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Theorem i1fd 24951
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 723 . . . . . . . 8 (((𝜑𝑥 ∈ ran (,)) ∧ 0 ∈ 𝑥) → 𝐹:ℝ⟶ℝ)
3 ffun 6654 . . . . . . . 8 (𝐹:ℝ⟶ℝ → Fun 𝐹)
4 funcnvcnv 6551 . . . . . . . 8 (Fun 𝐹 → Fun 𝐹)
5 imadif 6568 . . . . . . . 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 13280 . . . . . . . . . . . . 13 (,):(ℝ* × ℝ*)⟶𝒫 ℝ
8 frn 6658 . . . . . . . . . . . . 13 ((,):(ℝ* × ℝ*)⟶𝒫 ℝ → ran (,) ⊆ 𝒫 ℝ)
97, 8ax-mp 5 . . . . . . . . . . . 12 ran (,) ⊆ 𝒫 ℝ
109sseli 3928 . . . . . . . . . . 11 (𝑥 ∈ ran (,) → 𝑥 ∈ 𝒫 ℝ)
1110elpwid 4556 . . . . . . . . . 10 (𝑥 ∈ ran (,) → 𝑥 ⊆ ℝ)
1211ad2antlr 724 . . . . . . . . 9 (((𝜑𝑥 ∈ ran (,)) ∧ 0 ∈ 𝑥) → 𝑥 ⊆ ℝ)
13 dfss4 4205 . . . . . . . . 9 (𝑥 ⊆ ℝ ↔ (ℝ ∖ (ℝ ∖ 𝑥)) = 𝑥)
1412, 13sylib 217 . . . . . . . 8 (((𝜑𝑥 ∈ ran (,)) ∧ 0 ∈ 𝑥) → (ℝ ∖ (ℝ ∖ 𝑥)) = 𝑥)
1514imaeq2d 5999 . . . . . . 7 (((𝜑𝑥 ∈ ran (,)) ∧ 0 ∈ 𝑥) → (𝐹 “ (ℝ ∖ (ℝ ∖ 𝑥))) = (𝐹𝑥))
166, 15eqtr3d 2778 . . . . . 6 (((𝜑𝑥 ∈ ran (,)) ∧ 0 ∈ 𝑥) → ((𝐹 “ ℝ) ∖ (𝐹 “ (ℝ ∖ 𝑥))) = (𝐹𝑥))
17 fimacnv 6673 . . . . . . . . 9 (𝐹:ℝ⟶ℝ → (𝐹 “ ℝ) = ℝ)
182, 17syl 17 . . . . . . . 8 (((𝜑𝑥 ∈ ran (,)) ∧ 0 ∈ 𝑥) → (𝐹 “ ℝ) = ℝ)
19 rembl 24810 . . . . . . . 8 ℝ ∈ dom vol
2018, 19eqeltrdi 2845 . . . . . . 7 (((𝜑𝑥 ∈ ran (,)) ∧ 0 ∈ 𝑥) → (𝐹 “ ℝ) ∈ dom vol)
211adantr 481 . . . . . . . . . . . . 13 ((𝜑 ∧ ¬ 0 ∈ 𝑦) → 𝐹:ℝ⟶ℝ)
22 inpreima 6997 . . . . . . . . . . . . . 14 (Fun 𝐹 → (𝐹 “ (𝑦 ∩ ran 𝐹)) = ((𝐹𝑦) ∩ (𝐹 “ ran 𝐹)))
23 iunid 5007 . . . . . . . . . . . . . . . 16 𝑥 ∈ (𝑦 ∩ ran 𝐹){𝑥} = (𝑦 ∩ ran 𝐹)
2423imaeq2i 5997 . . . . . . . . . . . . . . 15 (𝐹 𝑥 ∈ (𝑦 ∩ ran 𝐹){𝑥}) = (𝐹 “ (𝑦 ∩ ran 𝐹))
25 imaiun 7174 . . . . . . . . . . . . . . 15 (𝐹 𝑥 ∈ (𝑦 ∩ ran 𝐹){𝑥}) = 𝑥 ∈ (𝑦 ∩ ran 𝐹)(𝐹 “ {𝑥})
2624, 25eqtr3i 2766 . . . . . . . . . . . . . 14 (𝐹 “ (𝑦 ∩ ran 𝐹)) = 𝑥 ∈ (𝑦 ∩ ran 𝐹)(𝐹 “ {𝑥})
27 cnvimass 6019 . . . . . . . . . . . . . . . 16 (𝐹𝑦) ⊆ dom 𝐹
28 cnvimarndm 6020 . . . . . . . . . . . . . . . 16 (𝐹 “ ran 𝐹) = dom 𝐹
2927, 28sseqtrri 3969 . . . . . . . . . . . . . . 15 (𝐹𝑦) ⊆ (𝐹 “ ran 𝐹)
30 df-ss 3915 . . . . . . . . . . . . . . 15 ((𝐹𝑦) ⊆ (𝐹 “ ran 𝐹) ↔ ((𝐹𝑦) ∩ (𝐹 “ ran 𝐹)) = (𝐹𝑦))
3129, 30mpbi 229 . . . . . . . . . . . . . 14 ((𝐹𝑦) ∩ (𝐹 “ ran 𝐹)) = (𝐹𝑦)
3222, 26, 313eqtr3g 2799 . . . . . . . . . . . . 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 481 . . . . . . . . . . . . . 14 ((𝜑 ∧ ¬ 0 ∈ 𝑦) → ran 𝐹 ∈ Fin)
36 inss2 4176 . . . . . . . . . . . . . 14 (𝑦 ∩ ran 𝐹) ⊆ ran 𝐹
37 ssfi 9038 . . . . . . . . . . . . . 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 764 . . . . . . . . . . . . . . 15 (((𝜑 ∧ ¬ 0 ∈ 𝑦) ∧ 𝑥 ∈ (𝑦 ∩ ran 𝐹)) → 𝜑)
40 elinel1 4142 . . . . . . . . . . . . . . . . . . . 20 (0 ∈ (𝑦 ∩ ran 𝐹) → 0 ∈ 𝑦)
4140con3i 154 . . . . . . . . . . . . . . . . . . 19 (¬ 0 ∈ 𝑦 → ¬ 0 ∈ (𝑦 ∩ ran 𝐹))
4241adantl 482 . . . . . . . . . . . . . . . . . 18 ((𝜑 ∧ ¬ 0 ∈ 𝑦) → ¬ 0 ∈ (𝑦 ∩ ran 𝐹))
43 disjsn 4659 . . . . . . . . . . . . . . . . . 18 (((𝑦 ∩ ran 𝐹) ∩ {0}) = ∅ ↔ ¬ 0 ∈ (𝑦 ∩ ran 𝐹))
4442, 43sylibr 233 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ ¬ 0 ∈ 𝑦) → ((𝑦 ∩ ran 𝐹) ∩ {0}) = ∅)
45 reldisj 4398 . . . . . . . . . . . . . . . . . 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 217 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ ¬ 0 ∈ 𝑦) → (𝑦 ∩ ran 𝐹) ⊆ (ran 𝐹 ∖ {0}))
4847sselda 3932 . . . . . . . . . . . . . . 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 3139 . . . . . . . . . . . . 13 ((𝜑 ∧ ¬ 0 ∈ 𝑦) → ∀𝑥 ∈ (𝑦 ∩ ran 𝐹)(𝐹 “ {𝑥}) ∈ dom vol)
52 finiunmbl 24814 . . . . . . . . . . . . 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 2838 . . . . . . . . . . 11 ((𝜑 ∧ ¬ 0 ∈ 𝑦) → (𝐹𝑦) ∈ dom vol)
5554ex 413 . . . . . . . . . 10 (𝜑 → (¬ 0 ∈ 𝑦 → (𝐹𝑦) ∈ dom vol))
5655alrimiv 1929 . . . . . . . . 9 (𝜑 → ∀𝑦(¬ 0 ∈ 𝑦 → (𝐹𝑦) ∈ dom vol))
5756ad2antrr 723 . . . . . . . 8 (((𝜑𝑥 ∈ ran (,)) ∧ 0 ∈ 𝑥) → ∀𝑦(¬ 0 ∈ 𝑦 → (𝐹𝑦) ∈ dom vol))
58 elndif 4075 . . . . . . . . 9 (0 ∈ 𝑥 → ¬ 0 ∈ (ℝ ∖ 𝑥))
5958adantl 482 . . . . . . . 8 (((𝜑𝑥 ∈ ran (,)) ∧ 0 ∈ 𝑥) → ¬ 0 ∈ (ℝ ∖ 𝑥))
60 reex 11063 . . . . . . . . . 10 ℝ ∈ V
6160difexi 5272 . . . . . . . . 9 (ℝ ∖ 𝑥) ∈ V
62 eleq2 2825 . . . . . . . . . . 11 (𝑦 = (ℝ ∖ 𝑥) → (0 ∈ 𝑦 ↔ 0 ∈ (ℝ ∖ 𝑥)))
6362notbid 317 . . . . . . . . . 10 (𝑦 = (ℝ ∖ 𝑥) → (¬ 0 ∈ 𝑦 ↔ ¬ 0 ∈ (ℝ ∖ 𝑥)))
64 imaeq2 5995 . . . . . . . . . . 11 (𝑦 = (ℝ ∖ 𝑥) → (𝐹𝑦) = (𝐹 “ (ℝ ∖ 𝑥)))
6564eleq1d 2821 . . . . . . . . . 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 3553 . . . . . . . 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 24813 . . . . . . 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 2838 . . . . 5 (((𝜑𝑥 ∈ ran (,)) ∧ 0 ∈ 𝑥) → (𝐹𝑥) ∈ dom vol)
72 eleq2 2825 . . . . . . . . . . 11 (𝑦 = 𝑥 → (0 ∈ 𝑦 ↔ 0 ∈ 𝑥))
7372notbid 317 . . . . . . . . . 10 (𝑦 = 𝑥 → (¬ 0 ∈ 𝑦 ↔ ¬ 0 ∈ 𝑥))
74 imaeq2 5995 . . . . . . . . . . 11 (𝑦 = 𝑥 → (𝐹𝑦) = (𝐹𝑥))
7574eleq1d 2821 . . . . . . . . . 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 1999 . . . . . . . 8 (∀𝑦(¬ 0 ∈ 𝑦 → (𝐹𝑦) ∈ dom vol) → (¬ 0 ∈ 𝑥 → (𝐹𝑥) ∈ dom vol))
7856, 77syl 17 . . . . . . 7 (𝜑 → (¬ 0 ∈ 𝑥 → (𝐹𝑥) ∈ dom vol))
7978imp 407 . . . . . 6 ((𝜑 ∧ ¬ 0 ∈ 𝑥) → (𝐹𝑥) ∈ dom vol)
8079adantlr 712 . . . . 5 (((𝜑𝑥 ∈ ran (,)) ∧ ¬ 0 ∈ 𝑥) → (𝐹𝑥) ∈ dom vol)
8171, 80pm2.61dan 810 . . . 4 ((𝜑𝑥 ∈ ran (,)) → (𝐹𝑥) ∈ dom vol)
8281ralrimiva 3139 . . 3 (𝜑 → ∀𝑥 ∈ ran (,)(𝐹𝑥) ∈ dom vol)
83 ismbf 24898 . . . 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 256 . 2 (𝜑𝐹 ∈ MblFn)
86 mblvol 24800 . . . . . . . 8 ((𝐹𝑦) ∈ dom vol → (vol‘(𝐹𝑦)) = (vol*‘(𝐹𝑦)))
8754, 86syl 17 . . . . . . 7 ((𝜑 ∧ ¬ 0 ∈ 𝑦) → (vol‘(𝐹𝑦)) = (vol*‘(𝐹𝑦)))
88 mblss 24801 . . . . . . . . 9 ((𝐹𝑦) ∈ dom vol → (𝐹𝑦) ⊆ ℝ)
8954, 88syl 17 . . . . . . . 8 ((𝜑 ∧ ¬ 0 ∈ 𝑦) → (𝐹𝑦) ⊆ ℝ)
90 mblvol 24800 . . . . . . . . . . 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 2838 . . . . . . . . 9 (((𝜑 ∧ ¬ 0 ∈ 𝑦) ∧ 𝑥 ∈ (𝑦 ∩ ran 𝐹)) → (vol*‘(𝐹 “ {𝑥})) ∈ ℝ)
9538, 94fsumrecl 15545 . . . . . . . 8 ((𝜑 ∧ ¬ 0 ∈ 𝑦) → Σ𝑥 ∈ (𝑦 ∩ ran 𝐹)(vol*‘(𝐹 “ {𝑥})) ∈ ℝ)
9633fveq2d 6829 . . . . . . . . 9 ((𝜑 ∧ ¬ 0 ∈ 𝑦) → (vol*‘ 𝑥 ∈ (𝑦 ∩ ran 𝐹)(𝐹 “ {𝑥})) = (vol*‘(𝐹𝑦)))
97 mblss 24801 . . . . . . . . . . . . 13 ((𝐹 “ {𝑥}) ∈ dom vol → (𝐹 “ {𝑥}) ⊆ ℝ)
9850, 97syl 17 . . . . . . . . . . . 12 (((𝜑 ∧ ¬ 0 ∈ 𝑦) ∧ 𝑥 ∈ (𝑦 ∩ ran 𝐹)) → (𝐹 “ {𝑥}) ⊆ ℝ)
9998, 94jca 512 . . . . . . . . . . 11 (((𝜑 ∧ ¬ 0 ∈ 𝑦) ∧ 𝑥 ∈ (𝑦 ∩ ran 𝐹)) → ((𝐹 “ {𝑥}) ⊆ ℝ ∧ (vol*‘(𝐹 “ {𝑥})) ∈ ℝ))
10099ralrimiva 3139 . . . . . . . . . 10 ((𝜑 ∧ ¬ 0 ∈ 𝑦) → ∀𝑥 ∈ (𝑦 ∩ ran 𝐹)((𝐹 “ {𝑥}) ⊆ ℝ ∧ (vol*‘(𝐹 “ {𝑥})) ∈ ℝ))
101 ovolfiniun 24771 . . . . . . . . . 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 5116 . . . . . . . 8 ((𝜑 ∧ ¬ 0 ∈ 𝑦) → (vol*‘(𝐹𝑦)) ≤ Σ𝑥 ∈ (𝑦 ∩ ran 𝐹)(vol*‘(𝐹 “ {𝑥})))
104 ovollecl 24753 . . . . . . . 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 2837 . . . . . 6 ((𝜑 ∧ ¬ 0 ∈ 𝑦) → (vol‘(𝐹𝑦)) ∈ ℝ)
107106ex 413 . . . . 5 (𝜑 → (¬ 0 ∈ 𝑦 → (vol‘(𝐹𝑦)) ∈ ℝ))
108107alrimiv 1929 . . . 4 (𝜑 → ∀𝑦(¬ 0 ∈ 𝑦 → (vol‘(𝐹𝑦)) ∈ ℝ))
109 neldifsn 4739 . . . 4 ¬ 0 ∈ (ℝ ∖ {0})
11060difexi 5272 . . . . 5 (ℝ ∖ {0}) ∈ V
111 eleq2 2825 . . . . . . 7 (𝑦 = (ℝ ∖ {0}) → (0 ∈ 𝑦 ↔ 0 ∈ (ℝ ∖ {0})))
112111notbid 317 . . . . . 6 (𝑦 = (ℝ ∖ {0}) → (¬ 0 ∈ 𝑦 ↔ ¬ 0 ∈ (ℝ ∖ {0})))
113 imaeq2 5995 . . . . . . . 8 (𝑦 = (ℝ ∖ {0}) → (𝐹𝑦) = (𝐹 “ (ℝ ∖ {0})))
114113fveq2d 6829 . . . . . . 7 (𝑦 = (ℝ ∖ {0}) → (vol‘(𝐹𝑦)) = (vol‘(𝐹 “ (ℝ ∖ {0}))))
115114eleq1d 2821 . . . . . 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 3553 . . . 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 24944 . 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 205  wa 396  w3a 1086  wal 1538   = wceq 1540  wcel 2105  wral 3061  cdif 3895  cin 3897  wss 3898  c0 4269  𝒫 cpw 4547  {csn 4573   ciun 4941   class class class wbr 5092   × cxp 5618  ccnv 5619  dom cdm 5620  ran crn 5621  cima 5623  Fun wfun 6473  wf 6475  cfv 6479  Fincfn 8804  cr 10971  0cc0 10972  *cxr 11109  cle 11111  (,)cioo 13180  Σcsu 15496  vol*covol 24732  volcvol 24733  MblFncmbf 24884  1citg1 24885
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2707  ax-rep 5229  ax-sep 5243  ax-nul 5250  ax-pow 5308  ax-pr 5372  ax-un 7650  ax-inf2 9498  ax-cnex 11028  ax-resscn 11029  ax-1cn 11030  ax-icn 11031  ax-addcl 11032  ax-addrcl 11033  ax-mulcl 11034  ax-mulrcl 11035  ax-mulcom 11036  ax-addass 11037  ax-mulass 11038  ax-distr 11039  ax-i2m1 11040  ax-1ne0 11041  ax-1rid 11042  ax-rnegex 11043  ax-rrecex 11044  ax-cnre 11045  ax-pre-lttri 11046  ax-pre-lttrn 11047  ax-pre-ltadd 11048  ax-pre-mulgt0 11049  ax-pre-sup 11050
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2886  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3349  df-reu 3350  df-rab 3404  df-v 3443  df-sbc 3728  df-csb 3844  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-pss 3917  df-nul 4270  df-if 4474  df-pw 4549  df-sn 4574  df-pr 4576  df-op 4580  df-uni 4853  df-int 4895  df-iun 4943  df-br 5093  df-opab 5155  df-mpt 5176  df-tr 5210  df-id 5518  df-eprel 5524  df-po 5532  df-so 5533  df-fr 5575  df-se 5576  df-we 5577  df-xp 5626  df-rel 5627  df-cnv 5628  df-co 5629  df-dm 5630  df-rn 5631  df-res 5632  df-ima 5633  df-pred 6238  df-ord 6305  df-on 6306  df-lim 6307  df-suc 6308  df-iota 6431  df-fun 6481  df-fn 6482  df-f 6483  df-f1 6484  df-fo 6485  df-f1o 6486  df-fv 6487  df-isom 6488  df-riota 7293  df-ov 7340  df-oprab 7341  df-mpo 7342  df-of 7595  df-om 7781  df-1st 7899  df-2nd 7900  df-frecs 8167  df-wrecs 8198  df-recs 8272  df-rdg 8311  df-1o 8367  df-2o 8368  df-er 8569  df-map 8688  df-pm 8689  df-en 8805  df-dom 8806  df-sdom 8807  df-fin 8808  df-sup 9299  df-inf 9300  df-oi 9367  df-dju 9758  df-card 9796  df-pnf 11112  df-mnf 11113  df-xr 11114  df-ltxr 11115  df-le 11116  df-sub 11308  df-neg 11309  df-div 11734  df-nn 12075  df-2 12137  df-3 12138  df-n0 12335  df-z 12421  df-uz 12684  df-q 12790  df-rp 12832  df-xadd 12950  df-ioo 13184  df-ico 13186  df-icc 13187  df-fz 13341  df-fzo 13484  df-fl 13613  df-seq 13823  df-exp 13884  df-hash 14146  df-cj 14909  df-re 14910  df-im 14911  df-sqrt 15045  df-abs 15046  df-clim 15296  df-sum 15497  df-xmet 20696  df-met 20697  df-ovol 24734  df-vol 24735  df-mbf 24889  df-itg1 24890
This theorem is referenced by:  i1f0  24957  i1f1  24960  i1fadd  24965  i1fmul  24966  i1fmulc  24974  i1fres  24976  mbfi1fseqlem4  24989  itg2addnclem2  35934  ftc1anclem3  35957
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