Theorem fvvolicof 40208

Description: The function value of the Lebesgue measure of a left-closed right-open interval composed with a function. (Contributed by Glauco Siliprandi, 3-Mar-2021.)

Hypotheses

Ref	Expression
fvvolicof.f	|- ( ph -> F : A --> ( RR* X. RR* ) )
fvvolicof.x	|- ( ph -> X e. A )

Assertion

Ref	Expression
fvvolicof	|- ( ph -> ( ( ( vol o. [,) ) o. F ) ` X ) = ( vol ` ( ( 1st ` ( F ` X ) ) [,) ( 2nd ` ( F ` X ) ) ) ) )

Proof of Theorem fvvolicof

Step	Hyp	Ref	Expression
1		fvvolicof.f	. . . 4 |- ( ph -> F : A --> ( RR* X. RR* ) )
2		ffun 6048	. . . 4 |- ( F : A --> ( RR* X. RR* ) -> Fun F )
3	1, 2	syl 17	. . 3 |- ( ph -> Fun F )
4		fvvolicof.x	. . . 4 |- ( ph -> X e. A )
5		fdm 6051	. . . . . 6 |- ( F : A --> ( RR* X. RR* ) -> dom F = A )
6	1, 5	syl 17	. . . . 5 |- ( ph -> dom F = A )
7	6	eqcomd 2628	. . . 4 |- ( ph -> A = dom F )
8	4, 7	eleqtrd 2703	. . 3 |- ( ph -> X e. dom F )
9		fvco 6274	. . 3 |- ( ( Fun F /\ X e. dom F ) -> ( ( ( vol o. [,) ) o. F ) ` X ) = ( ( vol o. [,) ) ` ( F ` X ) ) )
10	3, 8, 9	syl2anc 693	. 2 |- ( ph -> ( ( ( vol o. [,) ) o. F ) ` X ) = ( ( vol o. [,) ) ` ( F ` X ) ) )
11		icof 39411	. . . . 5 |- [,) : ( RR* X. RR* ) --> ~P RR*
12		ffun 6048	. . . . 5 |- ( [,) : ( RR* X. RR* ) --> ~P RR* -> Fun [,) )
13	11, 12	ax-mp 5	. . . 4 |- Fun [,)
14	13	a1i 11	. . 3 |- ( ph -> Fun [,) )
15	1, 4	ffvelrnd 6360	. . . 4 |- ( ph -> ( F ` X ) e. ( RR* X. RR* ) )
16	11	fdmi 6052	. . . 4 |- dom [,) = ( RR* X. RR* )
17	15, 16	syl6eleqr 2712	. . 3 |- ( ph -> ( F ` X ) e. dom [,) )
18		fvco 6274	. . 3 |- ( ( Fun [,) /\ ( F ` X ) e. dom [,) ) -> ( ( vol o. [,) ) ` ( F ` X ) ) = ( vol ` ( [,) ` ( F ` X ) ) ) )
19	14, 17, 18	syl2anc 693	. 2 |- ( ph -> ( ( vol o. [,) ) ` ( F ` X ) ) = ( vol ` ( [,) ` ( F ` X ) ) ) )
20		df-ov 6653	. . . . 5 |- ( ( 1st ` ( F ` X ) ) [,) ( 2nd ` ( F ` X ) ) ) = ( [,) ` <. ( 1st ` ( F ` X ) ) , ( 2nd ` ( F ` X ) ) >. )
21	20	a1i 11	. . . 4 |- ( ph -> ( ( 1st ` ( F ` X ) ) [,) ( 2nd ` ( F ` X ) ) ) = ( [,) ` <. ( 1st ` ( F ` X ) ) , ( 2nd ` ( F ` X ) ) >. ) )
22		1st2nd2 7205	. . . . . . 7 |- ( ( F ` X ) e. ( RR* X. RR* ) -> ( F ` X ) = <. ( 1st ` ( F ` X ) ) , ( 2nd ` ( F ` X ) ) >. )
23	15, 22	syl 17	. . . . . 6 |- ( ph -> ( F ` X ) = <. ( 1st ` ( F ` X ) ) , ( 2nd ` ( F ` X ) ) >. )
24	23	eqcomd 2628	. . . . 5 |- ( ph -> <. ( 1st ` ( F ` X ) ) , ( 2nd ` ( F ` X ) ) >. = ( F ` X ) )
25	24	fveq2d 6195	. . . 4 |- ( ph -> ( [,) ` <. ( 1st ` ( F ` X ) ) , ( 2nd ` ( F ` X ) ) >. ) = ( [,) ` ( F ` X ) ) )
26	21, 25	eqtr2d 2657	. . 3 |- ( ph -> ( [,) ` ( F ` X ) ) = ( ( 1st ` ( F ` X ) ) [,) ( 2nd ` ( F ` X ) ) ) )
27	26	fveq2d 6195	. 2 |- ( ph -> ( vol ` ( [,) ` ( F ` X ) ) ) = ( vol ` ( ( 1st ` ( F ` X ) ) [,) ( 2nd ` ( F ` X ) ) ) ) )
28	10, 19, 27	3eqtrd 2660	1 |- ( ph -> ( ( ( vol o. [,) ) o. F ) ` X ) = ( vol ` ( ( 1st ` ( F ` X ) ) [,) ( 2nd ` ( F ` X ) ) ) ) )

Syntax hints:   -> wi 4   = wceq 1483   e. wcel 1990   ~Pcpw 4158   <.cop 4183   X. cxp 5112   dom cdm 5114   o. ccom 5118   Fun wfun 5882   -->wf 5884   ` cfv 5888  (class class class)co 6650   1stc1st 7166   2ndc2nd 7167   RR*cxr 10073   [,)cico 12177   volcvol 23232

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  ax-cnex 9992  ax-resscn 9993

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-xr 10078  df-ico 12181

This theorem is referenced by:  voliooicof  40213  volicofmpt  40214