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Theorem ismbfm 27860
 Description: The predicate " is a measurable function from the measurable space to the measurable space ". Cf. ismbf 21769 (Contributed by Thierry Arnoux, 23-Jan-2017.)
Hypotheses
Ref Expression
ismbfm.1 sigAlgebra
ismbfm.2 sigAlgebra
Assertion
Ref Expression
ismbfm MblFnM
Distinct variable groups:   ,   ,   ,
Allowed substitution hint:   ()

Proof of Theorem ismbfm
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ismbfm.1 . . . 4 sigAlgebra
2 ismbfm.2 . . . 4 sigAlgebra
3 unieq 4253 . . . . . . 7
43oveq2d 6298 . . . . . 6
5 eleq2 2540 . . . . . . 7
65ralbidv 2903 . . . . . 6
74, 6rabeqbidv 3108 . . . . 5
8 unieq 4253 . . . . . . 7
98oveq1d 6297 . . . . . 6
10 raleq 3058 . . . . . 6
119, 10rabeqbidv 3108 . . . . 5
12 df-mbfm 27859 . . . . 5 MblFnM sigAlgebra sigAlgebra
13 ovex 6307 . . . . . 6
1413rabex 4598 . . . . 5
157, 11, 12, 14ovmpt2 6420 . . . 4 sigAlgebra sigAlgebra MblFnM
161, 2, 15syl2anc 661 . . 3 MblFnM
1716eleq2d 2537 . 2 MblFnM
18 cnveq 5174 . . . . . 6
1918imaeq1d 5334 . . . . 5
2019eleq1d 2536 . . . 4
2120ralbidv 2903 . . 3
2221elrab 3261 . 2
2317, 22syl6bb 261 1 MblFnM
 Colors of variables: wff setvar class Syntax hints:   wi 4   wb 184   wa 369   wceq 1379   wcel 1767  wral 2814  crab 2818  cuni 4245  ccnv 4998   crn 5000  cima 5002  (class class class)co 6282   cmap 7417  sigAlgebracsiga 27744  MblFnMcmbfm 27858 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pr 4686 This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fv 5594  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-mbfm 27859 This theorem is referenced by:  elunirnmbfm  27861  mbfmf  27863  isanmbfm  27864  mbfmcnvima  27865  mbfmcst  27867  1stmbfm  27868  2ndmbfm  27869  imambfm  27870  mbfmco  27872  elmbfmvol2  27875  mbfmcnt  27876  sibfof  27919  isrrvv  28019
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