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Theorem i1fd 25730
Description: A simplified set of assumptions to show that a given function is simple. (Contributed by Mario Carneiro, 26-Jun-2014.)
Hypotheses
Ref Expression
i1fd.1 (𝜑𝐹:ℝ⟶ℝ)
i1fd.2 (𝜑 → ran 𝐹 ∈ Fin)
i1fd.3 ((𝜑𝑥 ∈ (ran 𝐹 ∖ {0})) → (𝐹 “ {𝑥}) ∈ dom vol)
i1fd.4 ((𝜑𝑥 ∈ (ran 𝐹 ∖ {0})) → (vol‘(𝐹 “ {𝑥})) ∈ ℝ)
Assertion
Ref Expression
i1fd (𝜑𝐹 ∈ dom ∫1)
Distinct variable groups:   𝑥,𝐹   𝜑,𝑥

Proof of Theorem i1fd
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 i1fd.1 . . . . . . . . 9 (𝜑𝐹:ℝ⟶ℝ)
21ad2antrr 726 . . . . . . . 8 (((𝜑𝑥 ∈ ran (,)) ∧ 0 ∈ 𝑥) → 𝐹:ℝ⟶ℝ)
3 ffun 6740 . . . . . . . 8 (𝐹:ℝ⟶ℝ → Fun 𝐹)
4 funcnvcnv 6635 . . . . . . . 8 (Fun 𝐹 → Fun 𝐹)
5 imadif 6652 . . . . . . . 8 (Fun 𝐹 → (𝐹 “ (ℝ ∖ (ℝ ∖ 𝑥))) = ((𝐹 “ ℝ) ∖ (𝐹 “ (ℝ ∖ 𝑥))))
62, 3, 4, 54syl 19 . . . . . . 7 (((𝜑𝑥 ∈ ran (,)) ∧ 0 ∈ 𝑥) → (𝐹 “ (ℝ ∖ (ℝ ∖ 𝑥))) = ((𝐹 “ ℝ) ∖ (𝐹 “ (ℝ ∖ 𝑥))))
7 ioof 13484 . . . . . . . . . . . . 13 (,):(ℝ* × ℝ*)⟶𝒫 ℝ
8 frn 6744 . . . . . . . . . . . . 13 ((,):(ℝ* × ℝ*)⟶𝒫 ℝ → ran (,) ⊆ 𝒫 ℝ)
97, 8ax-mp 5 . . . . . . . . . . . 12 ran (,) ⊆ 𝒫 ℝ
109sseli 3991 . . . . . . . . . . 11 (𝑥 ∈ ran (,) → 𝑥 ∈ 𝒫 ℝ)
1110elpwid 4614 . . . . . . . . . 10 (𝑥 ∈ ran (,) → 𝑥 ⊆ ℝ)
1211ad2antlr 727 . . . . . . . . 9 (((𝜑𝑥 ∈ ran (,)) ∧ 0 ∈ 𝑥) → 𝑥 ⊆ ℝ)
13 dfss4 4275 . . . . . . . . 9 (𝑥 ⊆ ℝ ↔ (ℝ ∖ (ℝ ∖ 𝑥)) = 𝑥)
1412, 13sylib 218 . . . . . . . 8 (((𝜑𝑥 ∈ ran (,)) ∧ 0 ∈ 𝑥) → (ℝ ∖ (ℝ ∖ 𝑥)) = 𝑥)
1514imaeq2d 6080 . . . . . . 7 (((𝜑𝑥 ∈ ran (,)) ∧ 0 ∈ 𝑥) → (𝐹 “ (ℝ ∖ (ℝ ∖ 𝑥))) = (𝐹𝑥))
166, 15eqtr3d 2777 . . . . . 6 (((𝜑𝑥 ∈ ran (,)) ∧ 0 ∈ 𝑥) → ((𝐹 “ ℝ) ∖ (𝐹 “ (ℝ ∖ 𝑥))) = (𝐹𝑥))
17 fimacnv 6759 . . . . . . . . 9 (𝐹:ℝ⟶ℝ → (𝐹 “ ℝ) = ℝ)
182, 17syl 17 . . . . . . . 8 (((𝜑𝑥 ∈ ran (,)) ∧ 0 ∈ 𝑥) → (𝐹 “ ℝ) = ℝ)
19 rembl 25589 . . . . . . . 8 ℝ ∈ dom vol
2018, 19eqeltrdi 2847 . . . . . . 7 (((𝜑𝑥 ∈ ran (,)) ∧ 0 ∈ 𝑥) → (𝐹 “ ℝ) ∈ dom vol)
211adantr 480 . . . . . . . . . . . . 13 ((𝜑 ∧ ¬ 0 ∈ 𝑦) → 𝐹:ℝ⟶ℝ)
22 inpreima 7084 . . . . . . . . . . . . . 14 (Fun 𝐹 → (𝐹 “ (𝑦 ∩ ran 𝐹)) = ((𝐹𝑦) ∩ (𝐹 “ ran 𝐹)))
23 iunid 5065 . . . . . . . . . . . . . . . 16 𝑥 ∈ (𝑦 ∩ ran 𝐹){𝑥} = (𝑦 ∩ ran 𝐹)
2423imaeq2i 6078 . . . . . . . . . . . . . . 15 (𝐹 𝑥 ∈ (𝑦 ∩ ran 𝐹){𝑥}) = (𝐹 “ (𝑦 ∩ ran 𝐹))
25 imaiun 7265 . . . . . . . . . . . . . . 15 (𝐹 𝑥 ∈ (𝑦 ∩ ran 𝐹){𝑥}) = 𝑥 ∈ (𝑦 ∩ ran 𝐹)(𝐹 “ {𝑥})
2624, 25eqtr3i 2765 . . . . . . . . . . . . . 14 (𝐹 “ (𝑦 ∩ ran 𝐹)) = 𝑥 ∈ (𝑦 ∩ ran 𝐹)(𝐹 “ {𝑥})
27 cnvimass 6102 . . . . . . . . . . . . . . . 16 (𝐹𝑦) ⊆ dom 𝐹
28 cnvimarndm 6103 . . . . . . . . . . . . . . . 16 (𝐹 “ ran 𝐹) = dom 𝐹
2927, 28sseqtrri 4033 . . . . . . . . . . . . . . 15 (𝐹𝑦) ⊆ (𝐹 “ ran 𝐹)
30 dfss2 3981 . . . . . . . . . . . . . . 15 ((𝐹𝑦) ⊆ (𝐹 “ ran 𝐹) ↔ ((𝐹𝑦) ∩ (𝐹 “ ran 𝐹)) = (𝐹𝑦))
3129, 30mpbi 230 . . . . . . . . . . . . . 14 ((𝐹𝑦) ∩ (𝐹 “ ran 𝐹)) = (𝐹𝑦)
3222, 26, 313eqtr3g 2798 . . . . . . . . . . . . 13 (Fun 𝐹 𝑥 ∈ (𝑦 ∩ ran 𝐹)(𝐹 “ {𝑥}) = (𝐹𝑦))
3321, 3, 323syl 18 . . . . . . . . . . . 12 ((𝜑 ∧ ¬ 0 ∈ 𝑦) → 𝑥 ∈ (𝑦 ∩ ran 𝐹)(𝐹 “ {𝑥}) = (𝐹𝑦))
34 i1fd.2 . . . . . . . . . . . . . . 15 (𝜑 → ran 𝐹 ∈ Fin)
3534adantr 480 . . . . . . . . . . . . . 14 ((𝜑 ∧ ¬ 0 ∈ 𝑦) → ran 𝐹 ∈ Fin)
36 inss2 4246 . . . . . . . . . . . . . 14 (𝑦 ∩ ran 𝐹) ⊆ ran 𝐹
37 ssfi 9212 . . . . . . . . . . . . . 14 ((ran 𝐹 ∈ Fin ∧ (𝑦 ∩ ran 𝐹) ⊆ ran 𝐹) → (𝑦 ∩ ran 𝐹) ∈ Fin)
3835, 36, 37sylancl 586 . . . . . . . . . . . . 13 ((𝜑 ∧ ¬ 0 ∈ 𝑦) → (𝑦 ∩ ran 𝐹) ∈ Fin)
39 simpll 767 . . . . . . . . . . . . . . 15 (((𝜑 ∧ ¬ 0 ∈ 𝑦) ∧ 𝑥 ∈ (𝑦 ∩ ran 𝐹)) → 𝜑)
40 elinel1 4211 . . . . . . . . . . . . . . . . . . . 20 (0 ∈ (𝑦 ∩ ran 𝐹) → 0 ∈ 𝑦)
4140con3i 154 . . . . . . . . . . . . . . . . . . 19 (¬ 0 ∈ 𝑦 → ¬ 0 ∈ (𝑦 ∩ ran 𝐹))
4241adantl 481 . . . . . . . . . . . . . . . . . 18 ((𝜑 ∧ ¬ 0 ∈ 𝑦) → ¬ 0 ∈ (𝑦 ∩ ran 𝐹))
43 disjsn 4716 . . . . . . . . . . . . . . . . . 18 (((𝑦 ∩ ran 𝐹) ∩ {0}) = ∅ ↔ ¬ 0 ∈ (𝑦 ∩ ran 𝐹))
4442, 43sylibr 234 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ ¬ 0 ∈ 𝑦) → ((𝑦 ∩ ran 𝐹) ∩ {0}) = ∅)
45 reldisj 4459 . . . . . . . . . . . . . . . . . 18 ((𝑦 ∩ ran 𝐹) ⊆ ran 𝐹 → (((𝑦 ∩ ran 𝐹) ∩ {0}) = ∅ ↔ (𝑦 ∩ ran 𝐹) ⊆ (ran 𝐹 ∖ {0})))
4636, 45ax-mp 5 . . . . . . . . . . . . . . . . 17 (((𝑦 ∩ ran 𝐹) ∩ {0}) = ∅ ↔ (𝑦 ∩ ran 𝐹) ⊆ (ran 𝐹 ∖ {0}))
4744, 46sylib 218 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ ¬ 0 ∈ 𝑦) → (𝑦 ∩ ran 𝐹) ⊆ (ran 𝐹 ∖ {0}))
4847sselda 3995 . . . . . . . . . . . . . . 15 (((𝜑 ∧ ¬ 0 ∈ 𝑦) ∧ 𝑥 ∈ (𝑦 ∩ ran 𝐹)) → 𝑥 ∈ (ran 𝐹 ∖ {0}))
49 i1fd.3 . . . . . . . . . . . . . . 15 ((𝜑𝑥 ∈ (ran 𝐹 ∖ {0})) → (𝐹 “ {𝑥}) ∈ dom vol)
5039, 48, 49syl2anc 584 . . . . . . . . . . . . . 14 (((𝜑 ∧ ¬ 0 ∈ 𝑦) ∧ 𝑥 ∈ (𝑦 ∩ ran 𝐹)) → (𝐹 “ {𝑥}) ∈ dom vol)
5150ralrimiva 3144 . . . . . . . . . . . . 13 ((𝜑 ∧ ¬ 0 ∈ 𝑦) → ∀𝑥 ∈ (𝑦 ∩ ran 𝐹)(𝐹 “ {𝑥}) ∈ dom vol)
52 finiunmbl 25593 . . . . . . . . . . . . 13 (((𝑦 ∩ ran 𝐹) ∈ Fin ∧ ∀𝑥 ∈ (𝑦 ∩ ran 𝐹)(𝐹 “ {𝑥}) ∈ dom vol) → 𝑥 ∈ (𝑦 ∩ ran 𝐹)(𝐹 “ {𝑥}) ∈ dom vol)
5338, 51, 52syl2anc 584 . . . . . . . . . . . 12 ((𝜑 ∧ ¬ 0 ∈ 𝑦) → 𝑥 ∈ (𝑦 ∩ ran 𝐹)(𝐹 “ {𝑥}) ∈ dom vol)
5433, 53eqeltrrd 2840 . . . . . . . . . . 11 ((𝜑 ∧ ¬ 0 ∈ 𝑦) → (𝐹𝑦) ∈ dom vol)
5554ex 412 . . . . . . . . . 10 (𝜑 → (¬ 0 ∈ 𝑦 → (𝐹𝑦) ∈ dom vol))
5655alrimiv 1925 . . . . . . . . 9 (𝜑 → ∀𝑦(¬ 0 ∈ 𝑦 → (𝐹𝑦) ∈ dom vol))
5756ad2antrr 726 . . . . . . . 8 (((𝜑𝑥 ∈ ran (,)) ∧ 0 ∈ 𝑥) → ∀𝑦(¬ 0 ∈ 𝑦 → (𝐹𝑦) ∈ dom vol))
58 elndif 4143 . . . . . . . . 9 (0 ∈ 𝑥 → ¬ 0 ∈ (ℝ ∖ 𝑥))
5958adantl 481 . . . . . . . 8 (((𝜑𝑥 ∈ ran (,)) ∧ 0 ∈ 𝑥) → ¬ 0 ∈ (ℝ ∖ 𝑥))
60 reex 11244 . . . . . . . . . 10 ℝ ∈ V
6160difexi 5336 . . . . . . . . 9 (ℝ ∖ 𝑥) ∈ V
62 eleq2 2828 . . . . . . . . . . 11 (𝑦 = (ℝ ∖ 𝑥) → (0 ∈ 𝑦 ↔ 0 ∈ (ℝ ∖ 𝑥)))
6362notbid 318 . . . . . . . . . 10 (𝑦 = (ℝ ∖ 𝑥) → (¬ 0 ∈ 𝑦 ↔ ¬ 0 ∈ (ℝ ∖ 𝑥)))
64 imaeq2 6076 . . . . . . . . . . 11 (𝑦 = (ℝ ∖ 𝑥) → (𝐹𝑦) = (𝐹 “ (ℝ ∖ 𝑥)))
6564eleq1d 2824 . . . . . . . . . 10 (𝑦 = (ℝ ∖ 𝑥) → ((𝐹𝑦) ∈ dom vol ↔ (𝐹 “ (ℝ ∖ 𝑥)) ∈ dom vol))
6663, 65imbi12d 344 . . . . . . . . 9 (𝑦 = (ℝ ∖ 𝑥) → ((¬ 0 ∈ 𝑦 → (𝐹𝑦) ∈ dom vol) ↔ (¬ 0 ∈ (ℝ ∖ 𝑥) → (𝐹 “ (ℝ ∖ 𝑥)) ∈ dom vol)))
6761, 66spcv 3605 . . . . . . . 8 (∀𝑦(¬ 0 ∈ 𝑦 → (𝐹𝑦) ∈ dom vol) → (¬ 0 ∈ (ℝ ∖ 𝑥) → (𝐹 “ (ℝ ∖ 𝑥)) ∈ dom vol))
6857, 59, 67sylc 65 . . . . . . 7 (((𝜑𝑥 ∈ ran (,)) ∧ 0 ∈ 𝑥) → (𝐹 “ (ℝ ∖ 𝑥)) ∈ dom vol)
69 difmbl 25592 . . . . . . 7 (((𝐹 “ ℝ) ∈ dom vol ∧ (𝐹 “ (ℝ ∖ 𝑥)) ∈ dom vol) → ((𝐹 “ ℝ) ∖ (𝐹 “ (ℝ ∖ 𝑥))) ∈ dom vol)
7020, 68, 69syl2anc 584 . . . . . 6 (((𝜑𝑥 ∈ ran (,)) ∧ 0 ∈ 𝑥) → ((𝐹 “ ℝ) ∖ (𝐹 “ (ℝ ∖ 𝑥))) ∈ dom vol)
7116, 70eqeltrrd 2840 . . . . 5 (((𝜑𝑥 ∈ ran (,)) ∧ 0 ∈ 𝑥) → (𝐹𝑥) ∈ dom vol)
72 eleq2 2828 . . . . . . . . . . 11 (𝑦 = 𝑥 → (0 ∈ 𝑦 ↔ 0 ∈ 𝑥))
7372notbid 318 . . . . . . . . . 10 (𝑦 = 𝑥 → (¬ 0 ∈ 𝑦 ↔ ¬ 0 ∈ 𝑥))
74 imaeq2 6076 . . . . . . . . . . 11 (𝑦 = 𝑥 → (𝐹𝑦) = (𝐹𝑥))
7574eleq1d 2824 . . . . . . . . . 10 (𝑦 = 𝑥 → ((𝐹𝑦) ∈ dom vol ↔ (𝐹𝑥) ∈ dom vol))
7673, 75imbi12d 344 . . . . . . . . 9 (𝑦 = 𝑥 → ((¬ 0 ∈ 𝑦 → (𝐹𝑦) ∈ dom vol) ↔ (¬ 0 ∈ 𝑥 → (𝐹𝑥) ∈ dom vol)))
7776spvv 1994 . . . . . . . 8 (∀𝑦(¬ 0 ∈ 𝑦 → (𝐹𝑦) ∈ dom vol) → (¬ 0 ∈ 𝑥 → (𝐹𝑥) ∈ dom vol))
7856, 77syl 17 . . . . . . 7 (𝜑 → (¬ 0 ∈ 𝑥 → (𝐹𝑥) ∈ dom vol))
7978imp 406 . . . . . 6 ((𝜑 ∧ ¬ 0 ∈ 𝑥) → (𝐹𝑥) ∈ dom vol)
8079adantlr 715 . . . . 5 (((𝜑𝑥 ∈ ran (,)) ∧ ¬ 0 ∈ 𝑥) → (𝐹𝑥) ∈ dom vol)
8171, 80pm2.61dan 813 . . . 4 ((𝜑𝑥 ∈ ran (,)) → (𝐹𝑥) ∈ dom vol)
8281ralrimiva 3144 . . 3 (𝜑 → ∀𝑥 ∈ ran (,)(𝐹𝑥) ∈ dom vol)
83 ismbf 25677 . . . 4 (𝐹:ℝ⟶ℝ → (𝐹 ∈ MblFn ↔ ∀𝑥 ∈ ran (,)(𝐹𝑥) ∈ dom vol))
841, 83syl 17 . . 3 (𝜑 → (𝐹 ∈ MblFn ↔ ∀𝑥 ∈ ran (,)(𝐹𝑥) ∈ dom vol))
8582, 84mpbird 257 . 2 (𝜑𝐹 ∈ MblFn)
86 mblvol 25579 . . . . . . . 8 ((𝐹𝑦) ∈ dom vol → (vol‘(𝐹𝑦)) = (vol*‘(𝐹𝑦)))
8754, 86syl 17 . . . . . . 7 ((𝜑 ∧ ¬ 0 ∈ 𝑦) → (vol‘(𝐹𝑦)) = (vol*‘(𝐹𝑦)))
88 mblss 25580 . . . . . . . . 9 ((𝐹𝑦) ∈ dom vol → (𝐹𝑦) ⊆ ℝ)
8954, 88syl 17 . . . . . . . 8 ((𝜑 ∧ ¬ 0 ∈ 𝑦) → (𝐹𝑦) ⊆ ℝ)
90 mblvol 25579 . . . . . . . . . . 11 ((𝐹 “ {𝑥}) ∈ dom vol → (vol‘(𝐹 “ {𝑥})) = (vol*‘(𝐹 “ {𝑥})))
9150, 90syl 17 . . . . . . . . . 10 (((𝜑 ∧ ¬ 0 ∈ 𝑦) ∧ 𝑥 ∈ (𝑦 ∩ ran 𝐹)) → (vol‘(𝐹 “ {𝑥})) = (vol*‘(𝐹 “ {𝑥})))
92 i1fd.4 . . . . . . . . . . 11 ((𝜑𝑥 ∈ (ran 𝐹 ∖ {0})) → (vol‘(𝐹 “ {𝑥})) ∈ ℝ)
9339, 48, 92syl2anc 584 . . . . . . . . . 10 (((𝜑 ∧ ¬ 0 ∈ 𝑦) ∧ 𝑥 ∈ (𝑦 ∩ ran 𝐹)) → (vol‘(𝐹 “ {𝑥})) ∈ ℝ)
9491, 93eqeltrrd 2840 . . . . . . . . 9 (((𝜑 ∧ ¬ 0 ∈ 𝑦) ∧ 𝑥 ∈ (𝑦 ∩ ran 𝐹)) → (vol*‘(𝐹 “ {𝑥})) ∈ ℝ)
9538, 94fsumrecl 15767 . . . . . . . 8 ((𝜑 ∧ ¬ 0 ∈ 𝑦) → Σ𝑥 ∈ (𝑦 ∩ ran 𝐹)(vol*‘(𝐹 “ {𝑥})) ∈ ℝ)
9633fveq2d 6911 . . . . . . . . 9 ((𝜑 ∧ ¬ 0 ∈ 𝑦) → (vol*‘ 𝑥 ∈ (𝑦 ∩ ran 𝐹)(𝐹 “ {𝑥})) = (vol*‘(𝐹𝑦)))
97 mblss 25580 . . . . . . . . . . . . 13 ((𝐹 “ {𝑥}) ∈ dom vol → (𝐹 “ {𝑥}) ⊆ ℝ)
9850, 97syl 17 . . . . . . . . . . . 12 (((𝜑 ∧ ¬ 0 ∈ 𝑦) ∧ 𝑥 ∈ (𝑦 ∩ ran 𝐹)) → (𝐹 “ {𝑥}) ⊆ ℝ)
9998, 94jca 511 . . . . . . . . . . 11 (((𝜑 ∧ ¬ 0 ∈ 𝑦) ∧ 𝑥 ∈ (𝑦 ∩ ran 𝐹)) → ((𝐹 “ {𝑥}) ⊆ ℝ ∧ (vol*‘(𝐹 “ {𝑥})) ∈ ℝ))
10099ralrimiva 3144 . . . . . . . . . 10 ((𝜑 ∧ ¬ 0 ∈ 𝑦) → ∀𝑥 ∈ (𝑦 ∩ ran 𝐹)((𝐹 “ {𝑥}) ⊆ ℝ ∧ (vol*‘(𝐹 “ {𝑥})) ∈ ℝ))
101 ovolfiniun 25550 . . . . . . . . . 10 (((𝑦 ∩ ran 𝐹) ∈ Fin ∧ ∀𝑥 ∈ (𝑦 ∩ ran 𝐹)((𝐹 “ {𝑥}) ⊆ ℝ ∧ (vol*‘(𝐹 “ {𝑥})) ∈ ℝ)) → (vol*‘ 𝑥 ∈ (𝑦 ∩ ran 𝐹)(𝐹 “ {𝑥})) ≤ Σ𝑥 ∈ (𝑦 ∩ ran 𝐹)(vol*‘(𝐹 “ {𝑥})))
10238, 100, 101syl2anc 584 . . . . . . . . 9 ((𝜑 ∧ ¬ 0 ∈ 𝑦) → (vol*‘ 𝑥 ∈ (𝑦 ∩ ran 𝐹)(𝐹 “ {𝑥})) ≤ Σ𝑥 ∈ (𝑦 ∩ ran 𝐹)(vol*‘(𝐹 “ {𝑥})))
10396, 102eqbrtrrd 5172 . . . . . . . 8 ((𝜑 ∧ ¬ 0 ∈ 𝑦) → (vol*‘(𝐹𝑦)) ≤ Σ𝑥 ∈ (𝑦 ∩ ran 𝐹)(vol*‘(𝐹 “ {𝑥})))
104 ovollecl 25532 . . . . . . . 8 (((𝐹𝑦) ⊆ ℝ ∧ Σ𝑥 ∈ (𝑦 ∩ ran 𝐹)(vol*‘(𝐹 “ {𝑥})) ∈ ℝ ∧ (vol*‘(𝐹𝑦)) ≤ Σ𝑥 ∈ (𝑦 ∩ ran 𝐹)(vol*‘(𝐹 “ {𝑥}))) → (vol*‘(𝐹𝑦)) ∈ ℝ)
10589, 95, 103, 104syl3anc 1370 . . . . . . 7 ((𝜑 ∧ ¬ 0 ∈ 𝑦) → (vol*‘(𝐹𝑦)) ∈ ℝ)
10687, 105eqeltrd 2839 . . . . . 6 ((𝜑 ∧ ¬ 0 ∈ 𝑦) → (vol‘(𝐹𝑦)) ∈ ℝ)
107106ex 412 . . . . 5 (𝜑 → (¬ 0 ∈ 𝑦 → (vol‘(𝐹𝑦)) ∈ ℝ))
108107alrimiv 1925 . . . 4 (𝜑 → ∀𝑦(¬ 0 ∈ 𝑦 → (vol‘(𝐹𝑦)) ∈ ℝ))
109 neldifsn 4797 . . . 4 ¬ 0 ∈ (ℝ ∖ {0})
11060difexi 5336 . . . . 5 (ℝ ∖ {0}) ∈ V
111 eleq2 2828 . . . . . . 7 (𝑦 = (ℝ ∖ {0}) → (0 ∈ 𝑦 ↔ 0 ∈ (ℝ ∖ {0})))
112111notbid 318 . . . . . 6 (𝑦 = (ℝ ∖ {0}) → (¬ 0 ∈ 𝑦 ↔ ¬ 0 ∈ (ℝ ∖ {0})))
113 imaeq2 6076 . . . . . . . 8 (𝑦 = (ℝ ∖ {0}) → (𝐹𝑦) = (𝐹 “ (ℝ ∖ {0})))
114113fveq2d 6911 . . . . . . 7 (𝑦 = (ℝ ∖ {0}) → (vol‘(𝐹𝑦)) = (vol‘(𝐹 “ (ℝ ∖ {0}))))
115114eleq1d 2824 . . . . . 6 (𝑦 = (ℝ ∖ {0}) → ((vol‘(𝐹𝑦)) ∈ ℝ ↔ (vol‘(𝐹 “ (ℝ ∖ {0}))) ∈ ℝ))
116112, 115imbi12d 344 . . . . 5 (𝑦 = (ℝ ∖ {0}) → ((¬ 0 ∈ 𝑦 → (vol‘(𝐹𝑦)) ∈ ℝ) ↔ (¬ 0 ∈ (ℝ ∖ {0}) → (vol‘(𝐹 “ (ℝ ∖ {0}))) ∈ ℝ)))
117110, 116spcv 3605 . . . 4 (∀𝑦(¬ 0 ∈ 𝑦 → (vol‘(𝐹𝑦)) ∈ ℝ) → (¬ 0 ∈ (ℝ ∖ {0}) → (vol‘(𝐹 “ (ℝ ∖ {0}))) ∈ ℝ))
118108, 109, 117mpisyl 21 . . 3 (𝜑 → (vol‘(𝐹 “ (ℝ ∖ {0}))) ∈ ℝ)
1191, 34, 1183jca 1127 . 2 (𝜑 → (𝐹:ℝ⟶ℝ ∧ ran 𝐹 ∈ Fin ∧ (vol‘(𝐹 “ (ℝ ∖ {0}))) ∈ ℝ))
120 isi1f 25723 . 2 (𝐹 ∈ dom ∫1 ↔ (𝐹 ∈ MblFn ∧ (𝐹:ℝ⟶ℝ ∧ ran 𝐹 ∈ Fin ∧ (vol‘(𝐹 “ (ℝ ∖ {0}))) ∈ ℝ)))
12185, 119, 120sylanbrc 583 1 (𝜑𝐹 ∈ dom ∫1)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 206  wa 395  w3a 1086  wal 1535   = wceq 1537  wcel 2106  wral 3059  cdif 3960  cin 3962  wss 3963  c0 4339  𝒫 cpw 4605  {csn 4631   ciun 4996   class class class wbr 5148   × cxp 5687  ccnv 5688  dom cdm 5689  ran crn 5690  cima 5692  Fun wfun 6557  wf 6559  cfv 6563  Fincfn 8984  cr 11152  0cc0 11153  *cxr 11292  cle 11294  (,)cioo 13384  Σcsu 15719  vol*covol 25511  volcvol 25512  MblFncmbf 25663  1citg1 25664
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-inf2 9679  ax-cnex 11209  ax-resscn 11210  ax-1cn 11211  ax-icn 11212  ax-addcl 11213  ax-addrcl 11214  ax-mulcl 11215  ax-mulrcl 11216  ax-mulcom 11217  ax-addass 11218  ax-mulass 11219  ax-distr 11220  ax-i2m1 11221  ax-1ne0 11222  ax-1rid 11223  ax-rnegex 11224  ax-rrecex 11225  ax-cnre 11226  ax-pre-lttri 11227  ax-pre-lttrn 11228  ax-pre-ltadd 11229  ax-pre-mulgt0 11230  ax-pre-sup 11231
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-int 4952  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-se 5642  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-isom 6572  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-of 7697  df-om 7888  df-1st 8013  df-2nd 8014  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-1o 8505  df-2o 8506  df-er 8744  df-map 8867  df-pm 8868  df-en 8985  df-dom 8986  df-sdom 8987  df-fin 8988  df-sup 9480  df-inf 9481  df-oi 9548  df-dju 9939  df-card 9977  df-pnf 11295  df-mnf 11296  df-xr 11297  df-ltxr 11298  df-le 11299  df-sub 11492  df-neg 11493  df-div 11919  df-nn 12265  df-2 12327  df-3 12328  df-n0 12525  df-z 12612  df-uz 12877  df-q 12989  df-rp 13033  df-xadd 13153  df-ioo 13388  df-ico 13390  df-icc 13391  df-fz 13545  df-fzo 13692  df-fl 13829  df-seq 14040  df-exp 14100  df-hash 14367  df-cj 15135  df-re 15136  df-im 15137  df-sqrt 15271  df-abs 15272  df-clim 15521  df-sum 15720  df-xmet 21375  df-met 21376  df-ovol 25513  df-vol 25514  df-mbf 25668  df-itg1 25669
This theorem is referenced by:  i1f0  25736  i1f1  25739  i1fadd  25744  i1fmul  25745  i1fmulc  25753  i1fres  25755  mbfi1fseqlem4  25768  itg2addnclem2  37659  ftc1anclem3  37682
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