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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 2735 | . . . . . 6 ⊢ dom (𝐹 ∖ I ) = dom (𝐹 ∖ I ) | |
4 | 1, 2, 3 | pmtrfrn 19491 | . . . . 5 ⊢ (𝐹 ∈ 𝑅 → ((𝐷 ∈ V ∧ dom (𝐹 ∖ I ) ⊆ 𝐷 ∧ dom (𝐹 ∖ I ) ≈ 2o) ∧ 𝐹 = (𝑇‘dom (𝐹 ∖ I )))) |
5 | 4 | simpld 494 | . . . 4 ⊢ (𝐹 ∈ 𝑅 → (𝐷 ∈ V ∧ dom (𝐹 ∖ I ) ⊆ 𝐷 ∧ dom (𝐹 ∖ I ) ≈ 2o)) |
6 | 1 | pmtrf 19488 | . . . 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 495 | . . . 4 ⊢ (𝐹 ∈ 𝑅 → 𝐹 = (𝑇‘dom (𝐹 ∖ I ))) |
9 | 8 | feq1d 6721 | . . 3 ⊢ (𝐹 ∈ 𝑅 → (𝐹:𝐷⟶𝐷 ↔ (𝑇‘dom (𝐹 ∖ I )):𝐷⟶𝐷)) |
10 | 7, 9 | mpbird 257 | . 2 ⊢ (𝐹 ∈ 𝑅 → 𝐹:𝐷⟶𝐷) |
11 | 1, 2 | pmtrfinv 19494 | . 2 ⊢ (𝐹 ∈ 𝑅 → (𝐹 ∘ 𝐹) = ( I ↾ 𝐷)) |
12 | 10, 10, 11, 11 | fcof1od 7314 | 1 ⊢ (𝐹 ∈ 𝑅 → 𝐹:𝐷–1-1-onto→𝐷) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1086 = wceq 1537 ∈ wcel 2106 Vcvv 3478 ∖ cdif 3960 ⊆ wss 3963 class class class wbr 5148 I cid 5582 dom cdm 5689 ran crn 5690 ⟶wf 6559 –1-1-onto→wf1o 6562 ‘cfv 6563 2oc2o 8499 ≈ cen 8981 pmTrspcpmtr 19474 |
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 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-om 7888 df-1o 8505 df-2o 8506 df-er 8744 df-en 8985 df-dom 8986 df-sdom 8987 df-fin 8988 df-pmtr 19475 |
This theorem is referenced by: pmtrfb 19498 pmtrfconj 19499 symgtrf 19502 psgnunilem1 19526 pmtrcnel 33092 pmtrcnel2 33093 fzo0pmtrlast 33095 pmtridf1o 33097 psgnfzto1stlem 33103 |
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