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Mirrors > Home > MPE Home > Th. List > i1f1lem | Structured version Visualization version GIF version |
Description: Lemma for i1f1 24293 and itg11 24294. (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 10639 | . . . . . 6 ⊢ 1 ∈ V | |
2 | 1 | prid2 4701 | . . . . 5 ⊢ 1 ∈ {0, 1} |
3 | c0ex 10637 | . . . . . 6 ⊢ 0 ∈ V | |
4 | 3 | prid1 4700 | . . . . 5 ⊢ 0 ∈ {0, 1} |
5 | 2, 4 | ifcli 4515 | . . . 4 ⊢ if(𝑥 ∈ 𝐴, 1, 0) ∈ {0, 1} |
6 | 5 | rgenw 3152 | . . 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 6876 | . . 3 ⊢ (∀𝑥 ∈ ℝ if(𝑥 ∈ 𝐴, 1, 0) ∈ {0, 1} ↔ 𝐹:ℝ⟶{0, 1}) |
9 | 6, 8 | mpbi 232 | . 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 6880 | . . . . . 6 ⊢ (𝐴 ∈ dom vol → 𝐹:ℝ⟶{0, 1}) |
12 | ffn 6516 | . . . . . 6 ⊢ (𝐹:ℝ⟶{0, 1} → 𝐹 Fn ℝ) | |
13 | elpreima 6830 | . . . . . 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 6685 | . . . . . . . 8 ⊢ (𝐹‘𝑦) ∈ V | |
16 | 15 | elsn 4584 | . . . . . . 7 ⊢ ((𝐹‘𝑦) ∈ {1} ↔ (𝐹‘𝑦) = 1) |
17 | eleq1w 2897 | . . . . . . . . . . 11 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ 𝐴 ↔ 𝑦 ∈ 𝐴)) | |
18 | 17 | ifbid 4491 | . . . . . . . . . 10 ⊢ (𝑥 = 𝑦 → if(𝑥 ∈ 𝐴, 1, 0) = if(𝑦 ∈ 𝐴, 1, 0)) |
19 | 1, 3 | ifex 4517 | . . . . . . . . . 10 ⊢ if(𝑦 ∈ 𝐴, 1, 0) ∈ V |
20 | 18, 7, 19 | fvmpt 6770 | . . . . . . . . 9 ⊢ (𝑦 ∈ ℝ → (𝐹‘𝑦) = if(𝑦 ∈ 𝐴, 1, 0)) |
21 | 20 | eqeq1d 2825 | . . . . . . . 8 ⊢ (𝑦 ∈ ℝ → ((𝐹‘𝑦) = 1 ↔ if(𝑦 ∈ 𝐴, 1, 0) = 1)) |
22 | 0ne1 11711 | . . . . . . . . . . 11 ⊢ 0 ≠ 1 | |
23 | iffalse 4478 | . . . . . . . . . . . . 13 ⊢ (¬ 𝑦 ∈ 𝐴 → if(𝑦 ∈ 𝐴, 1, 0) = 0) | |
24 | 23 | eqeq1d 2825 | . . . . . . . . . . . 12 ⊢ (¬ 𝑦 ∈ 𝐴 → (if(𝑦 ∈ 𝐴, 1, 0) = 1 ↔ 0 = 1)) |
25 | 24 | necon3bbid 3055 | . . . . . . . . . . 11 ⊢ (¬ 𝑦 ∈ 𝐴 → (¬ if(𝑦 ∈ 𝐴, 1, 0) = 1 ↔ 0 ≠ 1)) |
26 | 22, 25 | mpbiri 260 | . . . . . . . . . 10 ⊢ (¬ 𝑦 ∈ 𝐴 → ¬ if(𝑦 ∈ 𝐴, 1, 0) = 1) |
27 | 26 | con4i 114 | . . . . . . . . 9 ⊢ (if(𝑦 ∈ 𝐴, 1, 0) = 1 → 𝑦 ∈ 𝐴) |
28 | iftrue 4475 | . . . . . . . . 9 ⊢ (𝑦 ∈ 𝐴 → if(𝑦 ∈ 𝐴, 1, 0) = 1) | |
29 | 27, 28 | impbii 211 | . . . . . . . 8 ⊢ (if(𝑦 ∈ 𝐴, 1, 0) = 1 ↔ 𝑦 ∈ 𝐴) |
30 | 21, 29 | syl6bb 289 | . . . . . . 7 ⊢ (𝑦 ∈ ℝ → ((𝐹‘𝑦) = 1 ↔ 𝑦 ∈ 𝐴)) |
31 | 16, 30 | syl5bb 285 | . . . . . 6 ⊢ (𝑦 ∈ ℝ → ((𝐹‘𝑦) ∈ {1} ↔ 𝑦 ∈ 𝐴)) |
32 | 31 | pm5.32i 577 | . . . . 5 ⊢ ((𝑦 ∈ ℝ ∧ (𝐹‘𝑦) ∈ {1}) ↔ (𝑦 ∈ ℝ ∧ 𝑦 ∈ 𝐴)) |
33 | 14, 32 | syl6bb 289 | . . . 4 ⊢ (𝐴 ∈ dom vol → (𝑦 ∈ (◡𝐹 “ {1}) ↔ (𝑦 ∈ ℝ ∧ 𝑦 ∈ 𝐴))) |
34 | mblss 24134 | . . . . . 6 ⊢ (𝐴 ∈ dom vol → 𝐴 ⊆ ℝ) | |
35 | 34 | sseld 3968 | . . . . 5 ⊢ (𝐴 ∈ dom vol → (𝑦 ∈ 𝐴 → 𝑦 ∈ ℝ)) |
36 | 35 | pm4.71rd 565 | . . . 4 ⊢ (𝐴 ∈ dom vol → (𝑦 ∈ 𝐴 ↔ (𝑦 ∈ ℝ ∧ 𝑦 ∈ 𝐴))) |
37 | 33, 36 | bitr4d 284 | . . 3 ⊢ (𝐴 ∈ dom vol → (𝑦 ∈ (◡𝐹 “ {1}) ↔ 𝑦 ∈ 𝐴)) |
38 | 37 | eqrdv 2821 | . 2 ⊢ (𝐴 ∈ dom vol → (◡𝐹 “ {1}) = 𝐴) |
39 | 9, 38 | pm3.2i 473 | 1 ⊢ (𝐹:ℝ⟶{0, 1} ∧ (𝐴 ∈ dom vol → (◡𝐹 “ {1}) = 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1537 ∈ wcel 2114 ≠ wne 3018 ∀wral 3140 ifcif 4469 {csn 4569 {cpr 4571 ↦ cmpt 5148 ◡ccnv 5556 dom cdm 5557 “ cima 5560 Fn wfn 6352 ⟶wf 6353 ‘cfv 6357 ℝcr 10538 0cc0 10539 1c1 10540 volcvol 24066 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-cnex 10595 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615 ax-pre-mulgt0 10616 ax-pre-sup 10617 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rmo 3148 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-om 7583 df-1st 7691 df-2nd 7692 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-er 8291 df-map 8410 df-en 8512 df-dom 8513 df-sdom 8514 df-sup 8908 df-inf 8909 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-sub 10874 df-neg 10875 df-div 11300 df-nn 11641 df-2 11703 df-3 11704 df-n0 11901 df-z 11985 df-uz 12247 df-rp 12393 df-ico 12747 df-icc 12748 df-fz 12896 df-seq 13373 df-exp 13433 df-cj 14460 df-re 14461 df-im 14462 df-sqrt 14596 df-abs 14597 df-ovol 24067 df-vol 24068 |
This theorem is referenced by: i1f1 24293 itg11 24294 |
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