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Theorem pmtrffv 17860
Description: Mapping of a point under a transposition function. (Contributed by Stefan O'Rear, 22-Aug-2015.)
Hypotheses
Ref Expression
pmtrrn.t 𝑇 = (pmTrsp‘𝐷)
pmtrrn.r 𝑅 = ran 𝑇
pmtrfrn.p 𝑃 = dom (𝐹 ∖ I )
Assertion
Ref Expression
pmtrffv ((𝐹𝑅𝑍𝐷) → (𝐹𝑍) = if(𝑍𝑃, (𝑃 ∖ {𝑍}), 𝑍))

Proof of Theorem pmtrffv
StepHypRef Expression
1 pmtrrn.t . . . . . 6 𝑇 = (pmTrsp‘𝐷)
2 pmtrrn.r . . . . . 6 𝑅 = ran 𝑇
3 pmtrfrn.p . . . . . 6 𝑃 = dom (𝐹 ∖ I )
41, 2, 3pmtrfrn 17859 . . . . 5 (𝐹𝑅 → ((𝐷 ∈ V ∧ 𝑃𝐷𝑃 ≈ 2𝑜) ∧ 𝐹 = (𝑇𝑃)))
54simprd 479 . . . 4 (𝐹𝑅𝐹 = (𝑇𝑃))
65fveq1d 6180 . . 3 (𝐹𝑅 → (𝐹𝑍) = ((𝑇𝑃)‘𝑍))
76adantr 481 . 2 ((𝐹𝑅𝑍𝐷) → (𝐹𝑍) = ((𝑇𝑃)‘𝑍))
84simpld 475 . . 3 (𝐹𝑅 → (𝐷 ∈ V ∧ 𝑃𝐷𝑃 ≈ 2𝑜))
91pmtrfv 17853 . . 3 (((𝐷 ∈ V ∧ 𝑃𝐷𝑃 ≈ 2𝑜) ∧ 𝑍𝐷) → ((𝑇𝑃)‘𝑍) = if(𝑍𝑃, (𝑃 ∖ {𝑍}), 𝑍))
108, 9sylan 488 . 2 ((𝐹𝑅𝑍𝐷) → ((𝑇𝑃)‘𝑍) = if(𝑍𝑃, (𝑃 ∖ {𝑍}), 𝑍))
117, 10eqtrd 2654 1 ((𝐹𝑅𝑍𝐷) → (𝐹𝑍) = if(𝑍𝑃, (𝑃 ∖ {𝑍}), 𝑍))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384  w3a 1036   = wceq 1481  wcel 1988  Vcvv 3195  cdif 3564  wss 3567  ifcif 4077  {csn 4168   cuni 4427   class class class wbr 4644   I cid 5013  dom cdm 5104  ran crn 5105  cfv 5876  2𝑜c2o 7539  cen 7937  pmTrspcpmtr 17842
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-8 1990  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-rep 4762  ax-sep 4772  ax-nul 4780  ax-pow 4834  ax-pr 4897  ax-un 6934
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ne 2792  df-ral 2914  df-rex 2915  df-reu 2916  df-rab 2918  df-v 3197  df-sbc 3430  df-csb 3527  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-pss 3583  df-nul 3908  df-if 4078  df-pw 4151  df-sn 4169  df-pr 4171  df-tp 4173  df-op 4175  df-uni 4428  df-iun 4513  df-br 4645  df-opab 4704  df-mpt 4721  df-tr 4744  df-id 5014  df-eprel 5019  df-po 5025  df-so 5026  df-fr 5063  df-we 5065  df-xp 5110  df-rel 5111  df-cnv 5112  df-co 5113  df-dm 5114  df-rn 5115  df-res 5116  df-ima 5117  df-ord 5714  df-on 5715  df-lim 5716  df-suc 5717  df-iota 5839  df-fun 5878  df-fn 5879  df-f 5880  df-f1 5881  df-fo 5882  df-f1o 5883  df-fv 5884  df-om 7051  df-1o 7545  df-2o 7546  df-er 7727  df-en 7941  df-fin 7944  df-pmtr 17843
This theorem is referenced by:  pmtrfinv  17862  pmtrdifellem3  17879  pmtrdifellem4  17880  psgnunilem1  17894
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