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Theorem fvvolicof 45253
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 6712 . . 3 (πœ‘ β†’ Fun 𝐹)
3 fvvolicof.x . . . 4 (πœ‘ β†’ 𝑋 ∈ 𝐴)
41fdmd 6719 . . . . 5 (πœ‘ β†’ dom 𝐹 = 𝐴)
54eqcomd 2730 . . . 4 (πœ‘ β†’ 𝐴 = dom 𝐹)
63, 5eleqtrd 2827 . . 3 (πœ‘ β†’ 𝑋 ∈ dom 𝐹)
7 fvco 6980 . . 3 ((Fun 𝐹 ∧ 𝑋 ∈ dom 𝐹) β†’ (((vol ∘ [,)) ∘ 𝐹)β€˜π‘‹) = ((vol ∘ [,))β€˜(πΉβ€˜π‘‹)))
82, 6, 7syl2anc 583 . 2 (πœ‘ β†’ (((vol ∘ [,)) ∘ 𝐹)β€˜π‘‹) = ((vol ∘ [,))β€˜(πΉβ€˜π‘‹)))
9 icof 44464 . . . . 5 [,):(ℝ* Γ— ℝ*)βŸΆπ’« ℝ*
10 ffun 6711 . . . . 5 ([,):(ℝ* Γ— ℝ*)βŸΆπ’« ℝ* β†’ Fun [,))
119, 10ax-mp 5 . . . 4 Fun [,)
1211a1i 11 . . 3 (πœ‘ β†’ Fun [,))
131, 3ffvelcdmd 7078 . . . 4 (πœ‘ β†’ (πΉβ€˜π‘‹) ∈ (ℝ* Γ— ℝ*))
149fdmi 6720 . . . 4 dom [,) = (ℝ* Γ— ℝ*)
1513, 14eleqtrrdi 2836 . . 3 (πœ‘ β†’ (πΉβ€˜π‘‹) ∈ dom [,))
16 fvco 6980 . . 3 ((Fun [,) ∧ (πΉβ€˜π‘‹) ∈ dom [,)) β†’ ((vol ∘ [,))β€˜(πΉβ€˜π‘‹)) = (volβ€˜([,)β€˜(πΉβ€˜π‘‹))))
1712, 15, 16syl2anc 583 . 2 (πœ‘ β†’ ((vol ∘ [,))β€˜(πΉβ€˜π‘‹)) = (volβ€˜([,)β€˜(πΉβ€˜π‘‹))))
18 df-ov 7405 . . . . 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 8008 . . . . . . 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 2730 . . . . 5 (πœ‘ β†’ ⟨(1st β€˜(πΉβ€˜π‘‹)), (2nd β€˜(πΉβ€˜π‘‹))⟩ = (πΉβ€˜π‘‹))
2322fveq2d 6886 . . . 4 (πœ‘ β†’ ([,)β€˜βŸ¨(1st β€˜(πΉβ€˜π‘‹)), (2nd β€˜(πΉβ€˜π‘‹))⟩) = ([,)β€˜(πΉβ€˜π‘‹)))
2419, 23eqtr2d 2765 . . 3 (πœ‘ β†’ ([,)β€˜(πΉβ€˜π‘‹)) = ((1st β€˜(πΉβ€˜π‘‹))[,)(2nd β€˜(πΉβ€˜π‘‹))))
2524fveq2d 6886 . 2 (πœ‘ β†’ (volβ€˜([,)β€˜(πΉβ€˜π‘‹))) = (volβ€˜((1st β€˜(πΉβ€˜π‘‹))[,)(2nd β€˜(πΉβ€˜π‘‹)))))
268, 17, 253eqtrd 2768 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 1533   ∈ wcel 2098  π’« cpw 4595  βŸ¨cop 4627   Γ— cxp 5665  dom cdm 5667   ∘ ccom 5671  Fun wfun 6528  βŸΆwf 6530  β€˜cfv 6534  (class class class)co 7402  1st c1st 7967  2nd c2nd 7968  β„*cxr 11246  [,)cico 13327  volcvol 25336
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-sep 5290  ax-nul 5297  ax-pr 5418  ax-un 7719  ax-cnex 11163  ax-resscn 11164
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-ral 3054  df-rex 3063  df-rab 3425  df-v 3468  df-sbc 3771  df-csb 3887  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-nul 4316  df-if 4522  df-pw 4597  df-sn 4622  df-pr 4624  df-op 4628  df-uni 4901  df-iun 4990  df-br 5140  df-opab 5202  df-mpt 5223  df-id 5565  df-xp 5673  df-rel 5674  df-cnv 5675  df-co 5676  df-dm 5677  df-rn 5678  df-res 5679  df-ima 5680  df-iota 6486  df-fun 6536  df-fn 6537  df-f 6538  df-fv 6542  df-ov 7405  df-oprab 7406  df-mpo 7407  df-1st 7969  df-2nd 7970  df-xr 11251  df-ico 13331
This theorem is referenced by:  voliooicof  45258  volicofmpt  45259
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