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Theorem i1fd 23667
 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 764 . . . . . . . 8 (((𝜑𝑥 ∈ ran (,)) ∧ 0 ∈ 𝑥) → 𝐹:ℝ⟶ℝ)
3 ffun 6209 . . . . . . . 8 (𝐹:ℝ⟶ℝ → Fun 𝐹)
4 funcnvcnv 6117 . . . . . . . 8 (Fun 𝐹 → Fun 𝐹)
5 imadif 6134 . . . . . . . 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 12484 . . . . . . . . . . . . 13 (,):(ℝ* × ℝ*)⟶𝒫 ℝ
8 frn 6214 . . . . . . . . . . . . 13 ((,):(ℝ* × ℝ*)⟶𝒫 ℝ → ran (,) ⊆ 𝒫 ℝ)
97, 8ax-mp 5 . . . . . . . . . . . 12 ran (,) ⊆ 𝒫 ℝ
109sseli 3740 . . . . . . . . . . 11 (𝑥 ∈ ran (,) → 𝑥 ∈ 𝒫 ℝ)
1110elpwid 4314 . . . . . . . . . 10 (𝑥 ∈ ran (,) → 𝑥 ⊆ ℝ)
1211ad2antlr 765 . . . . . . . . 9 (((𝜑𝑥 ∈ ran (,)) ∧ 0 ∈ 𝑥) → 𝑥 ⊆ ℝ)
13 dfss4 4001 . . . . . . . . 9 (𝑥 ⊆ ℝ ↔ (ℝ ∖ (ℝ ∖ 𝑥)) = 𝑥)
1412, 13sylib 208 . . . . . . . 8 (((𝜑𝑥 ∈ ran (,)) ∧ 0 ∈ 𝑥) → (ℝ ∖ (ℝ ∖ 𝑥)) = 𝑥)
1514imaeq2d 5624 . . . . . . 7 (((𝜑𝑥 ∈ ran (,)) ∧ 0 ∈ 𝑥) → (𝐹 “ (ℝ ∖ (ℝ ∖ 𝑥))) = (𝐹𝑥))
166, 15eqtr3d 2796 . . . . . 6 (((𝜑𝑥 ∈ ran (,)) ∧ 0 ∈ 𝑥) → ((𝐹 “ ℝ) ∖ (𝐹 “ (ℝ ∖ 𝑥))) = (𝐹𝑥))
17 fimacnv 6511 . . . . . . . . 9 (𝐹:ℝ⟶ℝ → (𝐹 “ ℝ) = ℝ)
182, 17syl 17 . . . . . . . 8 (((𝜑𝑥 ∈ ran (,)) ∧ 0 ∈ 𝑥) → (𝐹 “ ℝ) = ℝ)
19 rembl 23528 . . . . . . . 8 ℝ ∈ dom vol
2018, 19syl6eqel 2847 . . . . . . 7 (((𝜑𝑥 ∈ ran (,)) ∧ 0 ∈ 𝑥) → (𝐹 “ ℝ) ∈ dom vol)
211adantr 472 . . . . . . . . . . . . 13 ((𝜑 ∧ ¬ 0 ∈ 𝑦) → 𝐹:ℝ⟶ℝ)
22 inpreima 6506 . . . . . . . . . . . . . 14 (Fun 𝐹 → (𝐹 “ (𝑦 ∩ ran 𝐹)) = ((𝐹𝑦) ∩ (𝐹 “ ran 𝐹)))
23 iunid 4727 . . . . . . . . . . . . . . . 16 𝑥 ∈ (𝑦 ∩ ran 𝐹){𝑥} = (𝑦 ∩ ran 𝐹)
2423imaeq2i 5622 . . . . . . . . . . . . . . 15 (𝐹 𝑥 ∈ (𝑦 ∩ ran 𝐹){𝑥}) = (𝐹 “ (𝑦 ∩ ran 𝐹))
25 imaiun 6667 . . . . . . . . . . . . . . 15 (𝐹 𝑥 ∈ (𝑦 ∩ ran 𝐹){𝑥}) = 𝑥 ∈ (𝑦 ∩ ran 𝐹)(𝐹 “ {𝑥})
2624, 25eqtr3i 2784 . . . . . . . . . . . . . 14 (𝐹 “ (𝑦 ∩ ran 𝐹)) = 𝑥 ∈ (𝑦 ∩ ran 𝐹)(𝐹 “ {𝑥})
27 cnvimass 5643 . . . . . . . . . . . . . . . 16 (𝐹𝑦) ⊆ dom 𝐹
28 cnvimarndm 5644 . . . . . . . . . . . . . . . 16 (𝐹 “ ran 𝐹) = dom 𝐹
2927, 28sseqtr4i 3779 . . . . . . . . . . . . . . 15 (𝐹𝑦) ⊆ (𝐹 “ ran 𝐹)
30 df-ss 3729 . . . . . . . . . . . . . . 15 ((𝐹𝑦) ⊆ (𝐹 “ ran 𝐹) ↔ ((𝐹𝑦) ∩ (𝐹 “ ran 𝐹)) = (𝐹𝑦))
3129, 30mpbi 220 . . . . . . . . . . . . . 14 ((𝐹𝑦) ∩ (𝐹 “ ran 𝐹)) = (𝐹𝑦)
3222, 26, 313eqtr3g 2817 . . . . . . . . . . . . 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 472 . . . . . . . . . . . . . 14 ((𝜑 ∧ ¬ 0 ∈ 𝑦) → ran 𝐹 ∈ Fin)
36 inss2 3977 . . . . . . . . . . . . . 14 (𝑦 ∩ ran 𝐹) ⊆ ran 𝐹
37 ssfi 8347 . . . . . . . . . . . . . 14 ((ran 𝐹 ∈ Fin ∧ (𝑦 ∩ ran 𝐹) ⊆ ran 𝐹) → (𝑦 ∩ ran 𝐹) ∈ Fin)
3835, 36, 37sylancl 697 . . . . . . . . . . . . 13 ((𝜑 ∧ ¬ 0 ∈ 𝑦) → (𝑦 ∩ ran 𝐹) ∈ Fin)
39 simpll 807 . . . . . . . . . . . . . . 15 (((𝜑 ∧ ¬ 0 ∈ 𝑦) ∧ 𝑥 ∈ (𝑦 ∩ ran 𝐹)) → 𝜑)
40 inss1 3976 . . . . . . . . . . . . . . . . . . . . 21 (𝑦 ∩ ran 𝐹) ⊆ 𝑦
4140sseli 3740 . . . . . . . . . . . . . . . . . . . 20 (0 ∈ (𝑦 ∩ ran 𝐹) → 0 ∈ 𝑦)
4241con3i 150 . . . . . . . . . . . . . . . . . . 19 (¬ 0 ∈ 𝑦 → ¬ 0 ∈ (𝑦 ∩ ran 𝐹))
4342adantl 473 . . . . . . . . . . . . . . . . . 18 ((𝜑 ∧ ¬ 0 ∈ 𝑦) → ¬ 0 ∈ (𝑦 ∩ ran 𝐹))
44 disjsn 4390 . . . . . . . . . . . . . . . . . 18 (((𝑦 ∩ ran 𝐹) ∩ {0}) = ∅ ↔ ¬ 0 ∈ (𝑦 ∩ ran 𝐹))
4543, 44sylibr 224 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ ¬ 0 ∈ 𝑦) → ((𝑦 ∩ ran 𝐹) ∩ {0}) = ∅)
46 reldisj 4163 . . . . . . . . . . . . . . . . . 18 ((𝑦 ∩ ran 𝐹) ⊆ ran 𝐹 → (((𝑦 ∩ ran 𝐹) ∩ {0}) = ∅ ↔ (𝑦 ∩ ran 𝐹) ⊆ (ran 𝐹 ∖ {0})))
4736, 46ax-mp 5 . . . . . . . . . . . . . . . . 17 (((𝑦 ∩ ran 𝐹) ∩ {0}) = ∅ ↔ (𝑦 ∩ ran 𝐹) ⊆ (ran 𝐹 ∖ {0}))
4845, 47sylib 208 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ ¬ 0 ∈ 𝑦) → (𝑦 ∩ ran 𝐹) ⊆ (ran 𝐹 ∖ {0}))
4948sselda 3744 . . . . . . . . . . . . . . 15 (((𝜑 ∧ ¬ 0 ∈ 𝑦) ∧ 𝑥 ∈ (𝑦 ∩ ran 𝐹)) → 𝑥 ∈ (ran 𝐹 ∖ {0}))
50 i1fd.3 . . . . . . . . . . . . . . 15 ((𝜑𝑥 ∈ (ran 𝐹 ∖ {0})) → (𝐹 “ {𝑥}) ∈ dom vol)
5139, 49, 50syl2anc 696 . . . . . . . . . . . . . 14 (((𝜑 ∧ ¬ 0 ∈ 𝑦) ∧ 𝑥 ∈ (𝑦 ∩ ran 𝐹)) → (𝐹 “ {𝑥}) ∈ dom vol)
5251ralrimiva 3104 . . . . . . . . . . . . 13 ((𝜑 ∧ ¬ 0 ∈ 𝑦) → ∀𝑥 ∈ (𝑦 ∩ ran 𝐹)(𝐹 “ {𝑥}) ∈ dom vol)
53 finiunmbl 23532 . . . . . . . . . . . . 13 (((𝑦 ∩ ran 𝐹) ∈ Fin ∧ ∀𝑥 ∈ (𝑦 ∩ ran 𝐹)(𝐹 “ {𝑥}) ∈ dom vol) → 𝑥 ∈ (𝑦 ∩ ran 𝐹)(𝐹 “ {𝑥}) ∈ dom vol)
5438, 52, 53syl2anc 696 . . . . . . . . . . . 12 ((𝜑 ∧ ¬ 0 ∈ 𝑦) → 𝑥 ∈ (𝑦 ∩ ran 𝐹)(𝐹 “ {𝑥}) ∈ dom vol)
5533, 54eqeltrrd 2840 . . . . . . . . . . 11 ((𝜑 ∧ ¬ 0 ∈ 𝑦) → (𝐹𝑦) ∈ dom vol)
5655ex 449 . . . . . . . . . 10 (𝜑 → (¬ 0 ∈ 𝑦 → (𝐹𝑦) ∈ dom vol))
5756alrimiv 2004 . . . . . . . . 9 (𝜑 → ∀𝑦(¬ 0 ∈ 𝑦 → (𝐹𝑦) ∈ dom vol))
5857ad2antrr 764 . . . . . . . 8 (((𝜑𝑥 ∈ ran (,)) ∧ 0 ∈ 𝑥) → ∀𝑦(¬ 0 ∈ 𝑦 → (𝐹𝑦) ∈ dom vol))
59 elndif 3877 . . . . . . . . 9 (0 ∈ 𝑥 → ¬ 0 ∈ (ℝ ∖ 𝑥))
6059adantl 473 . . . . . . . 8 (((𝜑𝑥 ∈ ran (,)) ∧ 0 ∈ 𝑥) → ¬ 0 ∈ (ℝ ∖ 𝑥))
61 reex 10239 . . . . . . . . . 10 ℝ ∈ V
62 difexg 4960 . . . . . . . . . 10 (ℝ ∈ V → (ℝ ∖ 𝑥) ∈ V)
6361, 62ax-mp 5 . . . . . . . . 9 (ℝ ∖ 𝑥) ∈ V
64 eleq2 2828 . . . . . . . . . . 11 (𝑦 = (ℝ ∖ 𝑥) → (0 ∈ 𝑦 ↔ 0 ∈ (ℝ ∖ 𝑥)))
6564notbid 307 . . . . . . . . . 10 (𝑦 = (ℝ ∖ 𝑥) → (¬ 0 ∈ 𝑦 ↔ ¬ 0 ∈ (ℝ ∖ 𝑥)))
66 imaeq2 5620 . . . . . . . . . . 11 (𝑦 = (ℝ ∖ 𝑥) → (𝐹𝑦) = (𝐹 “ (ℝ ∖ 𝑥)))
6766eleq1d 2824 . . . . . . . . . 10 (𝑦 = (ℝ ∖ 𝑥) → ((𝐹𝑦) ∈ dom vol ↔ (𝐹 “ (ℝ ∖ 𝑥)) ∈ dom vol))
6865, 67imbi12d 333 . . . . . . . . 9 (𝑦 = (ℝ ∖ 𝑥) → ((¬ 0 ∈ 𝑦 → (𝐹𝑦) ∈ dom vol) ↔ (¬ 0 ∈ (ℝ ∖ 𝑥) → (𝐹 “ (ℝ ∖ 𝑥)) ∈ dom vol)))
6963, 68spcv 3439 . . . . . . . 8 (∀𝑦(¬ 0 ∈ 𝑦 → (𝐹𝑦) ∈ dom vol) → (¬ 0 ∈ (ℝ ∖ 𝑥) → (𝐹 “ (ℝ ∖ 𝑥)) ∈ dom vol))
7058, 60, 69sylc 65 . . . . . . 7 (((𝜑𝑥 ∈ ran (,)) ∧ 0 ∈ 𝑥) → (𝐹 “ (ℝ ∖ 𝑥)) ∈ dom vol)
71 difmbl 23531 . . . . . . 7 (((𝐹 “ ℝ) ∈ dom vol ∧ (𝐹 “ (ℝ ∖ 𝑥)) ∈ dom vol) → ((𝐹 “ ℝ) ∖ (𝐹 “ (ℝ ∖ 𝑥))) ∈ dom vol)
7220, 70, 71syl2anc 696 . . . . . 6 (((𝜑𝑥 ∈ ran (,)) ∧ 0 ∈ 𝑥) → ((𝐹 “ ℝ) ∖ (𝐹 “ (ℝ ∖ 𝑥))) ∈ dom vol)
7316, 72eqeltrrd 2840 . . . . 5 (((𝜑𝑥 ∈ ran (,)) ∧ 0 ∈ 𝑥) → (𝐹𝑥) ∈ dom vol)
74 eleq2 2828 . . . . . . . . . . 11 (𝑦 = 𝑥 → (0 ∈ 𝑦 ↔ 0 ∈ 𝑥))
7574notbid 307 . . . . . . . . . 10 (𝑦 = 𝑥 → (¬ 0 ∈ 𝑦 ↔ ¬ 0 ∈ 𝑥))
76 imaeq2 5620 . . . . . . . . . . 11 (𝑦 = 𝑥 → (𝐹𝑦) = (𝐹𝑥))
7776eleq1d 2824 . . . . . . . . . 10 (𝑦 = 𝑥 → ((𝐹𝑦) ∈ dom vol ↔ (𝐹𝑥) ∈ dom vol))
7875, 77imbi12d 333 . . . . . . . . 9 (𝑦 = 𝑥 → ((¬ 0 ∈ 𝑦 → (𝐹𝑦) ∈ dom vol) ↔ (¬ 0 ∈ 𝑥 → (𝐹𝑥) ∈ dom vol)))
7978spv 2405 . . . . . . . 8 (∀𝑦(¬ 0 ∈ 𝑦 → (𝐹𝑦) ∈ dom vol) → (¬ 0 ∈ 𝑥 → (𝐹𝑥) ∈ dom vol))
8057, 79syl 17 . . . . . . 7 (𝜑 → (¬ 0 ∈ 𝑥 → (𝐹𝑥) ∈ dom vol))
8180imp 444 . . . . . 6 ((𝜑 ∧ ¬ 0 ∈ 𝑥) → (𝐹𝑥) ∈ dom vol)
8281adantlr 753 . . . . 5 (((𝜑𝑥 ∈ ran (,)) ∧ ¬ 0 ∈ 𝑥) → (𝐹𝑥) ∈ dom vol)
8373, 82pm2.61dan 867 . . . 4 ((𝜑𝑥 ∈ ran (,)) → (𝐹𝑥) ∈ dom vol)
8483ralrimiva 3104 . . 3 (𝜑 → ∀𝑥 ∈ ran (,)(𝐹𝑥) ∈ dom vol)
85 ismbf 23616 . . . 4 (𝐹:ℝ⟶ℝ → (𝐹 ∈ MblFn ↔ ∀𝑥 ∈ ran (,)(𝐹𝑥) ∈ dom vol))
861, 85syl 17 . . 3 (𝜑 → (𝐹 ∈ MblFn ↔ ∀𝑥 ∈ ran (,)(𝐹𝑥) ∈ dom vol))
8784, 86mpbird 247 . 2 (𝜑𝐹 ∈ MblFn)
88 mblvol 23518 . . . . . . . 8 ((𝐹𝑦) ∈ dom vol → (vol‘(𝐹𝑦)) = (vol*‘(𝐹𝑦)))
8955, 88syl 17 . . . . . . 7 ((𝜑 ∧ ¬ 0 ∈ 𝑦) → (vol‘(𝐹𝑦)) = (vol*‘(𝐹𝑦)))
90 mblss 23519 . . . . . . . . 9 ((𝐹𝑦) ∈ dom vol → (𝐹𝑦) ⊆ ℝ)
9155, 90syl 17 . . . . . . . 8 ((𝜑 ∧ ¬ 0 ∈ 𝑦) → (𝐹𝑦) ⊆ ℝ)
92 mblvol 23518 . . . . . . . . . . 11 ((𝐹 “ {𝑥}) ∈ dom vol → (vol‘(𝐹 “ {𝑥})) = (vol*‘(𝐹 “ {𝑥})))
9351, 92syl 17 . . . . . . . . . 10 (((𝜑 ∧ ¬ 0 ∈ 𝑦) ∧ 𝑥 ∈ (𝑦 ∩ ran 𝐹)) → (vol‘(𝐹 “ {𝑥})) = (vol*‘(𝐹 “ {𝑥})))
94 i1fd.4 . . . . . . . . . . 11 ((𝜑𝑥 ∈ (ran 𝐹 ∖ {0})) → (vol‘(𝐹 “ {𝑥})) ∈ ℝ)
9539, 49, 94syl2anc 696 . . . . . . . . . 10 (((𝜑 ∧ ¬ 0 ∈ 𝑦) ∧ 𝑥 ∈ (𝑦 ∩ ran 𝐹)) → (vol‘(𝐹 “ {𝑥})) ∈ ℝ)
9693, 95eqeltrrd 2840 . . . . . . . . 9 (((𝜑 ∧ ¬ 0 ∈ 𝑦) ∧ 𝑥 ∈ (𝑦 ∩ ran 𝐹)) → (vol*‘(𝐹 “ {𝑥})) ∈ ℝ)
9738, 96fsumrecl 14684 . . . . . . . 8 ((𝜑 ∧ ¬ 0 ∈ 𝑦) → Σ𝑥 ∈ (𝑦 ∩ ran 𝐹)(vol*‘(𝐹 “ {𝑥})) ∈ ℝ)
9833fveq2d 6357 . . . . . . . . 9 ((𝜑 ∧ ¬ 0 ∈ 𝑦) → (vol*‘ 𝑥 ∈ (𝑦 ∩ ran 𝐹)(𝐹 “ {𝑥})) = (vol*‘(𝐹𝑦)))
99 mblss 23519 . . . . . . . . . . . . 13 ((𝐹 “ {𝑥}) ∈ dom vol → (𝐹 “ {𝑥}) ⊆ ℝ)
10051, 99syl 17 . . . . . . . . . . . 12 (((𝜑 ∧ ¬ 0 ∈ 𝑦) ∧ 𝑥 ∈ (𝑦 ∩ ran 𝐹)) → (𝐹 “ {𝑥}) ⊆ ℝ)
101100, 96jca 555 . . . . . . . . . . 11 (((𝜑 ∧ ¬ 0 ∈ 𝑦) ∧ 𝑥 ∈ (𝑦 ∩ ran 𝐹)) → ((𝐹 “ {𝑥}) ⊆ ℝ ∧ (vol*‘(𝐹 “ {𝑥})) ∈ ℝ))
102101ralrimiva 3104 . . . . . . . . . 10 ((𝜑 ∧ ¬ 0 ∈ 𝑦) → ∀𝑥 ∈ (𝑦 ∩ ran 𝐹)((𝐹 “ {𝑥}) ⊆ ℝ ∧ (vol*‘(𝐹 “ {𝑥})) ∈ ℝ))
103 ovolfiniun 23489 . . . . . . . . . 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})))
10438, 102, 103syl2anc 696 . . . . . . . . 9 ((𝜑 ∧ ¬ 0 ∈ 𝑦) → (vol*‘ 𝑥 ∈ (𝑦 ∩ ran 𝐹)(𝐹 “ {𝑥})) ≤ Σ𝑥 ∈ (𝑦 ∩ ran 𝐹)(vol*‘(𝐹 “ {𝑥})))
10598, 104eqbrtrrd 4828 . . . . . . . 8 ((𝜑 ∧ ¬ 0 ∈ 𝑦) → (vol*‘(𝐹𝑦)) ≤ Σ𝑥 ∈ (𝑦 ∩ ran 𝐹)(vol*‘(𝐹 “ {𝑥})))
106 ovollecl 23471 . . . . . . . 8 (((𝐹𝑦) ⊆ ℝ ∧ Σ𝑥 ∈ (𝑦 ∩ ran 𝐹)(vol*‘(𝐹 “ {𝑥})) ∈ ℝ ∧ (vol*‘(𝐹𝑦)) ≤ Σ𝑥 ∈ (𝑦 ∩ ran 𝐹)(vol*‘(𝐹 “ {𝑥}))) → (vol*‘(𝐹𝑦)) ∈ ℝ)
10791, 97, 105, 106syl3anc 1477 . . . . . . 7 ((𝜑 ∧ ¬ 0 ∈ 𝑦) → (vol*‘(𝐹𝑦)) ∈ ℝ)
10889, 107eqeltrd 2839 . . . . . 6 ((𝜑 ∧ ¬ 0 ∈ 𝑦) → (vol‘(𝐹𝑦)) ∈ ℝ)
109108ex 449 . . . . 5 (𝜑 → (¬ 0 ∈ 𝑦 → (vol‘(𝐹𝑦)) ∈ ℝ))
110109alrimiv 2004 . . . 4 (𝜑 → ∀𝑦(¬ 0 ∈ 𝑦 → (vol‘(𝐹𝑦)) ∈ ℝ))
111 neldifsn 4467 . . . 4 ¬ 0 ∈ (ℝ ∖ {0})
112 difexg 4960 . . . . . 6 (ℝ ∈ V → (ℝ ∖ {0}) ∈ V)
11361, 112ax-mp 5 . . . . 5 (ℝ ∖ {0}) ∈ V
114 eleq2 2828 . . . . . . 7 (𝑦 = (ℝ ∖ {0}) → (0 ∈ 𝑦 ↔ 0 ∈ (ℝ ∖ {0})))
115114notbid 307 . . . . . 6 (𝑦 = (ℝ ∖ {0}) → (¬ 0 ∈ 𝑦 ↔ ¬ 0 ∈ (ℝ ∖ {0})))
116 imaeq2 5620 . . . . . . . 8 (𝑦 = (ℝ ∖ {0}) → (𝐹𝑦) = (𝐹 “ (ℝ ∖ {0})))
117116fveq2d 6357 . . . . . . 7 (𝑦 = (ℝ ∖ {0}) → (vol‘(𝐹𝑦)) = (vol‘(𝐹 “ (ℝ ∖ {0}))))
118117eleq1d 2824 . . . . . 6 (𝑦 = (ℝ ∖ {0}) → ((vol‘(𝐹𝑦)) ∈ ℝ ↔ (vol‘(𝐹 “ (ℝ ∖ {0}))) ∈ ℝ))
119115, 118imbi12d 333 . . . . 5 (𝑦 = (ℝ ∖ {0}) → ((¬ 0 ∈ 𝑦 → (vol‘(𝐹𝑦)) ∈ ℝ) ↔ (¬ 0 ∈ (ℝ ∖ {0}) → (vol‘(𝐹 “ (ℝ ∖ {0}))) ∈ ℝ)))
120113, 119spcv 3439 . . . 4 (∀𝑦(¬ 0 ∈ 𝑦 → (vol‘(𝐹𝑦)) ∈ ℝ) → (¬ 0 ∈ (ℝ ∖ {0}) → (vol‘(𝐹 “ (ℝ ∖ {0}))) ∈ ℝ))
121110, 111, 120mpisyl 21 . . 3 (𝜑 → (vol‘(𝐹 “ (ℝ ∖ {0}))) ∈ ℝ)
1221, 34, 1213jca 1123 . 2 (𝜑 → (𝐹:ℝ⟶ℝ ∧ ran 𝐹 ∈ Fin ∧ (vol‘(𝐹 “ (ℝ ∖ {0}))) ∈ ℝ))
123 isi1f 23660 . 2 (𝐹 ∈ dom ∫1 ↔ (𝐹 ∈ MblFn ∧ (𝐹:ℝ⟶ℝ ∧ ran 𝐹 ∈ Fin ∧ (vol‘(𝐹 “ (ℝ ∖ {0}))) ∈ ℝ)))
12487, 122, 123sylanbrc 701 1 (𝜑𝐹 ∈ dom ∫1)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 196   ∧ wa 383   ∧ w3a 1072  ∀wal 1630   = wceq 1632   ∈ wcel 2139  ∀wral 3050  Vcvv 3340   ∖ cdif 3712   ∩ cin 3714   ⊆ wss 3715  ∅c0 4058  𝒫 cpw 4302  {csn 4321  ∪ ciun 4672   class class class wbr 4804   × cxp 5264  ◡ccnv 5265  dom cdm 5266  ran crn 5267   “ cima 5269  Fun wfun 6043  ⟶wf 6045  ‘cfv 6049  Fincfn 8123  ℝcr 10147  0cc0 10148  ℝ*cxr 10285   ≤ cle 10287  (,)cioo 12388  Σcsu 14635  vol*covol 23451  volcvol 23452  MblFncmbf 23602  ∫1citg1 23603 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7115  ax-inf2 8713  ax-cnex 10204  ax-resscn 10205  ax-1cn 10206  ax-icn 10207  ax-addcl 10208  ax-addrcl 10209  ax-mulcl 10210  ax-mulrcl 10211  ax-mulcom 10212  ax-addass 10213  ax-mulass 10214  ax-distr 10215  ax-i2m1 10216  ax-1ne0 10217  ax-1rid 10218  ax-rnegex 10219  ax-rrecex 10220  ax-cnre 10221  ax-pre-lttri 10222  ax-pre-lttrn 10223  ax-pre-ltadd 10224  ax-pre-mulgt0 10225  ax-pre-sup 10226 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1635  df-fal 1638  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-nel 3036  df-ral 3055  df-rex 3056  df-reu 3057  df-rmo 3058  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-pss 3731  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-tp 4326  df-op 4328  df-uni 4589  df-int 4628  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-tr 4905  df-id 5174  df-eprel 5179  df-po 5187  df-so 5188  df-fr 5225  df-se 5226  df-we 5227  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-pred 5841  df-ord 5887  df-on 5888  df-lim 5889  df-suc 5890  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-isom 6058  df-riota 6775  df-ov 6817  df-oprab 6818  df-mpt2 6819  df-of 7063  df-om 7232  df-1st 7334  df-2nd 7335  df-wrecs 7577  df-recs 7638  df-rdg 7676  df-1o 7730  df-2o 7731  df-oadd 7734  df-er 7913  df-map 8027  df-pm 8028  df-en 8124  df-dom 8125  df-sdom 8126  df-fin 8127  df-sup 8515  df-inf 8516  df-oi 8582  df-card 8975  df-cda 9202  df-pnf 10288  df-mnf 10289  df-xr 10290  df-ltxr 10291  df-le 10292  df-sub 10480  df-neg 10481  df-div 10897  df-nn 11233  df-2 11291  df-3 11292  df-n0 11505  df-z 11590  df-uz 11900  df-q 12002  df-rp 12046  df-xadd 12160  df-ioo 12392  df-ico 12394  df-icc 12395  df-fz 12540  df-fzo 12680  df-fl 12807  df-seq 13016  df-exp 13075  df-hash 13332  df-cj 14058  df-re 14059  df-im 14060  df-sqrt 14194  df-abs 14195  df-clim 14438  df-sum 14636  df-xmet 19961  df-met 19962  df-ovol 23453  df-vol 23454  df-mbf 23607  df-itg1 23608 This theorem is referenced by:  i1f0  23673  i1f1  23676  i1fadd  23681  i1fmul  23682  i1fmulc  23689  i1fres  23691  mbfi1fseqlem4  23704  itg2addnclem2  33793  ftc1anclem3  33818
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