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Theorem brcart 25778
 Description: Binary relationship form of the cartesian product operator. (Contributed by Scott Fenton, 11-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
Hypotheses
Ref Expression
brcart.1
brcart.2
brcart.3
Assertion
Ref Expression
brcart Cart

Proof of Theorem brcart
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 opex 4428 . 2
2 brcart.3 . 2
3 df-cart 25710 . 2 Cart (++)pprod
4 brcart.1 . . . 4
5 brcart.2 . . . 4
64, 5opelvv 4925 . . 3
7 brxp 4910 . . 3
86, 2, 7mpbir2an 888 . 2
9 3anass 941 . . . . 5
104epelc 4497 . . . . . . 7
115epelc 4497 . . . . . . 7
1210, 11anbi12i 680 . . . . . 6
1312anbi2i 677 . . . . 5
149, 13bitri 242 . . . 4
15142exbii 1594 . . 3
16 vex 2960 . . . 4
1716, 4, 5brpprod3b 25733 . . 3 pprod
18 elxp 4896 . . 3
1915, 17, 183bitr4ri 271 . 2 pprod
201, 2, 3, 8, 19brtxpsd3 25742 1 Cart
 Colors of variables: wff set class Syntax hints:   wb 178   wa 360   w3a 937  wex 1551   wceq 1653   wcel 1726  cvv 2957  cop 3818   class class class wbr 4213   cep 4493   cxp 4877  pprodcpprod 25676  Cartccart 25686 This theorem is referenced by:  brimg  25783  brrestrict  25795 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2418  ax-sep 4331  ax-nul 4339  ax-pow 4378  ax-pr 4404  ax-un 4702 This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2286  df-mo 2287  df-clab 2424  df-cleq 2430  df-clel 2433  df-nfc 2562  df-ne 2602  df-ral 2711  df-rex 2712  df-rab 2715  df-v 2959  df-sbc 3163  df-dif 3324  df-un 3326  df-in 3328  df-ss 3335  df-nul 3630  df-if 3741  df-sn 3821  df-pr 3822  df-op 3824  df-uni 4017  df-br 4214  df-opab 4268  df-mpt 4269  df-eprel 4495  df-id 4499  df-xp 4885  df-rel 4886  df-cnv 4887  df-co 4888  df-dm 4889  df-rn 4890  df-res 4891  df-iota 5419  df-fun 5457  df-fn 5458  df-f 5459  df-fo 5461  df-fv 5463  df-1st 6350  df-2nd 6351  df-symdif 25664  df-txp 25699  df-pprod 25700  df-cart 25710
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