Theorem sibfima 30400

Description: Any preimage of a singleton by a simple function is measurable. (Contributed by Thierry Arnoux, 19-Feb-2018.)

Hypotheses

Ref	Expression
sitgval.b	|- B = ( Base ` W )
sitgval.j	|- J = ( TopOpen ` W )
sitgval.s	|- S = (sigaGen ` J )
sitgval.0	|- .0. = ( 0g ` W )
sitgval.x	|- .x. = ( .s ` W )
sitgval.h	|- H = (RRHom ` (Scalar ` W ) )
sitgval.1	|- ( ph -> W e. V )
sitgval.2	|- ( ph -> M e. U. ran measures )
sibfmbl.1	|- ( ph -> F e. dom ( Wsitg M ) )

Assertion

Ref	Expression
sibfima	|- ( ( ph /\ A e. ( ran F \ { .0. } ) ) -> ( M ` ( `' F " { A } ) ) e. ( 0 [,) +oo ) )

Proof of Theorem sibfima

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		sibfmbl.1	. . . 4 |- ( ph -> F e. dom ( Wsitg M ) )
2		sitgval.b	. . . . 5 |- B = ( Base ` W )
3		sitgval.j	. . . . 5 |- J = ( TopOpen ` W )
4		sitgval.s	. . . . 5 |- S = (sigaGen ` J )
5		sitgval.0	. . . . 5 |- .0. = ( 0g ` W )
6		sitgval.x	. . . . 5 |- .x. = ( .s ` W )
7		sitgval.h	. . . . 5 |- H = (RRHom ` (Scalar ` W ) )
8		sitgval.1	. . . . 5 |- ( ph -> W e. V )
9		sitgval.2	. . . . 5 |- ( ph -> M e. U. ran measures )
10	2, 3, 4, 5, 6, 7, 8, 9	issibf 30395	. . . 4 |- ( ph -> ( F e. dom ( Wsitg M ) <-> ( F e. ( dom MMblFnM S ) /\ ran F e. Fin /\ A. x e. ( ran F \ { .0. } ) ( M ` ( `' F " { x } ) ) e. ( 0 [,) +oo ) ) ) )
11	1, 10	mpbid 222	. . 3 |- ( ph -> ( F e. ( dom MMblFnM S ) /\ ran F e. Fin /\ A. x e. ( ran F \ { .0. } ) ( M ` ( `' F " { x } ) ) e. ( 0 [,) +oo ) ) )
12	11	simp3d 1075	. 2 |- ( ph -> A. x e. ( ran F \ { .0. } ) ( M ` ( `' F " { x } ) ) e. ( 0 [,) +oo ) )
13		sneq 4187	. . . . . 6 |- ( x = A -> { x } = { A } )
14	13	imaeq2d 5466	. . . . 5 |- ( x = A -> ( `' F " { x } ) = ( `' F " { A } ) )
15	14	fveq2d 6195	. . . 4 |- ( x = A -> ( M ` ( `' F " { x } ) ) = ( M ` ( `' F " { A } ) ) )
16	15	eleq1d 2686	. . 3 |- ( x = A -> ( ( M ` ( `' F " { x } ) ) e. ( 0 [,) +oo ) <-> ( M ` ( `' F " { A } ) ) e. ( 0 [,) +oo ) ) )
17	16	rspcv 3305	. 2 |- ( A e. ( ran F \ { .0. } ) -> ( A. x e. ( ran F \ { .0. } ) ( M ` ( `' F " { x } ) ) e. ( 0 [,) +oo ) -> ( M ` ( `' F " { A } ) ) e. ( 0 [,) +oo ) ) )
18	12, 17	mpan9 486	1 |- ( ( ph /\ A e. ( ran F \ { .0. } ) ) -> ( M ` ( `' F " { A } ) ) e. ( 0 [,) +oo ) )

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   A.wral 2912   \ cdif 3571   {csn 4177   U.cuni 4436   `'ccnv 5113   dom cdm 5114   ran crn 5115   "cima 5117   ` cfv 5888  (class class class)co 6650   Fincfn 7955   0cc0 9936   +oocpnf 10071   [,)cico 12177   Basecbs 15857  Scalarcsca 15944   .scvsca 15945   TopOpenctopn 16082   0gc0g 16100  RRHomcrrh 30037  sigaGencsigagen 30201  measurescmeas 30258  MblFnMcmbfm 30312  sitgcsitg 30391

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-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pr 4906

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-reu 2919  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-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-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-sitg 30392

This theorem is referenced by:  sibfinima  30401  sitgfval  30403  sitgclg  30404