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Theorem uniiccdif 24945
Description: A union of closed intervals differs from the equivalent union of open intervals by a nullset. (Contributed by Mario Carneiro, 25-Mar-2015.)
Hypothesis
Ref Expression
uniioombl.1 (πœ‘ β†’ 𝐹:β„•βŸΆ( ≀ ∩ (ℝ Γ— ℝ)))
Assertion
Ref Expression
uniiccdif (πœ‘ β†’ (βˆͺ ran ((,) ∘ 𝐹) βŠ† βˆͺ ran ([,] ∘ 𝐹) ∧ (vol*β€˜(βˆͺ ran ([,] ∘ 𝐹) βˆ– βˆͺ ran ((,) ∘ 𝐹))) = 0))

Proof of Theorem uniiccdif
Dummy variable π‘₯ is distinct from all other variables.
StepHypRef Expression
1 ssun1 4133 . . 3 βˆͺ ran ((,) ∘ 𝐹) βŠ† (βˆͺ ran ((,) ∘ 𝐹) βˆͺ ((1st β€œ ran 𝐹) βˆͺ (2nd β€œ ran 𝐹)))
2 uniioombl.1 . . . . . . . 8 (πœ‘ β†’ 𝐹:β„•βŸΆ( ≀ ∩ (ℝ Γ— ℝ)))
3 ovolfcl 24833 . . . . . . . 8 ((𝐹:β„•βŸΆ( ≀ ∩ (ℝ Γ— ℝ)) ∧ π‘₯ ∈ β„•) β†’ ((1st β€˜(πΉβ€˜π‘₯)) ∈ ℝ ∧ (2nd β€˜(πΉβ€˜π‘₯)) ∈ ℝ ∧ (1st β€˜(πΉβ€˜π‘₯)) ≀ (2nd β€˜(πΉβ€˜π‘₯))))
42, 3sylan 581 . . . . . . 7 ((πœ‘ ∧ π‘₯ ∈ β„•) β†’ ((1st β€˜(πΉβ€˜π‘₯)) ∈ ℝ ∧ (2nd β€˜(πΉβ€˜π‘₯)) ∈ ℝ ∧ (1st β€˜(πΉβ€˜π‘₯)) ≀ (2nd β€˜(πΉβ€˜π‘₯))))
5 rexr 11202 . . . . . . . 8 ((1st β€˜(πΉβ€˜π‘₯)) ∈ ℝ β†’ (1st β€˜(πΉβ€˜π‘₯)) ∈ ℝ*)
6 rexr 11202 . . . . . . . 8 ((2nd β€˜(πΉβ€˜π‘₯)) ∈ ℝ β†’ (2nd β€˜(πΉβ€˜π‘₯)) ∈ ℝ*)
7 id 22 . . . . . . . 8 ((1st β€˜(πΉβ€˜π‘₯)) ≀ (2nd β€˜(πΉβ€˜π‘₯)) β†’ (1st β€˜(πΉβ€˜π‘₯)) ≀ (2nd β€˜(πΉβ€˜π‘₯)))
8 prunioo 13399 . . . . . . . 8 (((1st β€˜(πΉβ€˜π‘₯)) ∈ ℝ* ∧ (2nd β€˜(πΉβ€˜π‘₯)) ∈ ℝ* ∧ (1st β€˜(πΉβ€˜π‘₯)) ≀ (2nd β€˜(πΉβ€˜π‘₯))) β†’ (((1st β€˜(πΉβ€˜π‘₯))(,)(2nd β€˜(πΉβ€˜π‘₯))) βˆͺ {(1st β€˜(πΉβ€˜π‘₯)), (2nd β€˜(πΉβ€˜π‘₯))}) = ((1st β€˜(πΉβ€˜π‘₯))[,](2nd β€˜(πΉβ€˜π‘₯))))
95, 6, 7, 8syl3an 1161 . . . . . . 7 (((1st β€˜(πΉβ€˜π‘₯)) ∈ ℝ ∧ (2nd β€˜(πΉβ€˜π‘₯)) ∈ ℝ ∧ (1st β€˜(πΉβ€˜π‘₯)) ≀ (2nd β€˜(πΉβ€˜π‘₯))) β†’ (((1st β€˜(πΉβ€˜π‘₯))(,)(2nd β€˜(πΉβ€˜π‘₯))) βˆͺ {(1st β€˜(πΉβ€˜π‘₯)), (2nd β€˜(πΉβ€˜π‘₯))}) = ((1st β€˜(πΉβ€˜π‘₯))[,](2nd β€˜(πΉβ€˜π‘₯))))
104, 9syl 17 . . . . . 6 ((πœ‘ ∧ π‘₯ ∈ β„•) β†’ (((1st β€˜(πΉβ€˜π‘₯))(,)(2nd β€˜(πΉβ€˜π‘₯))) βˆͺ {(1st β€˜(πΉβ€˜π‘₯)), (2nd β€˜(πΉβ€˜π‘₯))}) = ((1st β€˜(πΉβ€˜π‘₯))[,](2nd β€˜(πΉβ€˜π‘₯))))
11 fvco3 6941 . . . . . . . . 9 ((𝐹:β„•βŸΆ( ≀ ∩ (ℝ Γ— ℝ)) ∧ π‘₯ ∈ β„•) β†’ (((,) ∘ 𝐹)β€˜π‘₯) = ((,)β€˜(πΉβ€˜π‘₯)))
122, 11sylan 581 . . . . . . . 8 ((πœ‘ ∧ π‘₯ ∈ β„•) β†’ (((,) ∘ 𝐹)β€˜π‘₯) = ((,)β€˜(πΉβ€˜π‘₯)))
132ffvelcdmda 7036 . . . . . . . . . . . 12 ((πœ‘ ∧ π‘₯ ∈ β„•) β†’ (πΉβ€˜π‘₯) ∈ ( ≀ ∩ (ℝ Γ— ℝ)))
1413elin2d 4160 . . . . . . . . . . 11 ((πœ‘ ∧ π‘₯ ∈ β„•) β†’ (πΉβ€˜π‘₯) ∈ (ℝ Γ— ℝ))
15 1st2nd2 7961 . . . . . . . . . . 11 ((πΉβ€˜π‘₯) ∈ (ℝ Γ— ℝ) β†’ (πΉβ€˜π‘₯) = ⟨(1st β€˜(πΉβ€˜π‘₯)), (2nd β€˜(πΉβ€˜π‘₯))⟩)
1614, 15syl 17 . . . . . . . . . 10 ((πœ‘ ∧ π‘₯ ∈ β„•) β†’ (πΉβ€˜π‘₯) = ⟨(1st β€˜(πΉβ€˜π‘₯)), (2nd β€˜(πΉβ€˜π‘₯))⟩)
1716fveq2d 6847 . . . . . . . . 9 ((πœ‘ ∧ π‘₯ ∈ β„•) β†’ ((,)β€˜(πΉβ€˜π‘₯)) = ((,)β€˜βŸ¨(1st β€˜(πΉβ€˜π‘₯)), (2nd β€˜(πΉβ€˜π‘₯))⟩))
18 df-ov 7361 . . . . . . . . 9 ((1st β€˜(πΉβ€˜π‘₯))(,)(2nd β€˜(πΉβ€˜π‘₯))) = ((,)β€˜βŸ¨(1st β€˜(πΉβ€˜π‘₯)), (2nd β€˜(πΉβ€˜π‘₯))⟩)
1917, 18eqtr4di 2795 . . . . . . . 8 ((πœ‘ ∧ π‘₯ ∈ β„•) β†’ ((,)β€˜(πΉβ€˜π‘₯)) = ((1st β€˜(πΉβ€˜π‘₯))(,)(2nd β€˜(πΉβ€˜π‘₯))))
2012, 19eqtrd 2777 . . . . . . 7 ((πœ‘ ∧ π‘₯ ∈ β„•) β†’ (((,) ∘ 𝐹)β€˜π‘₯) = ((1st β€˜(πΉβ€˜π‘₯))(,)(2nd β€˜(πΉβ€˜π‘₯))))
21 df-pr 4590 . . . . . . . 8 {((1st ∘ 𝐹)β€˜π‘₯), ((2nd ∘ 𝐹)β€˜π‘₯)} = ({((1st ∘ 𝐹)β€˜π‘₯)} βˆͺ {((2nd ∘ 𝐹)β€˜π‘₯)})
22 fvco3 6941 . . . . . . . . . 10 ((𝐹:β„•βŸΆ( ≀ ∩ (ℝ Γ— ℝ)) ∧ π‘₯ ∈ β„•) β†’ ((1st ∘ 𝐹)β€˜π‘₯) = (1st β€˜(πΉβ€˜π‘₯)))
232, 22sylan 581 . . . . . . . . 9 ((πœ‘ ∧ π‘₯ ∈ β„•) β†’ ((1st ∘ 𝐹)β€˜π‘₯) = (1st β€˜(πΉβ€˜π‘₯)))
24 fvco3 6941 . . . . . . . . . 10 ((𝐹:β„•βŸΆ( ≀ ∩ (ℝ Γ— ℝ)) ∧ π‘₯ ∈ β„•) β†’ ((2nd ∘ 𝐹)β€˜π‘₯) = (2nd β€˜(πΉβ€˜π‘₯)))
252, 24sylan 581 . . . . . . . . 9 ((πœ‘ ∧ π‘₯ ∈ β„•) β†’ ((2nd ∘ 𝐹)β€˜π‘₯) = (2nd β€˜(πΉβ€˜π‘₯)))
2623, 25preq12d 4703 . . . . . . . 8 ((πœ‘ ∧ π‘₯ ∈ β„•) β†’ {((1st ∘ 𝐹)β€˜π‘₯), ((2nd ∘ 𝐹)β€˜π‘₯)} = {(1st β€˜(πΉβ€˜π‘₯)), (2nd β€˜(πΉβ€˜π‘₯))})
2721, 26eqtr3id 2791 . . . . . . 7 ((πœ‘ ∧ π‘₯ ∈ β„•) β†’ ({((1st ∘ 𝐹)β€˜π‘₯)} βˆͺ {((2nd ∘ 𝐹)β€˜π‘₯)}) = {(1st β€˜(πΉβ€˜π‘₯)), (2nd β€˜(πΉβ€˜π‘₯))})
2820, 27uneq12d 4125 . . . . . 6 ((πœ‘ ∧ π‘₯ ∈ β„•) β†’ ((((,) ∘ 𝐹)β€˜π‘₯) βˆͺ ({((1st ∘ 𝐹)β€˜π‘₯)} βˆͺ {((2nd ∘ 𝐹)β€˜π‘₯)})) = (((1st β€˜(πΉβ€˜π‘₯))(,)(2nd β€˜(πΉβ€˜π‘₯))) βˆͺ {(1st β€˜(πΉβ€˜π‘₯)), (2nd β€˜(πΉβ€˜π‘₯))}))
29 fvco3 6941 . . . . . . . 8 ((𝐹:β„•βŸΆ( ≀ ∩ (ℝ Γ— ℝ)) ∧ π‘₯ ∈ β„•) β†’ (([,] ∘ 𝐹)β€˜π‘₯) = ([,]β€˜(πΉβ€˜π‘₯)))
302, 29sylan 581 . . . . . . 7 ((πœ‘ ∧ π‘₯ ∈ β„•) β†’ (([,] ∘ 𝐹)β€˜π‘₯) = ([,]β€˜(πΉβ€˜π‘₯)))
3116fveq2d 6847 . . . . . . . 8 ((πœ‘ ∧ π‘₯ ∈ β„•) β†’ ([,]β€˜(πΉβ€˜π‘₯)) = ([,]β€˜βŸ¨(1st β€˜(πΉβ€˜π‘₯)), (2nd β€˜(πΉβ€˜π‘₯))⟩))
32 df-ov 7361 . . . . . . . 8 ((1st β€˜(πΉβ€˜π‘₯))[,](2nd β€˜(πΉβ€˜π‘₯))) = ([,]β€˜βŸ¨(1st β€˜(πΉβ€˜π‘₯)), (2nd β€˜(πΉβ€˜π‘₯))⟩)
3331, 32eqtr4di 2795 . . . . . . 7 ((πœ‘ ∧ π‘₯ ∈ β„•) β†’ ([,]β€˜(πΉβ€˜π‘₯)) = ((1st β€˜(πΉβ€˜π‘₯))[,](2nd β€˜(πΉβ€˜π‘₯))))
3430, 33eqtrd 2777 . . . . . 6 ((πœ‘ ∧ π‘₯ ∈ β„•) β†’ (([,] ∘ 𝐹)β€˜π‘₯) = ((1st β€˜(πΉβ€˜π‘₯))[,](2nd β€˜(πΉβ€˜π‘₯))))
3510, 28, 343eqtr4rd 2788 . . . . 5 ((πœ‘ ∧ π‘₯ ∈ β„•) β†’ (([,] ∘ 𝐹)β€˜π‘₯) = ((((,) ∘ 𝐹)β€˜π‘₯) βˆͺ ({((1st ∘ 𝐹)β€˜π‘₯)} βˆͺ {((2nd ∘ 𝐹)β€˜π‘₯)})))
3635iuneq2dv 4979 . . . 4 (πœ‘ β†’ βˆͺ π‘₯ ∈ β„• (([,] ∘ 𝐹)β€˜π‘₯) = βˆͺ π‘₯ ∈ β„• ((((,) ∘ 𝐹)β€˜π‘₯) βˆͺ ({((1st ∘ 𝐹)β€˜π‘₯)} βˆͺ {((2nd ∘ 𝐹)β€˜π‘₯)})))
37 iccf 13366 . . . . . . 7 [,]:(ℝ* Γ— ℝ*)βŸΆπ’« ℝ*
38 ffn 6669 . . . . . . 7 ([,]:(ℝ* Γ— ℝ*)βŸΆπ’« ℝ* β†’ [,] Fn (ℝ* Γ— ℝ*))
3937, 38ax-mp 5 . . . . . 6 [,] Fn (ℝ* Γ— ℝ*)
40 inss2 4190 . . . . . . . 8 ( ≀ ∩ (ℝ Γ— ℝ)) βŠ† (ℝ Γ— ℝ)
41 rexpssxrxp 11201 . . . . . . . 8 (ℝ Γ— ℝ) βŠ† (ℝ* Γ— ℝ*)
4240, 41sstri 3954 . . . . . . 7 ( ≀ ∩ (ℝ Γ— ℝ)) βŠ† (ℝ* Γ— ℝ*)
43 fss 6686 . . . . . . 7 ((𝐹:β„•βŸΆ( ≀ ∩ (ℝ Γ— ℝ)) ∧ ( ≀ ∩ (ℝ Γ— ℝ)) βŠ† (ℝ* Γ— ℝ*)) β†’ 𝐹:β„•βŸΆ(ℝ* Γ— ℝ*))
442, 42, 43sylancl 587 . . . . . 6 (πœ‘ β†’ 𝐹:β„•βŸΆ(ℝ* Γ— ℝ*))
45 fnfco 6708 . . . . . 6 (([,] Fn (ℝ* Γ— ℝ*) ∧ 𝐹:β„•βŸΆ(ℝ* Γ— ℝ*)) β†’ ([,] ∘ 𝐹) Fn β„•)
4639, 44, 45sylancr 588 . . . . 5 (πœ‘ β†’ ([,] ∘ 𝐹) Fn β„•)
47 fniunfv 7195 . . . . 5 (([,] ∘ 𝐹) Fn β„• β†’ βˆͺ π‘₯ ∈ β„• (([,] ∘ 𝐹)β€˜π‘₯) = βˆͺ ran ([,] ∘ 𝐹))
4846, 47syl 17 . . . 4 (πœ‘ β†’ βˆͺ π‘₯ ∈ β„• (([,] ∘ 𝐹)β€˜π‘₯) = βˆͺ ran ([,] ∘ 𝐹))
49 iunun 5054 . . . . 5 βˆͺ π‘₯ ∈ β„• ((((,) ∘ 𝐹)β€˜π‘₯) βˆͺ ({((1st ∘ 𝐹)β€˜π‘₯)} βˆͺ {((2nd ∘ 𝐹)β€˜π‘₯)})) = (βˆͺ π‘₯ ∈ β„• (((,) ∘ 𝐹)β€˜π‘₯) βˆͺ βˆͺ π‘₯ ∈ β„• ({((1st ∘ 𝐹)β€˜π‘₯)} βˆͺ {((2nd ∘ 𝐹)β€˜π‘₯)}))
50 ioof 13365 . . . . . . . . 9 (,):(ℝ* Γ— ℝ*)βŸΆπ’« ℝ
51 ffn 6669 . . . . . . . . 9 ((,):(ℝ* Γ— ℝ*)βŸΆπ’« ℝ β†’ (,) Fn (ℝ* Γ— ℝ*))
5250, 51ax-mp 5 . . . . . . . 8 (,) Fn (ℝ* Γ— ℝ*)
53 fnfco 6708 . . . . . . . 8 (((,) Fn (ℝ* Γ— ℝ*) ∧ 𝐹:β„•βŸΆ(ℝ* Γ— ℝ*)) β†’ ((,) ∘ 𝐹) Fn β„•)
5452, 44, 53sylancr 588 . . . . . . 7 (πœ‘ β†’ ((,) ∘ 𝐹) Fn β„•)
55 fniunfv 7195 . . . . . . 7 (((,) ∘ 𝐹) Fn β„• β†’ βˆͺ π‘₯ ∈ β„• (((,) ∘ 𝐹)β€˜π‘₯) = βˆͺ ran ((,) ∘ 𝐹))
5654, 55syl 17 . . . . . 6 (πœ‘ β†’ βˆͺ π‘₯ ∈ β„• (((,) ∘ 𝐹)β€˜π‘₯) = βˆͺ ran ((,) ∘ 𝐹))
57 iunun 5054 . . . . . . 7 βˆͺ π‘₯ ∈ β„• ({((1st ∘ 𝐹)β€˜π‘₯)} βˆͺ {((2nd ∘ 𝐹)β€˜π‘₯)}) = (βˆͺ π‘₯ ∈ β„• {((1st ∘ 𝐹)β€˜π‘₯)} βˆͺ βˆͺ π‘₯ ∈ β„• {((2nd ∘ 𝐹)β€˜π‘₯)})
58 fo1st 7942 . . . . . . . . . . . . . 14 1st :V–ontoβ†’V
59 fofn 6759 . . . . . . . . . . . . . 14 (1st :V–ontoβ†’V β†’ 1st Fn V)
6058, 59ax-mp 5 . . . . . . . . . . . . 13 1st Fn V
61 ssv 3969 . . . . . . . . . . . . . 14 ( ≀ ∩ (ℝ Γ— ℝ)) βŠ† V
62 fss 6686 . . . . . . . . . . . . . 14 ((𝐹:β„•βŸΆ( ≀ ∩ (ℝ Γ— ℝ)) ∧ ( ≀ ∩ (ℝ Γ— ℝ)) βŠ† V) β†’ 𝐹:β„•βŸΆV)
632, 61, 62sylancl 587 . . . . . . . . . . . . 13 (πœ‘ β†’ 𝐹:β„•βŸΆV)
64 fnfco 6708 . . . . . . . . . . . . 13 ((1st Fn V ∧ 𝐹:β„•βŸΆV) β†’ (1st ∘ 𝐹) Fn β„•)
6560, 63, 64sylancr 588 . . . . . . . . . . . 12 (πœ‘ β†’ (1st ∘ 𝐹) Fn β„•)
66 fnfun 6603 . . . . . . . . . . . 12 ((1st ∘ 𝐹) Fn β„• β†’ Fun (1st ∘ 𝐹))
6765, 66syl 17 . . . . . . . . . . 11 (πœ‘ β†’ Fun (1st ∘ 𝐹))
68 fndm 6606 . . . . . . . . . . . 12 ((1st ∘ 𝐹) Fn β„• β†’ dom (1st ∘ 𝐹) = β„•)
69 eqimss2 4002 . . . . . . . . . . . 12 (dom (1st ∘ 𝐹) = β„• β†’ β„• βŠ† dom (1st ∘ 𝐹))
7065, 68, 693syl 18 . . . . . . . . . . 11 (πœ‘ β†’ β„• βŠ† dom (1st ∘ 𝐹))
71 dfimafn2 6907 . . . . . . . . . . 11 ((Fun (1st ∘ 𝐹) ∧ β„• βŠ† dom (1st ∘ 𝐹)) β†’ ((1st ∘ 𝐹) β€œ β„•) = βˆͺ π‘₯ ∈ β„• {((1st ∘ 𝐹)β€˜π‘₯)})
7267, 70, 71syl2anc 585 . . . . . . . . . 10 (πœ‘ β†’ ((1st ∘ 𝐹) β€œ β„•) = βˆͺ π‘₯ ∈ β„• {((1st ∘ 𝐹)β€˜π‘₯)})
73 fnima 6632 . . . . . . . . . . 11 ((1st ∘ 𝐹) Fn β„• β†’ ((1st ∘ 𝐹) β€œ β„•) = ran (1st ∘ 𝐹))
7465, 73syl 17 . . . . . . . . . 10 (πœ‘ β†’ ((1st ∘ 𝐹) β€œ β„•) = ran (1st ∘ 𝐹))
7572, 74eqtr3d 2779 . . . . . . . . 9 (πœ‘ β†’ βˆͺ π‘₯ ∈ β„• {((1st ∘ 𝐹)β€˜π‘₯)} = ran (1st ∘ 𝐹))
76 rnco2 6206 . . . . . . . . 9 ran (1st ∘ 𝐹) = (1st β€œ ran 𝐹)
7775, 76eqtrdi 2793 . . . . . . . 8 (πœ‘ β†’ βˆͺ π‘₯ ∈ β„• {((1st ∘ 𝐹)β€˜π‘₯)} = (1st β€œ ran 𝐹))
78 fo2nd 7943 . . . . . . . . . . . . . 14 2nd :V–ontoβ†’V
79 fofn 6759 . . . . . . . . . . . . . 14 (2nd :V–ontoβ†’V β†’ 2nd Fn V)
8078, 79ax-mp 5 . . . . . . . . . . . . 13 2nd Fn V
81 fnfco 6708 . . . . . . . . . . . . 13 ((2nd Fn V ∧ 𝐹:β„•βŸΆV) β†’ (2nd ∘ 𝐹) Fn β„•)
8280, 63, 81sylancr 588 . . . . . . . . . . . 12 (πœ‘ β†’ (2nd ∘ 𝐹) Fn β„•)
83 fnfun 6603 . . . . . . . . . . . 12 ((2nd ∘ 𝐹) Fn β„• β†’ Fun (2nd ∘ 𝐹))
8482, 83syl 17 . . . . . . . . . . 11 (πœ‘ β†’ Fun (2nd ∘ 𝐹))
85 fndm 6606 . . . . . . . . . . . 12 ((2nd ∘ 𝐹) Fn β„• β†’ dom (2nd ∘ 𝐹) = β„•)
86 eqimss2 4002 . . . . . . . . . . . 12 (dom (2nd ∘ 𝐹) = β„• β†’ β„• βŠ† dom (2nd ∘ 𝐹))
8782, 85, 863syl 18 . . . . . . . . . . 11 (πœ‘ β†’ β„• βŠ† dom (2nd ∘ 𝐹))
88 dfimafn2 6907 . . . . . . . . . . 11 ((Fun (2nd ∘ 𝐹) ∧ β„• βŠ† dom (2nd ∘ 𝐹)) β†’ ((2nd ∘ 𝐹) β€œ β„•) = βˆͺ π‘₯ ∈ β„• {((2nd ∘ 𝐹)β€˜π‘₯)})
8984, 87, 88syl2anc 585 . . . . . . . . . 10 (πœ‘ β†’ ((2nd ∘ 𝐹) β€œ β„•) = βˆͺ π‘₯ ∈ β„• {((2nd ∘ 𝐹)β€˜π‘₯)})
90 fnima 6632 . . . . . . . . . . 11 ((2nd ∘ 𝐹) Fn β„• β†’ ((2nd ∘ 𝐹) β€œ β„•) = ran (2nd ∘ 𝐹))
9182, 90syl 17 . . . . . . . . . 10 (πœ‘ β†’ ((2nd ∘ 𝐹) β€œ β„•) = ran (2nd ∘ 𝐹))
9289, 91eqtr3d 2779 . . . . . . . . 9 (πœ‘ β†’ βˆͺ π‘₯ ∈ β„• {((2nd ∘ 𝐹)β€˜π‘₯)} = ran (2nd ∘ 𝐹))
93 rnco2 6206 . . . . . . . . 9 ran (2nd ∘ 𝐹) = (2nd β€œ ran 𝐹)
9492, 93eqtrdi 2793 . . . . . . . 8 (πœ‘ β†’ βˆͺ π‘₯ ∈ β„• {((2nd ∘ 𝐹)β€˜π‘₯)} = (2nd β€œ ran 𝐹))
9577, 94uneq12d 4125 . . . . . . 7 (πœ‘ β†’ (βˆͺ π‘₯ ∈ β„• {((1st ∘ 𝐹)β€˜π‘₯)} βˆͺ βˆͺ π‘₯ ∈ β„• {((2nd ∘ 𝐹)β€˜π‘₯)}) = ((1st β€œ ran 𝐹) βˆͺ (2nd β€œ ran 𝐹)))
9657, 95eqtrid 2789 . . . . . 6 (πœ‘ β†’ βˆͺ π‘₯ ∈ β„• ({((1st ∘ 𝐹)β€˜π‘₯)} βˆͺ {((2nd ∘ 𝐹)β€˜π‘₯)}) = ((1st β€œ ran 𝐹) βˆͺ (2nd β€œ ran 𝐹)))
9756, 96uneq12d 4125 . . . . 5 (πœ‘ β†’ (βˆͺ π‘₯ ∈ β„• (((,) ∘ 𝐹)β€˜π‘₯) βˆͺ βˆͺ π‘₯ ∈ β„• ({((1st ∘ 𝐹)β€˜π‘₯)} βˆͺ {((2nd ∘ 𝐹)β€˜π‘₯)})) = (βˆͺ ran ((,) ∘ 𝐹) βˆͺ ((1st β€œ ran 𝐹) βˆͺ (2nd β€œ ran 𝐹))))
9849, 97eqtrid 2789 . . . 4 (πœ‘ β†’ βˆͺ π‘₯ ∈ β„• ((((,) ∘ 𝐹)β€˜π‘₯) βˆͺ ({((1st ∘ 𝐹)β€˜π‘₯)} βˆͺ {((2nd ∘ 𝐹)β€˜π‘₯)})) = (βˆͺ ran ((,) ∘ 𝐹) βˆͺ ((1st β€œ ran 𝐹) βˆͺ (2nd β€œ ran 𝐹))))
9936, 48, 983eqtr3d 2785 . . 3 (πœ‘ β†’ βˆͺ ran ([,] ∘ 𝐹) = (βˆͺ ran ((,) ∘ 𝐹) βˆͺ ((1st β€œ ran 𝐹) βˆͺ (2nd β€œ ran 𝐹))))
1001, 99sseqtrrid 3998 . 2 (πœ‘ β†’ βˆͺ ran ((,) ∘ 𝐹) βŠ† βˆͺ ran ([,] ∘ 𝐹))
101 ovolficcss 24836 . . . . 5 (𝐹:β„•βŸΆ( ≀ ∩ (ℝ Γ— ℝ)) β†’ βˆͺ ran ([,] ∘ 𝐹) βŠ† ℝ)
1022, 101syl 17 . . . 4 (πœ‘ β†’ βˆͺ ran ([,] ∘ 𝐹) βŠ† ℝ)
103102ssdifssd 4103 . . 3 (πœ‘ β†’ (βˆͺ ran ([,] ∘ 𝐹) βˆ– βˆͺ ran ((,) ∘ 𝐹)) βŠ† ℝ)
104 omelon 9583 . . . . . . . . . . 11 Ο‰ ∈ On
105 nnenom 13886 . . . . . . . . . . . 12 β„• β‰ˆ Ο‰
106105ensymi 8945 . . . . . . . . . . 11 Ο‰ β‰ˆ β„•
107 isnumi 9883 . . . . . . . . . . 11 ((Ο‰ ∈ On ∧ Ο‰ β‰ˆ β„•) β†’ β„• ∈ dom card)
108104, 106, 107mp2an 691 . . . . . . . . . 10 β„• ∈ dom card
109 fofun 6758 . . . . . . . . . . . . 13 (1st :V–ontoβ†’V β†’ Fun 1st )
11058, 109ax-mp 5 . . . . . . . . . . . 12 Fun 1st
111 ssv 3969 . . . . . . . . . . . . 13 ran 𝐹 βŠ† V
112 fof 6757 . . . . . . . . . . . . . . 15 (1st :V–ontoβ†’V β†’ 1st :V⟢V)
11358, 112ax-mp 5 . . . . . . . . . . . . . 14 1st :V⟢V
114113fdmi 6681 . . . . . . . . . . . . 13 dom 1st = V
115111, 114sseqtrri 3982 . . . . . . . . . . . 12 ran 𝐹 βŠ† dom 1st
116 fores 6767 . . . . . . . . . . . 12 ((Fun 1st ∧ ran 𝐹 βŠ† dom 1st ) β†’ (1st β†Ύ ran 𝐹):ran 𝐹–ontoβ†’(1st β€œ ran 𝐹))
117110, 115, 116mp2an 691 . . . . . . . . . . 11 (1st β†Ύ ran 𝐹):ran 𝐹–ontoβ†’(1st β€œ ran 𝐹)
1182ffnd 6670 . . . . . . . . . . . 12 (πœ‘ β†’ 𝐹 Fn β„•)
119 dffn4 6763 . . . . . . . . . . . 12 (𝐹 Fn β„• ↔ 𝐹:ℕ–ontoβ†’ran 𝐹)
120118, 119sylib 217 . . . . . . . . . . 11 (πœ‘ β†’ 𝐹:ℕ–ontoβ†’ran 𝐹)
121 foco 6771 . . . . . . . . . . 11 (((1st β†Ύ ran 𝐹):ran 𝐹–ontoβ†’(1st β€œ ran 𝐹) ∧ 𝐹:ℕ–ontoβ†’ran 𝐹) β†’ ((1st β†Ύ ran 𝐹) ∘ 𝐹):ℕ–ontoβ†’(1st β€œ ran 𝐹))
122117, 120, 121sylancr 588 . . . . . . . . . 10 (πœ‘ β†’ ((1st β†Ύ ran 𝐹) ∘ 𝐹):ℕ–ontoβ†’(1st β€œ ran 𝐹))
123 fodomnum 9994 . . . . . . . . . 10 (β„• ∈ dom card β†’ (((1st β†Ύ ran 𝐹) ∘ 𝐹):ℕ–ontoβ†’(1st β€œ ran 𝐹) β†’ (1st β€œ ran 𝐹) β‰Ό β„•))
124108, 122, 123mpsyl 68 . . . . . . . . 9 (πœ‘ β†’ (1st β€œ ran 𝐹) β‰Ό β„•)
125 domentr 8954 . . . . . . . . 9 (((1st β€œ ran 𝐹) β‰Ό β„• ∧ β„• β‰ˆ Ο‰) β†’ (1st β€œ ran 𝐹) β‰Ό Ο‰)
126124, 105, 125sylancl 587 . . . . . . . 8 (πœ‘ β†’ (1st β€œ ran 𝐹) β‰Ό Ο‰)
127 fofun 6758 . . . . . . . . . . . . 13 (2nd :V–ontoβ†’V β†’ Fun 2nd )
12878, 127ax-mp 5 . . . . . . . . . . . 12 Fun 2nd
129 fof 6757 . . . . . . . . . . . . . . 15 (2nd :V–ontoβ†’V β†’ 2nd :V⟢V)
13078, 129ax-mp 5 . . . . . . . . . . . . . 14 2nd :V⟢V
131130fdmi 6681 . . . . . . . . . . . . 13 dom 2nd = V
132111, 131sseqtrri 3982 . . . . . . . . . . . 12 ran 𝐹 βŠ† dom 2nd
133 fores 6767 . . . . . . . . . . . 12 ((Fun 2nd ∧ ran 𝐹 βŠ† dom 2nd ) β†’ (2nd β†Ύ ran 𝐹):ran 𝐹–ontoβ†’(2nd β€œ ran 𝐹))
134128, 132, 133mp2an 691 . . . . . . . . . . 11 (2nd β†Ύ ran 𝐹):ran 𝐹–ontoβ†’(2nd β€œ ran 𝐹)
135 foco 6771 . . . . . . . . . . 11 (((2nd β†Ύ ran 𝐹):ran 𝐹–ontoβ†’(2nd β€œ ran 𝐹) ∧ 𝐹:ℕ–ontoβ†’ran 𝐹) β†’ ((2nd β†Ύ ran 𝐹) ∘ 𝐹):ℕ–ontoβ†’(2nd β€œ ran 𝐹))
136134, 120, 135sylancr 588 . . . . . . . . . 10 (πœ‘ β†’ ((2nd β†Ύ ran 𝐹) ∘ 𝐹):ℕ–ontoβ†’(2nd β€œ ran 𝐹))
137 fodomnum 9994 . . . . . . . . . 10 (β„• ∈ dom card β†’ (((2nd β†Ύ ran 𝐹) ∘ 𝐹):ℕ–ontoβ†’(2nd β€œ ran 𝐹) β†’ (2nd β€œ ran 𝐹) β‰Ό β„•))
138108, 136, 137mpsyl 68 . . . . . . . . 9 (πœ‘ β†’ (2nd β€œ ran 𝐹) β‰Ό β„•)
139 domentr 8954 . . . . . . . . 9 (((2nd β€œ ran 𝐹) β‰Ό β„• ∧ β„• β‰ˆ Ο‰) β†’ (2nd β€œ ran 𝐹) β‰Ό Ο‰)
140138, 105, 139sylancl 587 . . . . . . . 8 (πœ‘ β†’ (2nd β€œ ran 𝐹) β‰Ό Ο‰)
141 unctb 10142 . . . . . . . 8 (((1st β€œ ran 𝐹) β‰Ό Ο‰ ∧ (2nd β€œ ran 𝐹) β‰Ό Ο‰) β†’ ((1st β€œ ran 𝐹) βˆͺ (2nd β€œ ran 𝐹)) β‰Ό Ο‰)
142126, 140, 141syl2anc 585 . . . . . . 7 (πœ‘ β†’ ((1st β€œ ran 𝐹) βˆͺ (2nd β€œ ran 𝐹)) β‰Ό Ο‰)
143 ctex 8904 . . . . . . 7 (((1st β€œ ran 𝐹) βˆͺ (2nd β€œ ran 𝐹)) β‰Ό Ο‰ β†’ ((1st β€œ ran 𝐹) βˆͺ (2nd β€œ ran 𝐹)) ∈ V)
144142, 143syl 17 . . . . . 6 (πœ‘ β†’ ((1st β€œ ran 𝐹) βˆͺ (2nd β€œ ran 𝐹)) ∈ V)
145 ssid 3967 . . . . . . . 8 βˆͺ ran ([,] ∘ 𝐹) βŠ† βˆͺ ran ([,] ∘ 𝐹)
146145, 99sseqtrid 3997 . . . . . . 7 (πœ‘ β†’ βˆͺ ran ([,] ∘ 𝐹) βŠ† (βˆͺ ran ((,) ∘ 𝐹) βˆͺ ((1st β€œ ran 𝐹) βˆͺ (2nd β€œ ran 𝐹))))
147 ssundif 4446 . . . . . . 7 (βˆͺ ran ([,] ∘ 𝐹) βŠ† (βˆͺ ran ((,) ∘ 𝐹) βˆͺ ((1st β€œ ran 𝐹) βˆͺ (2nd β€œ ran 𝐹))) ↔ (βˆͺ ran ([,] ∘ 𝐹) βˆ– βˆͺ ran ((,) ∘ 𝐹)) βŠ† ((1st β€œ ran 𝐹) βˆͺ (2nd β€œ ran 𝐹)))
148146, 147sylib 217 . . . . . 6 (πœ‘ β†’ (βˆͺ ran ([,] ∘ 𝐹) βˆ– βˆͺ ran ((,) ∘ 𝐹)) βŠ† ((1st β€œ ran 𝐹) βˆͺ (2nd β€œ ran 𝐹)))
149 ssdomg 8941 . . . . . 6 (((1st β€œ ran 𝐹) βˆͺ (2nd β€œ ran 𝐹)) ∈ V β†’ ((βˆͺ ran ([,] ∘ 𝐹) βˆ– βˆͺ ran ((,) ∘ 𝐹)) βŠ† ((1st β€œ ran 𝐹) βˆͺ (2nd β€œ ran 𝐹)) β†’ (βˆͺ ran ([,] ∘ 𝐹) βˆ– βˆͺ ran ((,) ∘ 𝐹)) β‰Ό ((1st β€œ ran 𝐹) βˆͺ (2nd β€œ ran 𝐹))))
150144, 148, 149sylc 65 . . . . 5 (πœ‘ β†’ (βˆͺ ran ([,] ∘ 𝐹) βˆ– βˆͺ ran ((,) ∘ 𝐹)) β‰Ό ((1st β€œ ran 𝐹) βˆͺ (2nd β€œ ran 𝐹)))
151 domtr 8948 . . . . 5 (((βˆͺ ran ([,] ∘ 𝐹) βˆ– βˆͺ ran ((,) ∘ 𝐹)) β‰Ό ((1st β€œ ran 𝐹) βˆͺ (2nd β€œ ran 𝐹)) ∧ ((1st β€œ ran 𝐹) βˆͺ (2nd β€œ ran 𝐹)) β‰Ό Ο‰) β†’ (βˆͺ ran ([,] ∘ 𝐹) βˆ– βˆͺ ran ((,) ∘ 𝐹)) β‰Ό Ο‰)
152150, 142, 151syl2anc 585 . . . 4 (πœ‘ β†’ (βˆͺ ran ([,] ∘ 𝐹) βˆ– βˆͺ ran ((,) ∘ 𝐹)) β‰Ό Ο‰)
153 domentr 8954 . . . 4 (((βˆͺ ran ([,] ∘ 𝐹) βˆ– βˆͺ ran ((,) ∘ 𝐹)) β‰Ό Ο‰ ∧ Ο‰ β‰ˆ β„•) β†’ (βˆͺ ran ([,] ∘ 𝐹) βˆ– βˆͺ ran ((,) ∘ 𝐹)) β‰Ό β„•)
154152, 106, 153sylancl 587 . . 3 (πœ‘ β†’ (βˆͺ ran ([,] ∘ 𝐹) βˆ– βˆͺ ran ((,) ∘ 𝐹)) β‰Ό β„•)
155 ovolctb2 24859 . . 3 (((βˆͺ ran ([,] ∘ 𝐹) βˆ– βˆͺ ran ((,) ∘ 𝐹)) βŠ† ℝ ∧ (βˆͺ ran ([,] ∘ 𝐹) βˆ– βˆͺ ran ((,) ∘ 𝐹)) β‰Ό β„•) β†’ (vol*β€˜(βˆͺ ran ([,] ∘ 𝐹) βˆ– βˆͺ ran ((,) ∘ 𝐹))) = 0)
156103, 154, 155syl2anc 585 . 2 (πœ‘ β†’ (vol*β€˜(βˆͺ ran ([,] ∘ 𝐹) βˆ– βˆͺ ran ((,) ∘ 𝐹))) = 0)
157100, 156jca 513 1 (πœ‘ β†’ (βˆͺ ran ((,) ∘ 𝐹) βŠ† βˆͺ ran ([,] ∘ 𝐹) ∧ (vol*β€˜(βˆͺ ran ([,] ∘ 𝐹) βˆ– βˆͺ ran ((,) ∘ 𝐹))) = 0))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 397   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107  Vcvv 3446   βˆ– cdif 3908   βˆͺ cun 3909   ∩ cin 3910   βŠ† wss 3911  π’« cpw 4561  {csn 4587  {cpr 4589  βŸ¨cop 4593  βˆͺ cuni 4866  βˆͺ ciun 4955   class class class wbr 5106   Γ— cxp 5632  dom cdm 5634  ran crn 5635   β†Ύ cres 5636   β€œ cima 5637   ∘ ccom 5638  Oncon0 6318  Fun wfun 6491   Fn wfn 6492  βŸΆwf 6493  β€“ontoβ†’wfo 6495  β€˜cfv 6497  (class class class)co 7358  Ο‰com 7803  1st c1st 7920  2nd c2nd 7921   β‰ˆ cen 8881   β‰Ό cdom 8882  cardccrd 9872  β„cr 11051  0cc0 11052  β„*cxr 11189   ≀ cle 11191  β„•cn 12154  (,)cioo 13265  [,]cicc 13268  vol*covol 24829
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 2708  ax-rep 5243  ax-sep 5257  ax-nul 5264  ax-pow 5321  ax-pr 5385  ax-un 7673  ax-inf2 9578  ax-cnex 11108  ax-resscn 11109  ax-1cn 11110  ax-icn 11111  ax-addcl 11112  ax-addrcl 11113  ax-mulcl 11114  ax-mulrcl 11115  ax-mulcom 11116  ax-addass 11117  ax-mulass 11118  ax-distr 11119  ax-i2m1 11120  ax-1ne0 11121  ax-1rid 11122  ax-rnegex 11123  ax-rrecex 11124  ax-cnre 11125  ax-pre-lttri 11126  ax-pre-lttrn 11127  ax-pre-ltadd 11128  ax-pre-mulgt0 11129  ax-pre-sup 11130
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ne 2945  df-nel 3051  df-ral 3066  df-rex 3075  df-rmo 3354  df-reu 3355  df-rab 3409  df-v 3448  df-sbc 3741  df-csb 3857  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-pss 3930  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-int 4909  df-iun 4957  df-br 5107  df-opab 5169  df-mpt 5190  df-tr 5224  df-id 5532  df-eprel 5538  df-po 5546  df-so 5547  df-fr 5589  df-se 5590  df-we 5591  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-pred 6254  df-ord 6321  df-on 6322  df-lim 6323  df-suc 6324  df-iota 6449  df-fun 6499  df-fn 6500  df-f 6501  df-f1 6502  df-fo 6503  df-f1o 6504  df-fv 6505  df-isom 6506  df-riota 7314  df-ov 7361  df-oprab 7362  df-mpo 7363  df-of 7618  df-om 7804  df-1st 7922  df-2nd 7923  df-frecs 8213  df-wrecs 8244  df-recs 8318  df-rdg 8357  df-1o 8413  df-2o 8414  df-er 8649  df-map 8768  df-en 8885  df-dom 8886  df-sdom 8887  df-fin 8888  df-sup 9379  df-inf 9380  df-oi 9447  df-dju 9838  df-card 9876  df-acn 9879  df-pnf 11192  df-mnf 11193  df-xr 11194  df-ltxr 11195  df-le 11196  df-sub 11388  df-neg 11389  df-div 11814  df-nn 12155  df-2 12217  df-3 12218  df-n0 12415  df-z 12501  df-uz 12765  df-q 12875  df-rp 12917  df-xadd 13035  df-ioo 13269  df-ico 13271  df-icc 13272  df-fz 13426  df-fzo 13569  df-seq 13908  df-exp 13969  df-hash 14232  df-cj 14985  df-re 14986  df-im 14987  df-sqrt 15121  df-abs 15122  df-clim 15371  df-sum 15572  df-xmet 20792  df-met 20793  df-ovol 24831
This theorem is referenced by:  uniioombllem3  24952  uniioombllem4  24953  uniioombllem5  24954  uniiccmbl  24957
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