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Mirrors > Home > MPE Home > Th. List > i1f1lem | Structured version Visualization version GIF version |
Description: Lemma for i1f1 23263 and itg11 23264. (Contributed by Mario Carneiro, 18-Jun-2014.) |
Ref | Expression |
---|---|
i1f1.1 | ⊢ 𝐹 = (𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, 1, 0)) |
Ref | Expression |
---|---|
i1f1lem | ⊢ (𝐹:ℝ⟶{0, 1} ∧ (𝐴 ∈ dom vol → (◡𝐹 “ {1}) = 𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 1ex 9914 | . . . . . 6 ⊢ 1 ∈ V | |
2 | 1 | prid2 4242 | . . . . 5 ⊢ 1 ∈ {0, 1} |
3 | c0ex 9913 | . . . . . 6 ⊢ 0 ∈ V | |
4 | 3 | prid1 4241 | . . . . 5 ⊢ 0 ∈ {0, 1} |
5 | 2, 4 | keepel 4105 | . . . 4 ⊢ if(𝑥 ∈ 𝐴, 1, 0) ∈ {0, 1} |
6 | 5 | rgenw 2908 | . . 3 ⊢ ∀𝑥 ∈ ℝ if(𝑥 ∈ 𝐴, 1, 0) ∈ {0, 1} |
7 | i1f1.1 | . . . 4 ⊢ 𝐹 = (𝑥 ∈ ℝ ↦ if(𝑥 ∈ 𝐴, 1, 0)) | |
8 | 7 | fmpt 6289 | . . 3 ⊢ (∀𝑥 ∈ ℝ if(𝑥 ∈ 𝐴, 1, 0) ∈ {0, 1} ↔ 𝐹:ℝ⟶{0, 1}) |
9 | 6, 8 | mpbi 219 | . 2 ⊢ 𝐹:ℝ⟶{0, 1} |
10 | 5 | a1i 11 | . . . . . . 7 ⊢ ((𝐴 ∈ dom vol ∧ 𝑥 ∈ ℝ) → if(𝑥 ∈ 𝐴, 1, 0) ∈ {0, 1}) |
11 | 10, 7 | fmptd 6292 | . . . . . 6 ⊢ (𝐴 ∈ dom vol → 𝐹:ℝ⟶{0, 1}) |
12 | ffn 5958 | . . . . . 6 ⊢ (𝐹:ℝ⟶{0, 1} → 𝐹 Fn ℝ) | |
13 | elpreima 6245 | . . . . . 6 ⊢ (𝐹 Fn ℝ → (𝑦 ∈ (◡𝐹 “ {1}) ↔ (𝑦 ∈ ℝ ∧ (𝐹‘𝑦) ∈ {1}))) | |
14 | 11, 12, 13 | 3syl 18 | . . . . 5 ⊢ (𝐴 ∈ dom vol → (𝑦 ∈ (◡𝐹 “ {1}) ↔ (𝑦 ∈ ℝ ∧ (𝐹‘𝑦) ∈ {1}))) |
15 | fvex 6113 | . . . . . . . 8 ⊢ (𝐹‘𝑦) ∈ V | |
16 | 15 | elsn 4140 | . . . . . . 7 ⊢ ((𝐹‘𝑦) ∈ {1} ↔ (𝐹‘𝑦) = 1) |
17 | eleq1 2676 | . . . . . . . . . . 11 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ 𝐴 ↔ 𝑦 ∈ 𝐴)) | |
18 | 17 | ifbid 4058 | . . . . . . . . . 10 ⊢ (𝑥 = 𝑦 → if(𝑥 ∈ 𝐴, 1, 0) = if(𝑦 ∈ 𝐴, 1, 0)) |
19 | 1, 3 | ifex 4106 | . . . . . . . . . 10 ⊢ if(𝑦 ∈ 𝐴, 1, 0) ∈ V |
20 | 18, 7, 19 | fvmpt 6191 | . . . . . . . . 9 ⊢ (𝑦 ∈ ℝ → (𝐹‘𝑦) = if(𝑦 ∈ 𝐴, 1, 0)) |
21 | 20 | eqeq1d 2612 | . . . . . . . 8 ⊢ (𝑦 ∈ ℝ → ((𝐹‘𝑦) = 1 ↔ if(𝑦 ∈ 𝐴, 1, 0) = 1)) |
22 | 0ne1 10965 | . . . . . . . . . . 11 ⊢ 0 ≠ 1 | |
23 | iffalse 4045 | . . . . . . . . . . . . 13 ⊢ (¬ 𝑦 ∈ 𝐴 → if(𝑦 ∈ 𝐴, 1, 0) = 0) | |
24 | 23 | eqeq1d 2612 | . . . . . . . . . . . 12 ⊢ (¬ 𝑦 ∈ 𝐴 → (if(𝑦 ∈ 𝐴, 1, 0) = 1 ↔ 0 = 1)) |
25 | 24 | necon3bbid 2819 | . . . . . . . . . . 11 ⊢ (¬ 𝑦 ∈ 𝐴 → (¬ if(𝑦 ∈ 𝐴, 1, 0) = 1 ↔ 0 ≠ 1)) |
26 | 22, 25 | mpbiri 247 | . . . . . . . . . 10 ⊢ (¬ 𝑦 ∈ 𝐴 → ¬ if(𝑦 ∈ 𝐴, 1, 0) = 1) |
27 | 26 | con4i 112 | . . . . . . . . 9 ⊢ (if(𝑦 ∈ 𝐴, 1, 0) = 1 → 𝑦 ∈ 𝐴) |
28 | iftrue 4042 | . . . . . . . . 9 ⊢ (𝑦 ∈ 𝐴 → if(𝑦 ∈ 𝐴, 1, 0) = 1) | |
29 | 27, 28 | impbii 198 | . . . . . . . 8 ⊢ (if(𝑦 ∈ 𝐴, 1, 0) = 1 ↔ 𝑦 ∈ 𝐴) |
30 | 21, 29 | syl6bb 275 | . . . . . . 7 ⊢ (𝑦 ∈ ℝ → ((𝐹‘𝑦) = 1 ↔ 𝑦 ∈ 𝐴)) |
31 | 16, 30 | syl5bb 271 | . . . . . 6 ⊢ (𝑦 ∈ ℝ → ((𝐹‘𝑦) ∈ {1} ↔ 𝑦 ∈ 𝐴)) |
32 | 31 | pm5.32i 667 | . . . . 5 ⊢ ((𝑦 ∈ ℝ ∧ (𝐹‘𝑦) ∈ {1}) ↔ (𝑦 ∈ ℝ ∧ 𝑦 ∈ 𝐴)) |
33 | 14, 32 | syl6bb 275 | . . . 4 ⊢ (𝐴 ∈ dom vol → (𝑦 ∈ (◡𝐹 “ {1}) ↔ (𝑦 ∈ ℝ ∧ 𝑦 ∈ 𝐴))) |
34 | mblss 23106 | . . . . . 6 ⊢ (𝐴 ∈ dom vol → 𝐴 ⊆ ℝ) | |
35 | 34 | sseld 3567 | . . . . 5 ⊢ (𝐴 ∈ dom vol → (𝑦 ∈ 𝐴 → 𝑦 ∈ ℝ)) |
36 | 35 | pm4.71rd 665 | . . . 4 ⊢ (𝐴 ∈ dom vol → (𝑦 ∈ 𝐴 ↔ (𝑦 ∈ ℝ ∧ 𝑦 ∈ 𝐴))) |
37 | 33, 36 | bitr4d 270 | . . 3 ⊢ (𝐴 ∈ dom vol → (𝑦 ∈ (◡𝐹 “ {1}) ↔ 𝑦 ∈ 𝐴)) |
38 | 37 | eqrdv 2608 | . 2 ⊢ (𝐴 ∈ dom vol → (◡𝐹 “ {1}) = 𝐴) |
39 | 9, 38 | pm3.2i 470 | 1 ⊢ (𝐹:ℝ⟶{0, 1} ∧ (𝐴 ∈ dom vol → (◡𝐹 “ {1}) = 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 195 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ≠ wne 2780 ∀wral 2896 ifcif 4036 {csn 4125 {cpr 4127 ↦ cmpt 4643 ◡ccnv 5037 dom cdm 5038 “ cima 5041 Fn wfn 5799 ⟶wf 5800 ‘cfv 5804 ℝcr 9814 0cc0 9815 1c1 9816 volcvol 23039 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 ax-pre-sup 9893 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-1st 7059 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-er 7629 df-map 7746 df-en 7842 df-dom 7843 df-sdom 7844 df-sup 8231 df-inf 8232 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-div 10564 df-nn 10898 df-2 10956 df-3 10957 df-n0 11170 df-z 11255 df-uz 11564 df-rp 11709 df-ico 12052 df-icc 12053 df-fz 12198 df-seq 12664 df-exp 12723 df-cj 13687 df-re 13688 df-im 13689 df-sqrt 13823 df-abs 13824 df-ovol 23040 df-vol 23041 |
This theorem is referenced by: i1f1 23263 itg11 23264 |
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