Theorem mbfmcnvima 30319

Description: The preimage by a measurable function is a measurable set. (Contributed by Thierry Arnoux, 23-Jan-2017.)

Hypotheses

Ref	Expression
mbfmf.1	|- ( ph -> S e. U. ran sigAlgebra )
mbfmf.2	|- ( ph -> T e. U. ran sigAlgebra )
mbfmf.3	|- ( ph -> F e. ( SMblFnM T ) )
mbfmcnvima.4	|- ( ph -> A e. T )

Assertion

Ref	Expression
mbfmcnvima	|- ( ph -> ( `' F " A ) e. S )

Proof of Theorem mbfmcnvima

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		mbfmcnvima.4	. 2 |- ( ph -> A e. T )
2		mbfmf.3	. . . 4 |- ( ph -> F e. ( SMblFnM T ) )
3		mbfmf.1	. . . . 5 |- ( ph -> S e. U. ran sigAlgebra )
4		mbfmf.2	. . . . 5 |- ( ph -> T e. U. ran sigAlgebra )
5	3, 4	ismbfm 30314	. . . 4 |- ( ph -> ( F e. ( SMblFnM T ) <-> ( F e. ( U. T ^m U. S ) /\ A. x e. T ( `' F " x ) e. S ) ) )
6	2, 5	mpbid 222	. . 3 |- ( ph -> ( F e. ( U. T ^m U. S ) /\ A. x e. T ( `' F " x ) e. S ) )
7	6	simprd 479	. 2 |- ( ph -> A. x e. T ( `' F " x ) e. S )
8		imaeq2 5462	. . . 4 |- ( x = A -> ( `' F " x ) = ( `' F " A ) )
9	8	eleq1d 2686	. . 3 |- ( x = A -> ( ( `' F " x ) e. S <-> ( `' F " A ) e. S ) )
10	9	rspcv 3305	. 2 |- ( A e. T -> ( A. x e. T ( `' F " x ) e. S -> ( `' F " A ) e. S ) )
11	1, 7, 10	sylc 65	1 |- ( ph -> ( `' F " A ) e. S )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   U.cuni 4436   `'ccnv 5113   ran crn 5115   "cima 5117  (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-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  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-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  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-br 4654  df-opab 4713  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-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-mbfm 30313

This theorem is referenced by:  imambfm  30324  mbfmco  30326  mbfmco2  30327  sxbrsiga  30352  sibfinima  30401  sibfof  30402  orvcoel  30523  orvccel  30524