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Theorem mpstval 29961
 Description: A pre-statement is an ordered triple, whose first member is a symmetric set of dv conditions, whose second member is a finite set of expressions, and whose third member is an expression. (Contributed by Mario Carneiro, 18-Jul-2016.)
Hypotheses
Ref Expression
mpstval.v mDV
mpstval.e mEx
mpstval.p mPreSt
Assertion
Ref Expression
mpstval
Distinct variable groups:   ,   ,
Allowed substitution hints:   ()   ()

Proof of Theorem mpstval
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 mpstval.p . 2 mPreSt
2 fveq2 5881 . . . . . . . . 9 mDV mDV
3 mpstval.v . . . . . . . . 9 mDV
42, 3syl6eqr 2488 . . . . . . . 8 mDV
54pweqd 3990 . . . . . . 7 mDV
6 biidd 240 . . . . . . 7
75, 6rabeqbidv 3082 . . . . . 6 mDV
8 fveq2 5881 . . . . . . . . 9 mEx mEx
9 mpstval.e . . . . . . . . 9 mEx
108, 9syl6eqr 2488 . . . . . . . 8 mEx
1110pweqd 3990 . . . . . . 7 mEx
1211ineq1d 3669 . . . . . 6 mEx
137, 12xpeq12d 4879 . . . . 5 mDV mEx
1413, 10xpeq12d 4879 . . . 4 mDV mEx mEx
15 df-mpst 29919 . . . 4 mPreSt mDV mEx mEx
16 fvex 5891 . . . . . . . . 9 mDV
173, 16eqeltri 2513 . . . . . . . 8
1817pwex 4608 . . . . . . 7
1918rabex 4576 . . . . . 6
20 fvex 5891 . . . . . . . . 9 mEx
219, 20eqeltri 2513 . . . . . . . 8
2221pwex 4608 . . . . . . 7
2322inex1 4566 . . . . . 6
2419, 23xpex 6609 . . . . 5
2524, 21xpex 6609 . . . 4
2614, 15, 25fvmpt 5964 . . 3 mPreSt
27 xp0 5275 . . . . 5
2827eqcomi 2442 . . . 4
29 fvprc 5875 . . . 4 mPreSt
30 fvprc 5875 . . . . . 6 mEx
319, 30syl5eq 2482 . . . . 5
3231xpeq2d 4878 . . . 4
3328, 29, 323eqtr4a 2496 . . 3 mPreSt
3426, 33pm2.61i 167 . 2 mPreSt
351, 34eqtri 2458 1
 Colors of variables: wff setvar class Syntax hints:   wn 3   wceq 1437   wcel 1870  crab 2786  cvv 3087   cin 3441  c0 3767  cpw 3985   cxp 4852  ccnv 4853  cfv 5601  cfn 7577  mExcmex 29893  mDVcmdv 29894  mPreStcmpst 29899 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661  ax-un 6597 This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-ral 2787  df-rex 2788  df-rab 2791  df-v 3089  df-sbc 3306  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-nul 3768  df-if 3916  df-pw 3987  df-sn 4003  df-pr 4005  df-op 4009  df-uni 4223  df-br 4427  df-opab 4485  df-mpt 4486  df-id 4769  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-iota 5565  df-fun 5603  df-fv 5609  df-mpst 29919 This theorem is referenced by:  elmpst  29962  mpstssv  29965
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