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Theorem fvvolicof 44322
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 (πœ‘ β†’ 𝐹:𝐴⟢(ℝ* Γ— ℝ*))
fvvolicof.x (πœ‘ β†’ 𝑋 ∈ 𝐴)
Assertion
Ref Expression
fvvolicof (πœ‘ β†’ (((vol ∘ [,)) ∘ 𝐹)β€˜π‘‹) = (volβ€˜((1st β€˜(πΉβ€˜π‘‹))[,)(2nd β€˜(πΉβ€˜π‘‹)))))

Proof of Theorem fvvolicof
StepHypRef Expression
1 fvvolicof.f . . . 4 (πœ‘ β†’ 𝐹:𝐴⟢(ℝ* Γ— ℝ*))
21ffund 6676 . . 3 (πœ‘ β†’ Fun 𝐹)
3 fvvolicof.x . . . 4 (πœ‘ β†’ 𝑋 ∈ 𝐴)
41fdmd 6683 . . . . 5 (πœ‘ β†’ dom 𝐹 = 𝐴)
54eqcomd 2739 . . . 4 (πœ‘ β†’ 𝐴 = dom 𝐹)
63, 5eleqtrd 2836 . . 3 (πœ‘ β†’ 𝑋 ∈ dom 𝐹)
7 fvco 6943 . . 3 ((Fun 𝐹 ∧ 𝑋 ∈ dom 𝐹) β†’ (((vol ∘ [,)) ∘ 𝐹)β€˜π‘‹) = ((vol ∘ [,))β€˜(πΉβ€˜π‘‹)))
82, 6, 7syl2anc 585 . 2 (πœ‘ β†’ (((vol ∘ [,)) ∘ 𝐹)β€˜π‘‹) = ((vol ∘ [,))β€˜(πΉβ€˜π‘‹)))
9 icof 43531 . . . . 5 [,):(ℝ* Γ— ℝ*)βŸΆπ’« ℝ*
10 ffun 6675 . . . . 5 ([,):(ℝ* Γ— ℝ*)βŸΆπ’« ℝ* β†’ Fun [,))
119, 10ax-mp 5 . . . 4 Fun [,)
1211a1i 11 . . 3 (πœ‘ β†’ Fun [,))
131, 3ffvelcdmd 7040 . . . 4 (πœ‘ β†’ (πΉβ€˜π‘‹) ∈ (ℝ* Γ— ℝ*))
149fdmi 6684 . . . 4 dom [,) = (ℝ* Γ— ℝ*)
1513, 14eleqtrrdi 2845 . . 3 (πœ‘ β†’ (πΉβ€˜π‘‹) ∈ dom [,))
16 fvco 6943 . . 3 ((Fun [,) ∧ (πΉβ€˜π‘‹) ∈ dom [,)) β†’ ((vol ∘ [,))β€˜(πΉβ€˜π‘‹)) = (volβ€˜([,)β€˜(πΉβ€˜π‘‹))))
1712, 15, 16syl2anc 585 . 2 (πœ‘ β†’ ((vol ∘ [,))β€˜(πΉβ€˜π‘‹)) = (volβ€˜([,)β€˜(πΉβ€˜π‘‹))))
18 df-ov 7364 . . . . 5 ((1st β€˜(πΉβ€˜π‘‹))[,)(2nd β€˜(πΉβ€˜π‘‹))) = ([,)β€˜βŸ¨(1st β€˜(πΉβ€˜π‘‹)), (2nd β€˜(πΉβ€˜π‘‹))⟩)
1918a1i 11 . . . 4 (πœ‘ β†’ ((1st β€˜(πΉβ€˜π‘‹))[,)(2nd β€˜(πΉβ€˜π‘‹))) = ([,)β€˜βŸ¨(1st β€˜(πΉβ€˜π‘‹)), (2nd β€˜(πΉβ€˜π‘‹))⟩))
20 1st2nd2 7964 . . . . . . 7 ((πΉβ€˜π‘‹) ∈ (ℝ* Γ— ℝ*) β†’ (πΉβ€˜π‘‹) = ⟨(1st β€˜(πΉβ€˜π‘‹)), (2nd β€˜(πΉβ€˜π‘‹))⟩)
2113, 20syl 17 . . . . . 6 (πœ‘ β†’ (πΉβ€˜π‘‹) = ⟨(1st β€˜(πΉβ€˜π‘‹)), (2nd β€˜(πΉβ€˜π‘‹))⟩)
2221eqcomd 2739 . . . . 5 (πœ‘ β†’ ⟨(1st β€˜(πΉβ€˜π‘‹)), (2nd β€˜(πΉβ€˜π‘‹))⟩ = (πΉβ€˜π‘‹))
2322fveq2d 6850 . . . 4 (πœ‘ β†’ ([,)β€˜βŸ¨(1st β€˜(πΉβ€˜π‘‹)), (2nd β€˜(πΉβ€˜π‘‹))⟩) = ([,)β€˜(πΉβ€˜π‘‹)))
2419, 23eqtr2d 2774 . . 3 (πœ‘ β†’ ([,)β€˜(πΉβ€˜π‘‹)) = ((1st β€˜(πΉβ€˜π‘‹))[,)(2nd β€˜(πΉβ€˜π‘‹))))
2524fveq2d 6850 . 2 (πœ‘ β†’ (volβ€˜([,)β€˜(πΉβ€˜π‘‹))) = (volβ€˜((1st β€˜(πΉβ€˜π‘‹))[,)(2nd β€˜(πΉβ€˜π‘‹)))))
268, 17, 253eqtrd 2777 1 (πœ‘ β†’ (((vol ∘ [,)) ∘ 𝐹)β€˜π‘‹) = (volβ€˜((1st β€˜(πΉβ€˜π‘‹))[,)(2nd β€˜(πΉβ€˜π‘‹)))))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   = wceq 1542   ∈ wcel 2107  π’« cpw 4564  βŸ¨cop 4596   Γ— cxp 5635  dom cdm 5637   ∘ ccom 5641  Fun wfun 6494  βŸΆwf 6496  β€˜cfv 6500  (class class class)co 7361  1st c1st 7923  2nd c2nd 7924  β„*cxr 11196  [,)cico 13275  volcvol 24850
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5260  ax-nul 5267  ax-pr 5388  ax-un 7676  ax-cnex 11115  ax-resscn 11116
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3449  df-sbc 3744  df-csb 3860  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4287  df-if 4491  df-pw 4566  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4870  df-iun 4960  df-br 5110  df-opab 5172  df-mpt 5193  df-id 5535  df-xp 5643  df-rel 5644  df-cnv 5645  df-co 5646  df-dm 5647  df-rn 5648  df-res 5649  df-ima 5650  df-iota 6452  df-fun 6502  df-fn 6503  df-f 6504  df-fv 6508  df-ov 7364  df-oprab 7365  df-mpo 7366  df-1st 7925  df-2nd 7926  df-xr 11201  df-ico 13279
This theorem is referenced by:  voliooicof  44327  volicofmpt  44328
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