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Theorem mbfmco 27875
 Description: The composition of two measurable functions is measurable. ( cf. cnmpt11 19899) (Contributed by Thierry Arnoux, 4-Jun-2017.)
Hypotheses
Ref Expression
mbfmco.1 sigAlgebra
mbfmco.2 sigAlgebra
mbfmco.3 sigAlgebra
mbfmco.4 MblFnM
mbfmco.5 MblFnM
Assertion
Ref Expression
mbfmco MblFnM

Proof of Theorem mbfmco
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 mbfmco.2 . . . . 5 sigAlgebra
2 mbfmco.3 . . . . 5 sigAlgebra
3 mbfmco.5 . . . . 5 MblFnM
41, 2, 3mbfmf 27866 . . . 4
5 mbfmco.1 . . . . 5 sigAlgebra
6 mbfmco.4 . . . . 5 MblFnM
75, 1, 6mbfmf 27866 . . . 4
8 fco 5739 . . . 4
94, 7, 8syl2anc 661 . . 3
10 unielsiga 27768 . . . . 5 sigAlgebra
112, 10syl 16 . . . 4
12 unielsiga 27768 . . . . 5 sigAlgebra
135, 12syl 16 . . . 4
14 elmapg 7430 . . . 4
1511, 13, 14syl2anc 661 . . 3
169, 15mpbird 232 . 2
17 cnvco 5186 . . . . . 6
1817imaeq1i 5332 . . . . 5
19 imaco 5510 . . . . 5
2018, 19eqtri 2496 . . . 4
215adantr 465 . . . . 5 sigAlgebra
221adantr 465 . . . . 5 sigAlgebra
236adantr 465 . . . . 5 MblFnM
242adantr 465 . . . . . 6 sigAlgebra
253adantr 465 . . . . . 6 MblFnM
26 simpr 461 . . . . . 6
2722, 24, 25, 26mbfmcnvima 27868 . . . . 5
2821, 22, 23, 27mbfmcnvima 27868 . . . 4
2920, 28syl5eqel 2559 . . 3
3029ralrimiva 2878 . 2
315, 2ismbfm 27863 . 2 MblFnM
3216, 30, 31mpbir2and 920 1 MblFnM
 Colors of variables: wff setvar class Syntax hints:   wi 4   wb 184   wa 369   wcel 1767  wral 2814  cuni 4245  ccnv 4998   crn 5000  cima 5002   ccom 5003  wf 5582  (class class class)co 6282   cmap 7417  sigAlgebracsiga 27747  MblFnMcmbfm 27861 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-8 1769  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-pow 4625  ax-pr 4686  ax-un 6574 This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-fal 1385  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-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  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-fn 5589  df-f 5590  df-fv 5594  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-1st 6781  df-2nd 6782  df-map 7419  df-siga 27748  df-mbfm 27862 This theorem is referenced by:  rrvadd  28031  rrvmulc  28032
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