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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 19415 | . 2 ⊢ ((𝐷 ∈ 𝑉 ∧ 𝑃 ⊆ 𝐷 ∧ 𝑃 ≈ 2o) → (𝑇‘𝑃) = (𝑧 ∈ 𝐷 ↦ if(𝑧 ∈ 𝑃, ∪ (𝑃 ∖ {𝑧}), 𝑧))) |
| 3 | simpll2 1215 | . . . 4 ⊢ ((((𝐷 ∈ 𝑉 ∧ 𝑃 ⊆ 𝐷 ∧ 𝑃 ≈ 2o) ∧ 𝑧 ∈ 𝐷) ∧ 𝑧 ∈ 𝑃) → 𝑃 ⊆ 𝐷) | |
| 4 | 1onn 8565 | . . . . . 6 ⊢ 1o ∈ ω | |
| 5 | simpll3 1216 | . . . . . . 7 ⊢ ((((𝐷 ∈ 𝑉 ∧ 𝑃 ⊆ 𝐷 ∧ 𝑃 ≈ 2o) ∧ 𝑧 ∈ 𝐷) ∧ 𝑧 ∈ 𝑃) → 𝑃 ≈ 2o) | |
| 6 | df-2o 8395 | . . . . . . 7 ⊢ 2o = suc 1o | |
| 7 | 5, 6 | breqtrdi 5115 | . . . . . 6 ⊢ ((((𝐷 ∈ 𝑉 ∧ 𝑃 ⊆ 𝐷 ∧ 𝑃 ≈ 2o) ∧ 𝑧 ∈ 𝐷) ∧ 𝑧 ∈ 𝑃) → 𝑃 ≈ suc 1o) |
| 8 | simpr 484 | . . . . . 6 ⊢ ((((𝐷 ∈ 𝑉 ∧ 𝑃 ⊆ 𝐷 ∧ 𝑃 ≈ 2o) ∧ 𝑧 ∈ 𝐷) ∧ 𝑧 ∈ 𝑃) → 𝑧 ∈ 𝑃) | |
| 9 | dif1ennn 9086 | . . . . . 6 ⊢ ((1o ∈ ω ∧ 𝑃 ≈ suc 1o ∧ 𝑧 ∈ 𝑃) → (𝑃 ∖ {𝑧}) ≈ 1o) | |
| 10 | 4, 7, 8, 9 | mp3an2i 1469 | . . . . 5 ⊢ ((((𝐷 ∈ 𝑉 ∧ 𝑃 ⊆ 𝐷 ∧ 𝑃 ≈ 2o) ∧ 𝑧 ∈ 𝐷) ∧ 𝑧 ∈ 𝑃) → (𝑃 ∖ {𝑧}) ≈ 1o) |
| 11 | en1uniel 8965 | . . . . 5 ⊢ ((𝑃 ∖ {𝑧}) ≈ 1o → ∪ (𝑃 ∖ {𝑧}) ∈ (𝑃 ∖ {𝑧})) | |
| 12 | eldifi 4063 | . . . . 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 3918 | . . 3 ⊢ ((((𝐷 ∈ 𝑉 ∧ 𝑃 ⊆ 𝐷 ∧ 𝑃 ≈ 2o) ∧ 𝑧 ∈ 𝐷) ∧ 𝑧 ∈ 𝑃) → ∪ (𝑃 ∖ {𝑧}) ∈ 𝐷) |
| 15 | simplr 769 | . . 3 ⊢ ((((𝐷 ∈ 𝑉 ∧ 𝑃 ⊆ 𝐷 ∧ 𝑃 ≈ 2o) ∧ 𝑧 ∈ 𝐷) ∧ ¬ 𝑧 ∈ 𝑃) → 𝑧 ∈ 𝐷) | |
| 16 | 14, 15 | ifclda 4492 | . 2 ⊢ (((𝐷 ∈ 𝑉 ∧ 𝑃 ⊆ 𝐷 ∧ 𝑃 ≈ 2o) ∧ 𝑧 ∈ 𝐷) → if(𝑧 ∈ 𝑃, ∪ (𝑃 ∖ {𝑧}), 𝑧) ∈ 𝐷) |
| 17 | 2, 16 | fmpt3d 7057 | 1 ⊢ ((𝐷 ∈ 𝑉 ∧ 𝑃 ⊆ 𝐷 ∧ 𝑃 ≈ 2o) → (𝑇‘𝑃):𝐷⟶𝐷) |
| Colors of variables: wff setvar class |
| Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 ∧ w3a 1087 = wceq 1542 ∈ wcel 2114 ∖ cdif 3882 ⊆ wss 3885 ifcif 4456 {csn 4557 ∪ cuni 4840 class class class wbr 5074 suc csuc 6314 ⟶wf 6483 ‘cfv 6487 ωcom 7806 1oc1o 8387 2oc2o 8388 ≈ cen 8879 pmTrspcpmtr 19405 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2184 ax-ext 2707 ax-rep 5201 ax-sep 5220 ax-nul 5230 ax-pow 5296 ax-pr 5364 ax-un 7678 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2538 df-eu 2568 df-clab 2714 df-cleq 2727 df-clel 2810 df-nfc 2884 df-ne 2931 df-ral 3050 df-rex 3060 df-reu 3341 df-rab 3388 df-v 3429 df-sbc 3726 df-csb 3834 df-dif 3888 df-un 3890 df-in 3892 df-ss 3902 df-pss 3905 df-nul 4264 df-if 4457 df-pw 4533 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4841 df-iun 4925 df-br 5075 df-opab 5137 df-mpt 5156 df-tr 5182 df-id 5515 df-eprel 5520 df-po 5528 df-so 5529 df-fr 5573 df-we 5575 df-xp 5626 df-rel 5627 df-cnv 5628 df-co 5629 df-dm 5630 df-rn 5631 df-res 5632 df-ima 5633 df-ord 6315 df-on 6316 df-lim 6317 df-suc 6318 df-iota 6443 df-fun 6489 df-fn 6490 df-f 6491 df-f1 6492 df-fo 6493 df-f1o 6494 df-fv 6495 df-om 7807 df-1o 8394 df-2o 8395 df-en 8883 df-pmtr 19406 |
| This theorem is referenced by: pmtrmvd 19420 pmtrfinv 19425 pmtrff1o 19427 pmtrfcnv 19428 pmtr3ncomlem1 19437 mdetralt 22561 mdetunilem7 22571 |
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