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Mirrors > Home > MPE Home > Th. List > pmtrf | Structured version Visualization version GIF version |
Description: Functionality of a transposition. (Contributed by Stefan O'Rear, 16-Aug-2015.) |
Ref | Expression |
---|---|
pmtrfval.t | ⊢ 𝑇 = (pmTrsp‘𝐷) |
Ref | Expression |
---|---|
pmtrf | ⊢ ((𝐷 ∈ 𝑉 ∧ 𝑃 ⊆ 𝐷 ∧ 𝑃 ≈ 2o) → (𝑇‘𝑃):𝐷⟶𝐷) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pmtrfval.t | . . 3 ⊢ 𝑇 = (pmTrsp‘𝐷) | |
2 | 1 | pmtrval 19241 | . 2 ⊢ ((𝐷 ∈ 𝑉 ∧ 𝑃 ⊆ 𝐷 ∧ 𝑃 ≈ 2o) → (𝑇‘𝑃) = (𝑧 ∈ 𝐷 ↦ if(𝑧 ∈ 𝑃, ∪ (𝑃 ∖ {𝑧}), 𝑧))) |
3 | simpll2 1214 | . . . 4 ⊢ ((((𝐷 ∈ 𝑉 ∧ 𝑃 ⊆ 𝐷 ∧ 𝑃 ≈ 2o) ∧ 𝑧 ∈ 𝐷) ∧ 𝑧 ∈ 𝑃) → 𝑃 ⊆ 𝐷) | |
4 | 1onn 8590 | . . . . . 6 ⊢ 1o ∈ ω | |
5 | simpll3 1215 | . . . . . . 7 ⊢ ((((𝐷 ∈ 𝑉 ∧ 𝑃 ⊆ 𝐷 ∧ 𝑃 ≈ 2o) ∧ 𝑧 ∈ 𝐷) ∧ 𝑧 ∈ 𝑃) → 𝑃 ≈ 2o) | |
6 | df-2o 8417 | . . . . . . 7 ⊢ 2o = suc 1o | |
7 | 5, 6 | breqtrdi 5150 | . . . . . 6 ⊢ ((((𝐷 ∈ 𝑉 ∧ 𝑃 ⊆ 𝐷 ∧ 𝑃 ≈ 2o) ∧ 𝑧 ∈ 𝐷) ∧ 𝑧 ∈ 𝑃) → 𝑃 ≈ suc 1o) |
8 | simpr 486 | . . . . . 6 ⊢ ((((𝐷 ∈ 𝑉 ∧ 𝑃 ⊆ 𝐷 ∧ 𝑃 ≈ 2o) ∧ 𝑧 ∈ 𝐷) ∧ 𝑧 ∈ 𝑃) → 𝑧 ∈ 𝑃) | |
9 | dif1ennn 9111 | . . . . . 6 ⊢ ((1o ∈ ω ∧ 𝑃 ≈ suc 1o ∧ 𝑧 ∈ 𝑃) → (𝑃 ∖ {𝑧}) ≈ 1o) | |
10 | 4, 7, 8, 9 | mp3an2i 1467 | . . . . 5 ⊢ ((((𝐷 ∈ 𝑉 ∧ 𝑃 ⊆ 𝐷 ∧ 𝑃 ≈ 2o) ∧ 𝑧 ∈ 𝐷) ∧ 𝑧 ∈ 𝑃) → (𝑃 ∖ {𝑧}) ≈ 1o) |
11 | en1uniel 8978 | . . . . 5 ⊢ ((𝑃 ∖ {𝑧}) ≈ 1o → ∪ (𝑃 ∖ {𝑧}) ∈ (𝑃 ∖ {𝑧})) | |
12 | eldifi 4090 | . . . . 5 ⊢ (∪ (𝑃 ∖ {𝑧}) ∈ (𝑃 ∖ {𝑧}) → ∪ (𝑃 ∖ {𝑧}) ∈ 𝑃) | |
13 | 10, 11, 12 | 3syl 18 | . . . 4 ⊢ ((((𝐷 ∈ 𝑉 ∧ 𝑃 ⊆ 𝐷 ∧ 𝑃 ≈ 2o) ∧ 𝑧 ∈ 𝐷) ∧ 𝑧 ∈ 𝑃) → ∪ (𝑃 ∖ {𝑧}) ∈ 𝑃) |
14 | 3, 13 | sseldd 3949 | . . 3 ⊢ ((((𝐷 ∈ 𝑉 ∧ 𝑃 ⊆ 𝐷 ∧ 𝑃 ≈ 2o) ∧ 𝑧 ∈ 𝐷) ∧ 𝑧 ∈ 𝑃) → ∪ (𝑃 ∖ {𝑧}) ∈ 𝐷) |
15 | simplr 768 | . . 3 ⊢ ((((𝐷 ∈ 𝑉 ∧ 𝑃 ⊆ 𝐷 ∧ 𝑃 ≈ 2o) ∧ 𝑧 ∈ 𝐷) ∧ ¬ 𝑧 ∈ 𝑃) → 𝑧 ∈ 𝐷) | |
16 | 14, 15 | ifclda 4525 | . 2 ⊢ (((𝐷 ∈ 𝑉 ∧ 𝑃 ⊆ 𝐷 ∧ 𝑃 ≈ 2o) ∧ 𝑧 ∈ 𝐷) → if(𝑧 ∈ 𝑃, ∪ (𝑃 ∖ {𝑧}), 𝑧) ∈ 𝐷) |
17 | 2, 16 | fmpt3d 7068 | 1 ⊢ ((𝐷 ∈ 𝑉 ∧ 𝑃 ⊆ 𝐷 ∧ 𝑃 ≈ 2o) → (𝑇‘𝑃):𝐷⟶𝐷) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 397 ∧ w3a 1088 = wceq 1542 ∈ wcel 2107 ∖ cdif 3911 ⊆ wss 3914 ifcif 4490 {csn 4590 ∪ cuni 4869 class class class wbr 5109 suc csuc 6323 ⟶wf 6496 ‘cfv 6500 ωcom 7806 1oc1o 8409 2oc2o 8410 ≈ cen 8886 pmTrspcpmtr 19231 |
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 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5246 ax-sep 5260 ax-nul 5267 ax-pow 5324 ax-pr 5388 ax-un 7676 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3353 df-rab 3407 df-v 3449 df-sbc 3744 df-csb 3860 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-pss 3933 df-nul 4287 df-if 4491 df-pw 4566 df-sn 4591 df-pr 4593 df-op 4597 df-uni 4870 df-iun 4960 df-br 5110 df-opab 5172 df-mpt 5193 df-tr 5227 df-id 5535 df-eprel 5541 df-po 5549 df-so 5550 df-fr 5592 df-we 5594 df-xp 5643 df-rel 5644 df-cnv 5645 df-co 5646 df-dm 5647 df-rn 5648 df-res 5649 df-ima 5650 df-ord 6324 df-on 6325 df-lim 6326 df-suc 6327 df-iota 6452 df-fun 6502 df-fn 6503 df-f 6504 df-f1 6505 df-fo 6506 df-f1o 6507 df-fv 6508 df-om 7807 df-1o 8416 df-2o 8417 df-en 8890 df-pmtr 19232 |
This theorem is referenced by: pmtrmvd 19246 pmtrfinv 19251 pmtrff1o 19253 pmtrfcnv 19254 pmtr3ncomlem1 19263 mdetralt 21980 mdetunilem7 21990 |
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