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Theorem fvvolicof 41840
 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 (𝜑𝐹:𝐴⟶(ℝ* × ℝ*))
fvvolicof.x (𝜑𝑋𝐴)
Assertion
Ref Expression
fvvolicof (𝜑 → (((vol ∘ [,)) ∘ 𝐹)‘𝑋) = (vol‘((1st ‘(𝐹𝑋))[,)(2nd ‘(𝐹𝑋)))))

Proof of Theorem fvvolicof
StepHypRef Expression
1 fvvolicof.f . . . 4 (𝜑𝐹:𝐴⟶(ℝ* × ℝ*))
21ffund 6393 . . 3 (𝜑 → Fun 𝐹)
3 fvvolicof.x . . . 4 (𝜑𝑋𝐴)
41fdmd 6398 . . . . 5 (𝜑 → dom 𝐹 = 𝐴)
54eqcomd 2803 . . . 4 (𝜑𝐴 = dom 𝐹)
63, 5eleqtrd 2887 . . 3 (𝜑𝑋 ∈ dom 𝐹)
7 fvco 6633 . . 3 ((Fun 𝐹𝑋 ∈ dom 𝐹) → (((vol ∘ [,)) ∘ 𝐹)‘𝑋) = ((vol ∘ [,))‘(𝐹𝑋)))
82, 6, 7syl2anc 584 . 2 (𝜑 → (((vol ∘ [,)) ∘ 𝐹)‘𝑋) = ((vol ∘ [,))‘(𝐹𝑋)))
9 icof 41043 . . . . 5 [,):(ℝ* × ℝ*)⟶𝒫 ℝ*
10 ffun 6392 . . . . 5 ([,):(ℝ* × ℝ*)⟶𝒫 ℝ* → Fun [,))
119, 10ax-mp 5 . . . 4 Fun [,)
1211a1i 11 . . 3 (𝜑 → Fun [,))
131, 3ffvelrnd 6724 . . . 4 (𝜑 → (𝐹𝑋) ∈ (ℝ* × ℝ*))
149fdmi 6399 . . . 4 dom [,) = (ℝ* × ℝ*)
1513, 14syl6eleqr 2896 . . 3 (𝜑 → (𝐹𝑋) ∈ dom [,))
16 fvco 6633 . . 3 ((Fun [,) ∧ (𝐹𝑋) ∈ dom [,)) → ((vol ∘ [,))‘(𝐹𝑋)) = (vol‘([,)‘(𝐹𝑋))))
1712, 15, 16syl2anc 584 . 2 (𝜑 → ((vol ∘ [,))‘(𝐹𝑋)) = (vol‘([,)‘(𝐹𝑋))))
18 df-ov 7026 . . . . 5 ((1st ‘(𝐹𝑋))[,)(2nd ‘(𝐹𝑋))) = ([,)‘⟨(1st ‘(𝐹𝑋)), (2nd ‘(𝐹𝑋))⟩)
1918a1i 11 . . . 4 (𝜑 → ((1st ‘(𝐹𝑋))[,)(2nd ‘(𝐹𝑋))) = ([,)‘⟨(1st ‘(𝐹𝑋)), (2nd ‘(𝐹𝑋))⟩))
20 1st2nd2 7591 . . . . . . 7 ((𝐹𝑋) ∈ (ℝ* × ℝ*) → (𝐹𝑋) = ⟨(1st ‘(𝐹𝑋)), (2nd ‘(𝐹𝑋))⟩)
2113, 20syl 17 . . . . . 6 (𝜑 → (𝐹𝑋) = ⟨(1st ‘(𝐹𝑋)), (2nd ‘(𝐹𝑋))⟩)
2221eqcomd 2803 . . . . 5 (𝜑 → ⟨(1st ‘(𝐹𝑋)), (2nd ‘(𝐹𝑋))⟩ = (𝐹𝑋))
2322fveq2d 6549 . . . 4 (𝜑 → ([,)‘⟨(1st ‘(𝐹𝑋)), (2nd ‘(𝐹𝑋))⟩) = ([,)‘(𝐹𝑋)))
2419, 23eqtr2d 2834 . . 3 (𝜑 → ([,)‘(𝐹𝑋)) = ((1st ‘(𝐹𝑋))[,)(2nd ‘(𝐹𝑋))))
2524fveq2d 6549 . 2 (𝜑 → (vol‘([,)‘(𝐹𝑋))) = (vol‘((1st ‘(𝐹𝑋))[,)(2nd ‘(𝐹𝑋)))))
268, 17, 253eqtrd 2837 1 (𝜑 → (((vol ∘ [,)) ∘ 𝐹)‘𝑋) = (vol‘((1st ‘(𝐹𝑋))[,)(2nd ‘(𝐹𝑋)))))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1525   ∈ wcel 2083  𝒫 cpw 4459  ⟨cop 4484   × cxp 5448  dom cdm 5450   ∘ ccom 5454  Fun wfun 6226  ⟶wf 6228  ‘cfv 6232  (class class class)co 7023  1st c1st 7550  2nd c2nd 7551  ℝ*cxr 10527  [,)cico 12594  volcvol 23751 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1781  ax-4 1795  ax-5 1892  ax-6 1951  ax-7 1996  ax-8 2085  ax-9 2093  ax-10 2114  ax-11 2128  ax-12 2143  ax-13 2346  ax-ext 2771  ax-sep 5101  ax-nul 5108  ax-pow 5164  ax-pr 5228  ax-un 7326  ax-cnex 10446  ax-resscn 10447 This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3an 1082  df-tru 1528  df-ex 1766  df-nf 1770  df-sb 2045  df-mo 2578  df-eu 2614  df-clab 2778  df-cleq 2790  df-clel 2865  df-nfc 2937  df-ne 2987  df-ral 3112  df-rex 3113  df-rab 3116  df-v 3442  df-sbc 3712  df-csb 3818  df-dif 3868  df-un 3870  df-in 3872  df-ss 3880  df-nul 4218  df-if 4388  df-pw 4461  df-sn 4479  df-pr 4481  df-op 4485  df-uni 4752  df-iun 4833  df-br 4969  df-opab 5031  df-mpt 5048  df-id 5355  df-xp 5456  df-rel 5457  df-cnv 5458  df-co 5459  df-dm 5460  df-rn 5461  df-res 5462  df-ima 5463  df-iota 6196  df-fun 6234  df-fn 6235  df-f 6236  df-fv 6240  df-ov 7026  df-oprab 7027  df-mpo 7028  df-1st 7552  df-2nd 7553  df-xr 10532  df-ico 12598 This theorem is referenced by:  voliooicof  41845  volicofmpt  41846
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