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Theorem domintrefb 25166
 Description: The domain of the intersection of two reflexive classes is the intersection of their domains. Compare with dmin 4902. (Contributed by FL, 30-Dec-2011.)
Assertion
Ref Expression
domintrefb
Distinct variable groups:   ,   ,

Proof of Theorem domintrefb
StepHypRef Expression
1 inss1 3402 . . . . . . 7
21sseli 3189 . . . . . 6
32imim1i 54 . . . . 5
43ralimi2 2628 . . . 4
5 inss2 3403 . . . . . . 7
65sseli 3189 . . . . . 6
76imim1i 54 . . . . 5
87ralimi2 2628 . . . 4
9 r19.26 2688 . . . . 5
10 brin 4086 . . . . . . . 8
11 vex 2804 . . . . . . . . 9
1211, 11breldm 4899 . . . . . . . 8
1310, 12sylbir 204 . . . . . . 7
1413ralimi 2631 . . . . . 6
15 dfss3 3183 . . . . . 6
1614, 15sylibr 203 . . . . 5
179, 16sylbir 204 . . . 4
184, 8, 17syl2an 463 . . 3
19 dmin 4902 . . 3
2018, 19jctil 523 . 2
21 eqss 3207 . 2
2220, 21sylibr 203 1
 Colors of variables: wff set class Syntax hints:   wi 4   wa 358   wceq 1632   wcel 1696  wral 2556   cin 3164   wss 3165   class class class wbr 4039   cdm 4705 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ral 2561  df-rab 2565  df-v 2803  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-br 4040  df-dm 4715
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