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Theorem pmtrprfv2 33108
Description: In a transposition of two given points, each maps to the other. (Contributed by Thierry Arnoux, 22-Aug-2020.)
Hypothesis
Ref Expression
pmtrprfv2.t 𝑇 = (pmTrsp‘𝐷)
Assertion
Ref Expression
pmtrprfv2 ((𝐷𝑉 ∧ (𝑋𝐷𝑌𝐷𝑋𝑌)) → ((𝑇‘{𝑋, 𝑌})‘𝑌) = 𝑋)

Proof of Theorem pmtrprfv2
StepHypRef Expression
1 prcom 4732 . . . 4 {𝑌, 𝑋} = {𝑋, 𝑌}
21fveq2i 6909 . . 3 (𝑇‘{𝑌, 𝑋}) = (𝑇‘{𝑋, 𝑌})
32fveq1i 6907 . 2 ((𝑇‘{𝑌, 𝑋})‘𝑌) = ((𝑇‘{𝑋, 𝑌})‘𝑌)
4 ancom 460 . . . . 5 ((𝑋𝐷𝑌𝐷) ↔ (𝑌𝐷𝑋𝐷))
5 necom 2994 . . . . 5 (𝑋𝑌𝑌𝑋)
64, 5anbi12i 628 . . . 4 (((𝑋𝐷𝑌𝐷) ∧ 𝑋𝑌) ↔ ((𝑌𝐷𝑋𝐷) ∧ 𝑌𝑋))
7 df-3an 1089 . . . 4 ((𝑋𝐷𝑌𝐷𝑋𝑌) ↔ ((𝑋𝐷𝑌𝐷) ∧ 𝑋𝑌))
8 df-3an 1089 . . . 4 ((𝑌𝐷𝑋𝐷𝑌𝑋) ↔ ((𝑌𝐷𝑋𝐷) ∧ 𝑌𝑋))
96, 7, 83bitr4i 303 . . 3 ((𝑋𝐷𝑌𝐷𝑋𝑌) ↔ (𝑌𝐷𝑋𝐷𝑌𝑋))
10 pmtrprfv2.t . . . 4 𝑇 = (pmTrsp‘𝐷)
1110pmtrprfv 19471 . . 3 ((𝐷𝑉 ∧ (𝑌𝐷𝑋𝐷𝑌𝑋)) → ((𝑇‘{𝑌, 𝑋})‘𝑌) = 𝑋)
129, 11sylan2b 594 . 2 ((𝐷𝑉 ∧ (𝑋𝐷𝑌𝐷𝑋𝑌)) → ((𝑇‘{𝑌, 𝑋})‘𝑌) = 𝑋)
133, 12eqtr3id 2791 1 ((𝐷𝑉 ∧ (𝑋𝐷𝑌𝐷𝑋𝑌)) → ((𝑇‘{𝑋, 𝑌})‘𝑌) = 𝑋)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  w3a 1087   = wceq 1540  wcel 2108  wne 2940  {cpr 4628  cfv 6561  pmTrspcpmtr 19459
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-1o 8506  df-2o 8507  df-en 8986  df-pmtr 19460
This theorem is referenced by:  pmtrcnel  33109  fzo0pmtrlast  33112  pmtridfv2  33116  psgnfzto1stlem  33120
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