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Mirrors > Home > MPE Home > Th. List > i1f1lem | Structured version Visualization version GIF version |
Description: Lemma for i1f1 24854 and itg11 24855. (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 10971 | . . . . . 6 ⊢ 1 ∈ V | |
2 | 1 | prid2 4699 | . . . . 5 ⊢ 1 ∈ {0, 1} |
3 | c0ex 10969 | . . . . . 6 ⊢ 0 ∈ V | |
4 | 3 | prid1 4698 | . . . . 5 ⊢ 0 ∈ {0, 1} |
5 | 2, 4 | ifcli 4506 | . . . 4 ⊢ if(𝑥 ∈ 𝐴, 1, 0) ∈ {0, 1} |
6 | 5 | rgenw 3076 | . . 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 6984 | . . 3 ⊢ (∀𝑥 ∈ ℝ if(𝑥 ∈ 𝐴, 1, 0) ∈ {0, 1} ↔ 𝐹:ℝ⟶{0, 1}) |
9 | 6, 8 | mpbi 229 | . 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 6988 | . . . . . 6 ⊢ (𝐴 ∈ dom vol → 𝐹:ℝ⟶{0, 1}) |
12 | ffn 6600 | . . . . . 6 ⊢ (𝐹:ℝ⟶{0, 1} → 𝐹 Fn ℝ) | |
13 | elpreima 6935 | . . . . . 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 6787 | . . . . . . . 8 ⊢ (𝐹‘𝑦) ∈ V | |
16 | 15 | elsn 4576 | . . . . . . 7 ⊢ ((𝐹‘𝑦) ∈ {1} ↔ (𝐹‘𝑦) = 1) |
17 | eleq1w 2821 | . . . . . . . . . . 11 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ 𝐴 ↔ 𝑦 ∈ 𝐴)) | |
18 | 17 | ifbid 4482 | . . . . . . . . . 10 ⊢ (𝑥 = 𝑦 → if(𝑥 ∈ 𝐴, 1, 0) = if(𝑦 ∈ 𝐴, 1, 0)) |
19 | 1, 3 | ifex 4509 | . . . . . . . . . 10 ⊢ if(𝑦 ∈ 𝐴, 1, 0) ∈ V |
20 | 18, 7, 19 | fvmpt 6875 | . . . . . . . . 9 ⊢ (𝑦 ∈ ℝ → (𝐹‘𝑦) = if(𝑦 ∈ 𝐴, 1, 0)) |
21 | 20 | eqeq1d 2740 | . . . . . . . 8 ⊢ (𝑦 ∈ ℝ → ((𝐹‘𝑦) = 1 ↔ if(𝑦 ∈ 𝐴, 1, 0) = 1)) |
22 | 0ne1 12044 | . . . . . . . . . . 11 ⊢ 0 ≠ 1 | |
23 | iffalse 4468 | . . . . . . . . . . . . 13 ⊢ (¬ 𝑦 ∈ 𝐴 → if(𝑦 ∈ 𝐴, 1, 0) = 0) | |
24 | 23 | eqeq1d 2740 | . . . . . . . . . . . 12 ⊢ (¬ 𝑦 ∈ 𝐴 → (if(𝑦 ∈ 𝐴, 1, 0) = 1 ↔ 0 = 1)) |
25 | 24 | necon3bbid 2981 | . . . . . . . . . . 11 ⊢ (¬ 𝑦 ∈ 𝐴 → (¬ if(𝑦 ∈ 𝐴, 1, 0) = 1 ↔ 0 ≠ 1)) |
26 | 22, 25 | mpbiri 257 | . . . . . . . . . 10 ⊢ (¬ 𝑦 ∈ 𝐴 → ¬ if(𝑦 ∈ 𝐴, 1, 0) = 1) |
27 | 26 | con4i 114 | . . . . . . . . 9 ⊢ (if(𝑦 ∈ 𝐴, 1, 0) = 1 → 𝑦 ∈ 𝐴) |
28 | iftrue 4465 | . . . . . . . . 9 ⊢ (𝑦 ∈ 𝐴 → if(𝑦 ∈ 𝐴, 1, 0) = 1) | |
29 | 27, 28 | impbii 208 | . . . . . . . 8 ⊢ (if(𝑦 ∈ 𝐴, 1, 0) = 1 ↔ 𝑦 ∈ 𝐴) |
30 | 21, 29 | bitrdi 287 | . . . . . . 7 ⊢ (𝑦 ∈ ℝ → ((𝐹‘𝑦) = 1 ↔ 𝑦 ∈ 𝐴)) |
31 | 16, 30 | bitrid 282 | . . . . . 6 ⊢ (𝑦 ∈ ℝ → ((𝐹‘𝑦) ∈ {1} ↔ 𝑦 ∈ 𝐴)) |
32 | 31 | pm5.32i 575 | . . . . 5 ⊢ ((𝑦 ∈ ℝ ∧ (𝐹‘𝑦) ∈ {1}) ↔ (𝑦 ∈ ℝ ∧ 𝑦 ∈ 𝐴)) |
33 | 14, 32 | bitrdi 287 | . . . 4 ⊢ (𝐴 ∈ dom vol → (𝑦 ∈ (◡𝐹 “ {1}) ↔ (𝑦 ∈ ℝ ∧ 𝑦 ∈ 𝐴))) |
34 | mblss 24695 | . . . . . 6 ⊢ (𝐴 ∈ dom vol → 𝐴 ⊆ ℝ) | |
35 | 34 | sseld 3920 | . . . . 5 ⊢ (𝐴 ∈ dom vol → (𝑦 ∈ 𝐴 → 𝑦 ∈ ℝ)) |
36 | 35 | pm4.71rd 563 | . . . 4 ⊢ (𝐴 ∈ dom vol → (𝑦 ∈ 𝐴 ↔ (𝑦 ∈ ℝ ∧ 𝑦 ∈ 𝐴))) |
37 | 33, 36 | bitr4d 281 | . . 3 ⊢ (𝐴 ∈ dom vol → (𝑦 ∈ (◡𝐹 “ {1}) ↔ 𝑦 ∈ 𝐴)) |
38 | 37 | eqrdv 2736 | . 2 ⊢ (𝐴 ∈ dom vol → (◡𝐹 “ {1}) = 𝐴) |
39 | 9, 38 | pm3.2i 471 | 1 ⊢ (𝐹:ℝ⟶{0, 1} ∧ (𝐴 ∈ dom vol → (◡𝐹 “ {1}) = 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 396 = wceq 1539 ∈ wcel 2106 ≠ wne 2943 ∀wral 3064 ifcif 4459 {csn 4561 {cpr 4563 ↦ cmpt 5157 ◡ccnv 5588 dom cdm 5589 “ cima 5592 Fn wfn 6428 ⟶wf 6429 ‘cfv 6433 ℝcr 10870 0cc0 10871 1c1 10872 volcvol 24627 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-sep 5223 ax-nul 5230 ax-pow 5288 ax-pr 5352 ax-un 7588 ax-cnex 10927 ax-resscn 10928 ax-1cn 10929 ax-icn 10930 ax-addcl 10931 ax-addrcl 10932 ax-mulcl 10933 ax-mulrcl 10934 ax-mulcom 10935 ax-addass 10936 ax-mulass 10937 ax-distr 10938 ax-i2m1 10939 ax-1ne0 10940 ax-1rid 10941 ax-rnegex 10942 ax-rrecex 10943 ax-cnre 10944 ax-pre-lttri 10945 ax-pre-lttrn 10946 ax-pre-ltadd 10947 ax-pre-mulgt0 10948 ax-pre-sup 10949 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3069 df-rex 3070 df-rmo 3071 df-reu 3072 df-rab 3073 df-v 3434 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-pss 3906 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4840 df-iun 4926 df-br 5075 df-opab 5137 df-mpt 5158 df-tr 5192 df-id 5489 df-eprel 5495 df-po 5503 df-so 5504 df-fr 5544 df-we 5546 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-pred 6202 df-ord 6269 df-on 6270 df-lim 6271 df-suc 6272 df-iota 6391 df-fun 6435 df-fn 6436 df-f 6437 df-f1 6438 df-fo 6439 df-f1o 6440 df-fv 6441 df-riota 7232 df-ov 7278 df-oprab 7279 df-mpo 7280 df-om 7713 df-1st 7831 df-2nd 7832 df-frecs 8097 df-wrecs 8128 df-recs 8202 df-rdg 8241 df-er 8498 df-map 8617 df-en 8734 df-dom 8735 df-sdom 8736 df-sup 9201 df-inf 9202 df-pnf 11011 df-mnf 11012 df-xr 11013 df-ltxr 11014 df-le 11015 df-sub 11207 df-neg 11208 df-div 11633 df-nn 11974 df-2 12036 df-3 12037 df-n0 12234 df-z 12320 df-uz 12583 df-rp 12731 df-ico 13085 df-icc 13086 df-fz 13240 df-seq 13722 df-exp 13783 df-cj 14810 df-re 14811 df-im 14812 df-sqrt 14946 df-abs 14947 df-ovol 24628 df-vol 24629 |
This theorem is referenced by: i1f1 24854 itg11 24855 |
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