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Theorem brcap 25503
 Description: Binary relationship form of the Cap function. (Contributed by Scott Fenton, 17-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
Hypotheses
Ref Expression
brcap.1
brcap.2
brcap.3
Assertion
Ref Expression
brcap Cap

Proof of Theorem brcap
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 opex 4368 . 2
2 brcap.3 . 2
3 df-cap 25435 . 2 Cap (++)
4 brcap.1 . . . 4
5 brcap.2 . . . 4
64, 5opelvv 4864 . . 3
7 brxp 4849 . . 3
86, 2, 7mpbir2an 887 . 2
9 vex 2902 . . . . . . 7
109, 1brco 4983 . . . . . 6
11 ancom 438 . . . . . . . 8
12 vex 2902 . . . . . . . . . . 11
1312, 1brcnv 4995 . . . . . . . . . 10
144, 5, 12br1steq 25154 . . . . . . . . . 10
1513, 14bitri 241 . . . . . . . . 9
1615anbi1i 677 . . . . . . . 8
1711, 16bitri 241 . . . . . . 7
1817exbii 1589 . . . . . 6
1910, 18bitri 241 . . . . 5
20 breq2 4157 . . . . . 6
214, 20ceqsexv 2934 . . . . 5
224epelc 4437 . . . . 5
2319, 21, 223bitri 263 . . . 4
249, 1brco 4983 . . . . . 6
25 ancom 438 . . . . . . . 8
2612, 1brcnv 4995 . . . . . . . . . 10
274, 5, 12br2ndeq 25155 . . . . . . . . . 10
2826, 27bitri 241 . . . . . . . . 9
2928anbi1i 677 . . . . . . . 8
3025, 29bitri 241 . . . . . . 7
3130exbii 1589 . . . . . 6
3224, 31bitri 241 . . . . 5
33 breq2 4157 . . . . . 6
345, 33ceqsexv 2934 . . . . 5
355epelc 4437 . . . . 5
3632, 34, 353bitri 263 . . . 4
3723, 36anbi12i 679 . . 3
38 brin 4200 . . 3
39 elin 3473 . . 3
4037, 38, 393bitr4ri 270 . 2
411, 2, 3, 8, 40brtxpsd3 25460 1 Cap
 Colors of variables: wff set class Syntax hints:   wb 177   wa 359  wex 1547   wceq 1649   wcel 1717  cvv 2899   cin 3262  cop 3760   class class class wbr 4153   cep 4433   cxp 4816  ccnv 4817   ccom 4822  c1st 6286  c2nd 6287  Capccap 25414 This theorem is referenced by:  brrestrict  25512 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368  ax-sep 4271  ax-nul 4279  ax-pow 4318  ax-pr 4344  ax-un 4641 This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2242  df-mo 2243  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-ral 2654  df-rex 2655  df-rab 2658  df-v 2901  df-sbc 3105  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-nul 3572  df-if 3683  df-sn 3763  df-pr 3764  df-op 3766  df-uni 3958  df-br 4154  df-opab 4208  df-mpt 4209  df-eprel 4435  df-id 4439  df-xp 4824  df-rel 4825  df-cnv 4826  df-co 4827  df-dm 4828  df-rn 4829  df-res 4830  df-iota 5358  df-fun 5396  df-fn 5397  df-f 5398  df-fo 5400  df-fv 5402  df-1st 6288  df-2nd 6289  df-symdif 25386  df-txp 25419  df-cap 25435
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