Theorem mbfdmssre 40217

Description: The domain of a measurable function is a subset of the Reals. (Contributed by Glauco Siliprandi, 26-Jun-2021.)

Assertion

Ref	Expression
mbfdmssre	|- ( F e. MblFn -> dom F C_ RR )

Proof of Theorem mbfdmssre

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ismbf1 23393	. . 3 |- ( F e. MblFn <-> ( F e. ( CC ^pm RR ) /\ A. x e. ran (,) ( ( `' ( Re o. F ) " x ) e. dom vol /\ ( `' ( Im o. F ) " x ) e. dom vol ) ) )
2	1	simplbi 476	. 2 |- ( F e. MblFn -> F e. ( CC ^pm RR ) )
3		elpmi2 39418	. 2 |- ( F e. ( CC ^pm RR ) -> dom F C_ RR )
4	2, 3	syl 17	1 |- ( F e. MblFn -> dom F C_ RR )

Syntax hints:   -> wi 4   /\ wa 384   e. wcel 1990   A.wral 2912   C_ wss 3574   `'ccnv 5113   dom cdm 5114   ran crn 5115   "cima 5117   o. ccom 5118  (class class class)co 6650   ^pm cpm 7858   CCcc 9934   RRcr 9935   (,)cioo 12175   Recre 13837   Imcim 13838   volcvol 23232  MblFncmbf 23383

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-1st 7168  df-2nd 7169  df-pm 7860  df-mbf 23388

This theorem is referenced by:  mbfresmf  40948