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Mirrors > Home > MPE Home > Th. List > i1f1lem | Structured version Visualization version GIF version |
Description: Lemma for i1f1 25739 and itg11 25740. (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 11255 | . . . . . 6 ⊢ 1 ∈ V | |
2 | 1 | prid2 4768 | . . . . 5 ⊢ 1 ∈ {0, 1} |
3 | c0ex 11253 | . . . . . 6 ⊢ 0 ∈ V | |
4 | 3 | prid1 4767 | . . . . 5 ⊢ 0 ∈ {0, 1} |
5 | 2, 4 | ifcli 4578 | . . . 4 ⊢ if(𝑥 ∈ 𝐴, 1, 0) ∈ {0, 1} |
6 | 5 | rgenw 3063 | . . 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 7130 | . . 3 ⊢ (∀𝑥 ∈ ℝ if(𝑥 ∈ 𝐴, 1, 0) ∈ {0, 1} ↔ 𝐹:ℝ⟶{0, 1}) |
9 | 6, 8 | mpbi 230 | . 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 7134 | . . . . . 6 ⊢ (𝐴 ∈ dom vol → 𝐹:ℝ⟶{0, 1}) |
12 | ffn 6737 | . . . . . 6 ⊢ (𝐹:ℝ⟶{0, 1} → 𝐹 Fn ℝ) | |
13 | elpreima 7078 | . . . . . 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 6920 | . . . . . . . 8 ⊢ (𝐹‘𝑦) ∈ V | |
16 | 15 | elsn 4646 | . . . . . . 7 ⊢ ((𝐹‘𝑦) ∈ {1} ↔ (𝐹‘𝑦) = 1) |
17 | eleq1w 2822 | . . . . . . . . . . 11 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ 𝐴 ↔ 𝑦 ∈ 𝐴)) | |
18 | 17 | ifbid 4554 | . . . . . . . . . 10 ⊢ (𝑥 = 𝑦 → if(𝑥 ∈ 𝐴, 1, 0) = if(𝑦 ∈ 𝐴, 1, 0)) |
19 | 1, 3 | ifex 4581 | . . . . . . . . . 10 ⊢ if(𝑦 ∈ 𝐴, 1, 0) ∈ V |
20 | 18, 7, 19 | fvmpt 7016 | . . . . . . . . 9 ⊢ (𝑦 ∈ ℝ → (𝐹‘𝑦) = if(𝑦 ∈ 𝐴, 1, 0)) |
21 | 20 | eqeq1d 2737 | . . . . . . . 8 ⊢ (𝑦 ∈ ℝ → ((𝐹‘𝑦) = 1 ↔ if(𝑦 ∈ 𝐴, 1, 0) = 1)) |
22 | 0ne1 12335 | . . . . . . . . . . 11 ⊢ 0 ≠ 1 | |
23 | iffalse 4540 | . . . . . . . . . . . . 13 ⊢ (¬ 𝑦 ∈ 𝐴 → if(𝑦 ∈ 𝐴, 1, 0) = 0) | |
24 | 23 | eqeq1d 2737 | . . . . . . . . . . . 12 ⊢ (¬ 𝑦 ∈ 𝐴 → (if(𝑦 ∈ 𝐴, 1, 0) = 1 ↔ 0 = 1)) |
25 | 24 | necon3bbid 2976 | . . . . . . . . . . 11 ⊢ (¬ 𝑦 ∈ 𝐴 → (¬ if(𝑦 ∈ 𝐴, 1, 0) = 1 ↔ 0 ≠ 1)) |
26 | 22, 25 | mpbiri 258 | . . . . . . . . . 10 ⊢ (¬ 𝑦 ∈ 𝐴 → ¬ if(𝑦 ∈ 𝐴, 1, 0) = 1) |
27 | 26 | con4i 114 | . . . . . . . . 9 ⊢ (if(𝑦 ∈ 𝐴, 1, 0) = 1 → 𝑦 ∈ 𝐴) |
28 | iftrue 4537 | . . . . . . . . 9 ⊢ (𝑦 ∈ 𝐴 → if(𝑦 ∈ 𝐴, 1, 0) = 1) | |
29 | 27, 28 | impbii 209 | . . . . . . . 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 283 | . . . . . 6 ⊢ (𝑦 ∈ ℝ → ((𝐹‘𝑦) ∈ {1} ↔ 𝑦 ∈ 𝐴)) |
32 | 31 | pm5.32i 574 | . . . . 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 25580 | . . . . . 6 ⊢ (𝐴 ∈ dom vol → 𝐴 ⊆ ℝ) | |
35 | 34 | sseld 3994 | . . . . 5 ⊢ (𝐴 ∈ dom vol → (𝑦 ∈ 𝐴 → 𝑦 ∈ ℝ)) |
36 | 35 | pm4.71rd 562 | . . . 4 ⊢ (𝐴 ∈ dom vol → (𝑦 ∈ 𝐴 ↔ (𝑦 ∈ ℝ ∧ 𝑦 ∈ 𝐴))) |
37 | 33, 36 | bitr4d 282 | . . 3 ⊢ (𝐴 ∈ dom vol → (𝑦 ∈ (◡𝐹 “ {1}) ↔ 𝑦 ∈ 𝐴)) |
38 | 37 | eqrdv 2733 | . 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 206 ∧ wa 395 = wceq 1537 ∈ wcel 2106 ≠ wne 2938 ∀wral 3059 ifcif 4531 {csn 4631 {cpr 4633 ↦ cmpt 5231 ◡ccnv 5688 dom cdm 5689 “ cima 5692 Fn wfn 6558 ⟶wf 6559 ‘cfv 6563 ℝcr 11152 0cc0 11153 1c1 11154 volcvol 25512 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230 ax-pre-sup 11231 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8013 df-2nd 8014 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-er 8744 df-map 8867 df-en 8985 df-dom 8986 df-sdom 8987 df-sup 9480 df-inf 9481 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-div 11919 df-nn 12265 df-2 12327 df-3 12328 df-n0 12525 df-z 12612 df-uz 12877 df-rp 13033 df-ico 13390 df-icc 13391 df-fz 13545 df-seq 14040 df-exp 14100 df-cj 15135 df-re 15136 df-im 15137 df-sqrt 15271 df-abs 15272 df-ovol 25513 df-vol 25514 |
This theorem is referenced by: i1f1 25739 itg11 25740 |
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