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Mirrors > Home > MPE Home > Th. List > pmtrfval | Structured version Visualization version GIF version |
Description: The function generating transpositions on a set. (Contributed by Stefan O'Rear, 16-Aug-2015.) |
Ref | Expression |
---|---|
pmtrfval.t | ⊢ 𝑇 = (pmTrsp‘𝐷) |
Ref | Expression |
---|---|
pmtrfval | ⊢ (𝐷 ∈ 𝑉 → 𝑇 = (𝑝 ∈ {𝑦 ∈ 𝒫 𝐷 ∣ 𝑦 ≈ 2o} ↦ (𝑧 ∈ 𝐷 ↦ if(𝑧 ∈ 𝑝, ∪ (𝑝 ∖ {𝑧}), 𝑧)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pmtrfval.t | . 2 ⊢ 𝑇 = (pmTrsp‘𝐷) | |
2 | elex 3429 | . . 3 ⊢ (𝐷 ∈ 𝑉 → 𝐷 ∈ V) | |
3 | pweq 4381 | . . . . . 6 ⊢ (𝑑 = 𝐷 → 𝒫 𝑑 = 𝒫 𝐷) | |
4 | 3 | rabeqdv 3407 | . . . . 5 ⊢ (𝑑 = 𝐷 → {𝑦 ∈ 𝒫 𝑑 ∣ 𝑦 ≈ 2o} = {𝑦 ∈ 𝒫 𝐷 ∣ 𝑦 ≈ 2o}) |
5 | mpteq1 4960 | . . . . 5 ⊢ (𝑑 = 𝐷 → (𝑧 ∈ 𝑑 ↦ if(𝑧 ∈ 𝑝, ∪ (𝑝 ∖ {𝑧}), 𝑧)) = (𝑧 ∈ 𝐷 ↦ if(𝑧 ∈ 𝑝, ∪ (𝑝 ∖ {𝑧}), 𝑧))) | |
6 | 4, 5 | mpteq12dv 4956 | . . . 4 ⊢ (𝑑 = 𝐷 → (𝑝 ∈ {𝑦 ∈ 𝒫 𝑑 ∣ 𝑦 ≈ 2o} ↦ (𝑧 ∈ 𝑑 ↦ if(𝑧 ∈ 𝑝, ∪ (𝑝 ∖ {𝑧}), 𝑧))) = (𝑝 ∈ {𝑦 ∈ 𝒫 𝐷 ∣ 𝑦 ≈ 2o} ↦ (𝑧 ∈ 𝐷 ↦ if(𝑧 ∈ 𝑝, ∪ (𝑝 ∖ {𝑧}), 𝑧)))) |
7 | df-pmtr 18212 | . . . 4 ⊢ pmTrsp = (𝑑 ∈ V ↦ (𝑝 ∈ {𝑦 ∈ 𝒫 𝑑 ∣ 𝑦 ≈ 2o} ↦ (𝑧 ∈ 𝑑 ↦ if(𝑧 ∈ 𝑝, ∪ (𝑝 ∖ {𝑧}), 𝑧)))) | |
8 | vpwex 5077 | . . . . 5 ⊢ 𝒫 𝑑 ∈ V | |
9 | 8 | mptrabex 6744 | . . . 4 ⊢ (𝑝 ∈ {𝑦 ∈ 𝒫 𝑑 ∣ 𝑦 ≈ 2o} ↦ (𝑧 ∈ 𝑑 ↦ if(𝑧 ∈ 𝑝, ∪ (𝑝 ∖ {𝑧}), 𝑧))) ∈ V |
10 | 6, 7, 9 | fvmpt3i 6534 | . . 3 ⊢ (𝐷 ∈ V → (pmTrsp‘𝐷) = (𝑝 ∈ {𝑦 ∈ 𝒫 𝐷 ∣ 𝑦 ≈ 2o} ↦ (𝑧 ∈ 𝐷 ↦ if(𝑧 ∈ 𝑝, ∪ (𝑝 ∖ {𝑧}), 𝑧)))) |
11 | 2, 10 | syl 17 | . 2 ⊢ (𝐷 ∈ 𝑉 → (pmTrsp‘𝐷) = (𝑝 ∈ {𝑦 ∈ 𝒫 𝐷 ∣ 𝑦 ≈ 2o} ↦ (𝑧 ∈ 𝐷 ↦ if(𝑧 ∈ 𝑝, ∪ (𝑝 ∖ {𝑧}), 𝑧)))) |
12 | 1, 11 | syl5eq 2873 | 1 ⊢ (𝐷 ∈ 𝑉 → 𝑇 = (𝑝 ∈ {𝑦 ∈ 𝒫 𝐷 ∣ 𝑦 ≈ 2o} ↦ (𝑧 ∈ 𝐷 ↦ if(𝑧 ∈ 𝑝, ∪ (𝑝 ∖ {𝑧}), 𝑧)))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1656 ∈ wcel 2164 {crab 3121 Vcvv 3414 ∖ cdif 3795 ifcif 4306 𝒫 cpw 4378 {csn 4397 ∪ cuni 4658 class class class wbr 4873 ↦ cmpt 4952 ‘cfv 6123 2oc2o 7820 ≈ cen 8219 pmTrspcpmtr 18211 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1894 ax-4 1908 ax-5 2009 ax-6 2075 ax-7 2112 ax-9 2173 ax-10 2192 ax-11 2207 ax-12 2220 ax-13 2389 ax-ext 2803 ax-rep 4994 ax-sep 5005 ax-nul 5013 ax-pow 5065 ax-pr 5127 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 879 df-3an 1113 df-tru 1660 df-ex 1879 df-nf 1883 df-sb 2068 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ne 3000 df-ral 3122 df-rex 3123 df-reu 3124 df-rab 3126 df-v 3416 df-sbc 3663 df-csb 3758 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-nul 4145 df-if 4307 df-pw 4380 df-sn 4398 df-pr 4400 df-op 4404 df-uni 4659 df-iun 4742 df-br 4874 df-opab 4936 df-mpt 4953 df-id 5250 df-xp 5348 df-rel 5349 df-cnv 5350 df-co 5351 df-dm 5352 df-rn 5353 df-res 5354 df-ima 5355 df-iota 6086 df-fun 6125 df-fn 6126 df-f 6127 df-f1 6128 df-fo 6129 df-f1o 6130 df-fv 6131 df-pmtr 18212 |
This theorem is referenced by: pmtrval 18221 pmtrrn 18227 pmtrfrn 18228 pmtrprfval 18257 pmtrsn 18290 |
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