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Mirrors > Home > MPE Home > Th. List > pmtrff1o | Structured version Visualization version GIF version |
Description: A transposition function is a permutation. (Contributed by Stefan O'Rear, 22-Aug-2015.) |
Ref | Expression |
---|---|
pmtrrn.t | ⊢ 𝑇 = (pmTrsp‘𝐷) |
pmtrrn.r | ⊢ 𝑅 = ran 𝑇 |
Ref | Expression |
---|---|
pmtrff1o | ⊢ (𝐹 ∈ 𝑅 → 𝐹:𝐷–1-1-onto→𝐷) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pmtrrn.t | . . . . . 6 ⊢ 𝑇 = (pmTrsp‘𝐷) | |
2 | pmtrrn.r | . . . . . 6 ⊢ 𝑅 = ran 𝑇 | |
3 | eqid 2736 | . . . . . 6 ⊢ dom (𝐹 ∖ I ) = dom (𝐹 ∖ I ) | |
4 | 1, 2, 3 | pmtrfrn 19236 | . . . . 5 ⊢ (𝐹 ∈ 𝑅 → ((𝐷 ∈ V ∧ dom (𝐹 ∖ I ) ⊆ 𝐷 ∧ dom (𝐹 ∖ I ) ≈ 2o) ∧ 𝐹 = (𝑇‘dom (𝐹 ∖ I )))) |
5 | 4 | simpld 495 | . . . 4 ⊢ (𝐹 ∈ 𝑅 → (𝐷 ∈ V ∧ dom (𝐹 ∖ I ) ⊆ 𝐷 ∧ dom (𝐹 ∖ I ) ≈ 2o)) |
6 | 1 | pmtrf 19233 | . . . 4 ⊢ ((𝐷 ∈ V ∧ dom (𝐹 ∖ I ) ⊆ 𝐷 ∧ dom (𝐹 ∖ I ) ≈ 2o) → (𝑇‘dom (𝐹 ∖ I )):𝐷⟶𝐷) |
7 | 5, 6 | syl 17 | . . 3 ⊢ (𝐹 ∈ 𝑅 → (𝑇‘dom (𝐹 ∖ I )):𝐷⟶𝐷) |
8 | 4 | simprd 496 | . . . 4 ⊢ (𝐹 ∈ 𝑅 → 𝐹 = (𝑇‘dom (𝐹 ∖ I ))) |
9 | 8 | feq1d 6651 | . . 3 ⊢ (𝐹 ∈ 𝑅 → (𝐹:𝐷⟶𝐷 ↔ (𝑇‘dom (𝐹 ∖ I )):𝐷⟶𝐷)) |
10 | 7, 9 | mpbird 256 | . 2 ⊢ (𝐹 ∈ 𝑅 → 𝐹:𝐷⟶𝐷) |
11 | 1, 2 | pmtrfinv 19239 | . 2 ⊢ (𝐹 ∈ 𝑅 → (𝐹 ∘ 𝐹) = ( I ↾ 𝐷)) |
12 | 10, 10, 11, 11 | fcof1od 7237 | 1 ⊢ (𝐹 ∈ 𝑅 → 𝐹:𝐷–1-1-onto→𝐷) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1087 = wceq 1541 ∈ wcel 2106 Vcvv 3444 ∖ cdif 3906 ⊆ wss 3909 class class class wbr 5104 I cid 5529 dom cdm 5632 ran crn 5633 ⟶wf 6490 –1-1-onto→wf1o 6493 ‘cfv 6494 2oc2o 8403 ≈ cen 8877 pmTrspcpmtr 19219 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-rep 5241 ax-sep 5255 ax-nul 5262 ax-pow 5319 ax-pr 5383 ax-un 7669 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2888 df-ne 2943 df-ral 3064 df-rex 3073 df-reu 3353 df-rab 3407 df-v 3446 df-sbc 3739 df-csb 3855 df-dif 3912 df-un 3914 df-in 3916 df-ss 3926 df-pss 3928 df-nul 4282 df-if 4486 df-pw 4561 df-sn 4586 df-pr 4588 df-op 4592 df-uni 4865 df-iun 4955 df-br 5105 df-opab 5167 df-mpt 5188 df-tr 5222 df-id 5530 df-eprel 5536 df-po 5544 df-so 5545 df-fr 5587 df-we 5589 df-xp 5638 df-rel 5639 df-cnv 5640 df-co 5641 df-dm 5642 df-rn 5643 df-res 5644 df-ima 5645 df-ord 6319 df-on 6320 df-lim 6321 df-suc 6322 df-iota 6446 df-fun 6496 df-fn 6497 df-f 6498 df-f1 6499 df-fo 6500 df-f1o 6501 df-fv 6502 df-om 7800 df-1o 8409 df-2o 8410 df-er 8645 df-en 8881 df-dom 8882 df-sdom 8883 df-fin 8884 df-pmtr 19220 |
This theorem is referenced by: pmtrfb 19243 pmtrfconj 19244 symgtrf 19247 psgnunilem1 19271 pmtrcnel 31835 pmtrcnel2 31836 pmtridf1o 31838 psgnfzto1stlem 31844 |
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