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Mirrors > Home > MPE Home > Th. List > pmtrfcnv | Structured version Visualization version GIF version |
Description: A transposition function is its own inverse. (Contributed by Stefan O'Rear, 22-Aug-2015.) |
Ref | Expression |
---|---|
pmtrrn.t | ⊢ 𝑇 = (pmTrsp‘𝐷) |
pmtrrn.r | ⊢ 𝑅 = ran 𝑇 |
Ref | Expression |
---|---|
pmtrfcnv | ⊢ (𝐹 ∈ 𝑅 → ◡𝐹 = 𝐹) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pmtrrn.t | . . . . . 6 ⊢ 𝑇 = (pmTrsp‘𝐷) | |
2 | pmtrrn.r | . . . . . 6 ⊢ 𝑅 = ran 𝑇 | |
3 | eqid 2821 | . . . . . 6 ⊢ dom (𝐹 ∖ I ) = dom (𝐹 ∖ I ) | |
4 | 1, 2, 3 | pmtrfrn 18580 | . . . . 5 ⊢ (𝐹 ∈ 𝑅 → ((𝐷 ∈ V ∧ dom (𝐹 ∖ I ) ⊆ 𝐷 ∧ dom (𝐹 ∖ I ) ≈ 2o) ∧ 𝐹 = (𝑇‘dom (𝐹 ∖ I )))) |
5 | 4 | simpld 497 | . . . 4 ⊢ (𝐹 ∈ 𝑅 → (𝐷 ∈ V ∧ dom (𝐹 ∖ I ) ⊆ 𝐷 ∧ dom (𝐹 ∖ I ) ≈ 2o)) |
6 | 1 | pmtrf 18577 | . . . 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 498 | . . . 4 ⊢ (𝐹 ∈ 𝑅 → 𝐹 = (𝑇‘dom (𝐹 ∖ I ))) |
9 | 8 | feq1d 6493 | . . 3 ⊢ (𝐹 ∈ 𝑅 → (𝐹:𝐷⟶𝐷 ↔ (𝑇‘dom (𝐹 ∖ I )):𝐷⟶𝐷)) |
10 | 7, 9 | mpbird 259 | . 2 ⊢ (𝐹 ∈ 𝑅 → 𝐹:𝐷⟶𝐷) |
11 | 1, 2 | pmtrfinv 18583 | . 2 ⊢ (𝐹 ∈ 𝑅 → (𝐹 ∘ 𝐹) = ( I ↾ 𝐷)) |
12 | 10, 10, 11, 11 | 2fcoidinvd 7045 | 1 ⊢ (𝐹 ∈ 𝑅 → ◡𝐹 = 𝐹) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1083 = wceq 1533 ∈ wcel 2110 Vcvv 3494 ∖ cdif 3932 ⊆ wss 3935 class class class wbr 5058 I cid 5453 ◡ccnv 5548 dom cdm 5549 ran crn 5550 ⟶wf 6345 ‘cfv 6349 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-dom 8505 df-sdom 8506 df-fin 8507 df-pmtr 18564 |
This theorem is referenced by: symgtrinv 18594 psgnunilem1 18615 pmtrcnel2 30729 |
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