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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 18573 | . 2 ⊢ ((𝐷 ∈ 𝑉 ∧ 𝑃 ⊆ 𝐷 ∧ 𝑃 ≈ 2o) → (𝑇‘𝑃) = (𝑧 ∈ 𝐷 ↦ if(𝑧 ∈ 𝑃, ∪ (𝑃 ∖ {𝑧}), 𝑧))) |
3 | simpll2 1209 | . . . 4 ⊢ ((((𝐷 ∈ 𝑉 ∧ 𝑃 ⊆ 𝐷 ∧ 𝑃 ≈ 2o) ∧ 𝑧 ∈ 𝐷) ∧ 𝑧 ∈ 𝑃) → 𝑃 ⊆ 𝐷) | |
4 | 1onn 8259 | . . . . . 6 ⊢ 1o ∈ ω | |
5 | simpll3 1210 | . . . . . . 7 ⊢ ((((𝐷 ∈ 𝑉 ∧ 𝑃 ⊆ 𝐷 ∧ 𝑃 ≈ 2o) ∧ 𝑧 ∈ 𝐷) ∧ 𝑧 ∈ 𝑃) → 𝑃 ≈ 2o) | |
6 | df-2o 8097 | . . . . . . 7 ⊢ 2o = suc 1o | |
7 | 5, 6 | breqtrdi 5099 | . . . . . 6 ⊢ ((((𝐷 ∈ 𝑉 ∧ 𝑃 ⊆ 𝐷 ∧ 𝑃 ≈ 2o) ∧ 𝑧 ∈ 𝐷) ∧ 𝑧 ∈ 𝑃) → 𝑃 ≈ suc 1o) |
8 | simpr 487 | . . . . . 6 ⊢ ((((𝐷 ∈ 𝑉 ∧ 𝑃 ⊆ 𝐷 ∧ 𝑃 ≈ 2o) ∧ 𝑧 ∈ 𝐷) ∧ 𝑧 ∈ 𝑃) → 𝑧 ∈ 𝑃) | |
9 | dif1en 8745 | . . . . . 6 ⊢ ((1o ∈ ω ∧ 𝑃 ≈ suc 1o ∧ 𝑧 ∈ 𝑃) → (𝑃 ∖ {𝑧}) ≈ 1o) | |
10 | 4, 7, 8, 9 | mp3an2i 1462 | . . . . 5 ⊢ ((((𝐷 ∈ 𝑉 ∧ 𝑃 ⊆ 𝐷 ∧ 𝑃 ≈ 2o) ∧ 𝑧 ∈ 𝐷) ∧ 𝑧 ∈ 𝑃) → (𝑃 ∖ {𝑧}) ≈ 1o) |
11 | en1uniel 8575 | . . . . 5 ⊢ ((𝑃 ∖ {𝑧}) ≈ 1o → ∪ (𝑃 ∖ {𝑧}) ∈ (𝑃 ∖ {𝑧})) | |
12 | eldifi 4102 | . . . . 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 3967 | . . 3 ⊢ ((((𝐷 ∈ 𝑉 ∧ 𝑃 ⊆ 𝐷 ∧ 𝑃 ≈ 2o) ∧ 𝑧 ∈ 𝐷) ∧ 𝑧 ∈ 𝑃) → ∪ (𝑃 ∖ {𝑧}) ∈ 𝐷) |
15 | simplr 767 | . . 3 ⊢ ((((𝐷 ∈ 𝑉 ∧ 𝑃 ⊆ 𝐷 ∧ 𝑃 ≈ 2o) ∧ 𝑧 ∈ 𝐷) ∧ ¬ 𝑧 ∈ 𝑃) → 𝑧 ∈ 𝐷) | |
16 | 14, 15 | ifclda 4500 | . 2 ⊢ (((𝐷 ∈ 𝑉 ∧ 𝑃 ⊆ 𝐷 ∧ 𝑃 ≈ 2o) ∧ 𝑧 ∈ 𝐷) → if(𝑧 ∈ 𝑃, ∪ (𝑃 ∖ {𝑧}), 𝑧) ∈ 𝐷) |
17 | 2, 16 | fmpt3d 6874 | 1 ⊢ ((𝐷 ∈ 𝑉 ∧ 𝑃 ⊆ 𝐷 ∧ 𝑃 ≈ 2o) → (𝑇‘𝑃):𝐷⟶𝐷) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 398 ∧ w3a 1083 = wceq 1533 ∈ wcel 2110 ∖ cdif 3932 ⊆ wss 3935 ifcif 4466 {csn 4560 ∪ cuni 4831 class class class wbr 5058 suc csuc 6187 ⟶wf 6345 ‘cfv 6349 ωcom 7574 1oc1o 8089 2oc2o 8090 ≈ cen 8500 pmTrspcpmtr 18563 |
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 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-rep 5182 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4561 df-pr 4563 df-tp 4565 df-op 4567 df-uni 4832 df-iun 4913 df-br 5059 df-opab 5121 df-mpt 5139 df-tr 5165 df-id 5454 df-eprel 5459 df-po 5468 df-so 5469 df-fr 5508 df-we 5510 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-ord 6188 df-on 6189 df-lim 6190 df-suc 6191 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-om 7575 df-1o 8096 df-2o 8097 df-er 8283 df-en 8504 df-fin 8507 df-pmtr 18564 |
This theorem is referenced by: pmtrmvd 18578 pmtrfinv 18583 pmtrff1o 18585 pmtrfcnv 18586 pmtr3ncomlem1 18595 mdetralt 21211 mdetunilem7 21221 |
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