Theorem mbfmfun 30316

Description: A measurable function is a function. (Contributed by Thierry Arnoux, 24-Jan-2017.)

Hypothesis

Ref	Expression
mbfmfun.1	|- ( ph -> F e. U. ran MblFnM )

Assertion

Ref	Expression
mbfmfun	|- ( ph -> Fun F )

Proof of Theorem mbfmfun

Dummy variables t s x are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		mbfmfun.1	. 2 |- ( ph -> F e. U. ran MblFnM )
2		elunirnmbfm 30315	. . 3 |- ( F e. U. ran MblFnM <-> E. s e. U. ran sigAlgebra E. t e. U. ran sigAlgebra ( F e. ( U. t ^m U. s ) /\ A. x e. t ( `' F " x ) e. s ) )
3	2	biimpi 206	. 2 |- ( F e. U. ran MblFnM -> E. s e. U. ran sigAlgebra E. t e. U. ran sigAlgebra ( F e. ( U. t ^m U. s ) /\ A. x e. t ( `' F " x ) e. s ) )
4		elmapfun 7881	. . . . 5 |- ( F e. ( U. t ^m U. s ) -> Fun F )
5	4	adantr 481	. . . 4 |- ( ( F e. ( U. t ^m U. s ) /\ A. x e. t ( `' F " x ) e. s ) -> Fun F )
6	5	rexlimivw 3029	. . 3 |- ( E. t e. U. ran sigAlgebra ( F e. ( U. t ^m U. s ) /\ A. x e. t ( `' F " x ) e. s ) -> Fun F )
7	6	rexlimivw 3029	. 2 |- ( E. s e. U. ran sigAlgebra E. t e. U. ran sigAlgebra ( F e. ( U. t ^m U. s ) /\ A. x e. t ( `' F " x ) e. s ) -> Fun F )
8	1, 3, 7	3syl 18	1 |- ( ph -> Fun F )

Syntax hints:   -> wi 4   /\ wa 384   e. wcel 1990   A.wral 2912   E.wrex 2913   U.cuni 4436   `'ccnv 5113   ran crn 5115   "cima 5117   Fun wfun 5882  (class class class)co 6650   ^m cmap 7857  sigAlgebracsiga 30170  MblFnMcmbfm 30312

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-map 7859  df-mbfm 30313

This theorem is referenced by:  orvcval4  30522