext 2435  ax-sep 4573  ax-nul 4581  ax-pr 4691
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1613  df-nf 1617  df-sb 1740  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-v 3111  df-dif 3478  df-un 3480  df-in 3482  df-ss 3489  df-nul 3785  df-if 3942  df-sn 4030  df-pr 4032  df-op 4036  df-opab 4511  df-xp 5010
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