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Mirrors > Home > MPE Home > Th. List > i1f1lem | Structured version Visualization version GIF version |
Description: Lemma for i1f1 24442 and itg11 24443. (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 10715 | . . . . . 6 ⊢ 1 ∈ V | |
2 | 1 | prid2 4654 | . . . . 5 ⊢ 1 ∈ {0, 1} |
3 | c0ex 10713 | . . . . . 6 ⊢ 0 ∈ V | |
4 | 3 | prid1 4653 | . . . . 5 ⊢ 0 ∈ {0, 1} |
5 | 2, 4 | ifcli 4461 | . . . 4 ⊢ if(𝑥 ∈ 𝐴, 1, 0) ∈ {0, 1} |
6 | 5 | rgenw 3065 | . . 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 6884 | . . 3 ⊢ (∀𝑥 ∈ ℝ if(𝑥 ∈ 𝐴, 1, 0) ∈ {0, 1} ↔ 𝐹:ℝ⟶{0, 1}) |
9 | 6, 8 | mpbi 233 | . 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 6888 | . . . . . 6 ⊢ (𝐴 ∈ dom vol → 𝐹:ℝ⟶{0, 1}) |
12 | ffn 6504 | . . . . . 6 ⊢ (𝐹:ℝ⟶{0, 1} → 𝐹 Fn ℝ) | |
13 | elpreima 6835 | . . . . . 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 6687 | . . . . . . . 8 ⊢ (𝐹‘𝑦) ∈ V | |
16 | 15 | elsn 4531 | . . . . . . 7 ⊢ ((𝐹‘𝑦) ∈ {1} ↔ (𝐹‘𝑦) = 1) |
17 | eleq1w 2815 | . . . . . . . . . . 11 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ 𝐴 ↔ 𝑦 ∈ 𝐴)) | |
18 | 17 | ifbid 4437 | . . . . . . . . . 10 ⊢ (𝑥 = 𝑦 → if(𝑥 ∈ 𝐴, 1, 0) = if(𝑦 ∈ 𝐴, 1, 0)) |
19 | 1, 3 | ifex 4464 | . . . . . . . . . 10 ⊢ if(𝑦 ∈ 𝐴, 1, 0) ∈ V |
20 | 18, 7, 19 | fvmpt 6775 | . . . . . . . . 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 11787 | . . . . . . . . . . 11 ⊢ 0 ≠ 1 | |
23 | iffalse 4423 | . . . . . . . . . . . . 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 2971 | . . . . . . . . . . 11 ⊢ (¬ 𝑦 ∈ 𝐴 → (¬ if(𝑦 ∈ 𝐴, 1, 0) = 1 ↔ 0 ≠ 1)) |
26 | 22, 25 | mpbiri 261 | . . . . . . . . . 10 ⊢ (¬ 𝑦 ∈ 𝐴 → ¬ if(𝑦 ∈ 𝐴, 1, 0) = 1) |
27 | 26 | con4i 114 | . . . . . . . . 9 ⊢ (if(𝑦 ∈ 𝐴, 1, 0) = 1 → 𝑦 ∈ 𝐴) |
28 | iftrue 4420 | . . . . . . . . 9 ⊢ (𝑦 ∈ 𝐴 → if(𝑦 ∈ 𝐴, 1, 0) = 1) | |
29 | 27, 28 | impbii 212 | . . . . . . . 8 ⊢ (if(𝑦 ∈ 𝐴, 1, 0) = 1 ↔ 𝑦 ∈ 𝐴) |
30 | 21, 29 | bitrdi 290 | . . . . . . 7 ⊢ (𝑦 ∈ ℝ → ((𝐹‘𝑦) = 1 ↔ 𝑦 ∈ 𝐴)) |
31 | 16, 30 | syl5bb 286 | . . . . . 6 ⊢ (𝑦 ∈ ℝ → ((𝐹‘𝑦) ∈ {1} ↔ 𝑦 ∈ 𝐴)) |
32 | 31 | pm5.32i 578 | . . . . 5 ⊢ ((𝑦 ∈ ℝ ∧ (𝐹‘𝑦) ∈ {1}) ↔ (𝑦 ∈ ℝ ∧ 𝑦 ∈ 𝐴)) |
33 | 14, 32 | bitrdi 290 | . . . 4 ⊢ (𝐴 ∈ dom vol → (𝑦 ∈ (◡𝐹 “ {1}) ↔ (𝑦 ∈ ℝ ∧ 𝑦 ∈ 𝐴))) |
34 | mblss 24283 | . . . . . 6 ⊢ (𝐴 ∈ dom vol → 𝐴 ⊆ ℝ) | |
35 | 34 | sseld 3876 | . . . . 5 ⊢ (𝐴 ∈ dom vol → (𝑦 ∈ 𝐴 → 𝑦 ∈ ℝ)) |
36 | 35 | pm4.71rd 566 | . . . 4 ⊢ (𝐴 ∈ dom vol → (𝑦 ∈ 𝐴 ↔ (𝑦 ∈ ℝ ∧ 𝑦 ∈ 𝐴))) |
37 | 33, 36 | bitr4d 285 | . . 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 474 | 1 ⊢ (𝐹:ℝ⟶{0, 1} ∧ (𝐴 ∈ dom vol → (◡𝐹 “ {1}) = 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 209 ∧ wa 399 = wceq 1542 ∈ wcel 2114 ≠ wne 2934 ∀wral 3053 ifcif 4414 {csn 4516 {cpr 4518 ↦ cmpt 5110 ◡ccnv 5524 dom cdm 5525 “ cima 5528 Fn wfn 6334 ⟶wf 6335 ‘cfv 6339 ℝcr 10614 0cc0 10615 1c1 10616 volcvol 24215 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-ext 2710 ax-sep 5167 ax-nul 5174 ax-pow 5232 ax-pr 5296 ax-un 7479 ax-cnex 10671 ax-resscn 10672 ax-1cn 10673 ax-icn 10674 ax-addcl 10675 ax-addrcl 10676 ax-mulcl 10677 ax-mulrcl 10678 ax-mulcom 10679 ax-addass 10680 ax-mulass 10681 ax-distr 10682 ax-i2m1 10683 ax-1ne0 10684 ax-1rid 10685 ax-rnegex 10686 ax-rrecex 10687 ax-cnre 10688 ax-pre-lttri 10689 ax-pre-lttrn 10690 ax-pre-ltadd 10691 ax-pre-mulgt0 10692 ax-pre-sup 10693 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2075 df-mo 2540 df-eu 2570 df-clab 2717 df-cleq 2730 df-clel 2811 df-nfc 2881 df-ne 2935 df-nel 3039 df-ral 3058 df-rex 3059 df-reu 3060 df-rmo 3061 df-rab 3062 df-v 3400 df-sbc 3681 df-csb 3791 df-dif 3846 df-un 3848 df-in 3850 df-ss 3860 df-pss 3862 df-nul 4212 df-if 4415 df-pw 4490 df-sn 4517 df-pr 4519 df-tp 4521 df-op 4523 df-uni 4797 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5429 df-eprel 5434 df-po 5442 df-so 5443 df-fr 5483 df-we 5485 df-xp 5531 df-rel 5532 df-cnv 5533 df-co 5534 df-dm 5535 df-rn 5536 df-res 5537 df-ima 5538 df-pred 6129 df-ord 6175 df-on 6176 df-lim 6177 df-suc 6178 df-iota 6297 df-fun 6341 df-fn 6342 df-f 6343 df-f1 6344 df-fo 6345 df-f1o 6346 df-fv 6347 df-riota 7127 df-ov 7173 df-oprab 7174 df-mpo 7175 df-om 7600 df-1st 7714 df-2nd 7715 df-wrecs 7976 df-recs 8037 df-rdg 8075 df-er 8320 df-map 8439 df-en 8556 df-dom 8557 df-sdom 8558 df-sup 8979 df-inf 8980 df-pnf 10755 df-mnf 10756 df-xr 10757 df-ltxr 10758 df-le 10759 df-sub 10950 df-neg 10951 df-div 11376 df-nn 11717 df-2 11779 df-3 11780 df-n0 11977 df-z 12063 df-uz 12325 df-rp 12473 df-ico 12827 df-icc 12828 df-fz 12982 df-seq 13461 df-exp 13522 df-cj 14548 df-re 14549 df-im 14550 df-sqrt 14684 df-abs 14685 df-ovol 24216 df-vol 24217 |
This theorem is referenced by: i1f1 24442 itg11 24443 |
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