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Theorem pmtrfv 19445
Description: General value of mapping a point under a transposition. (Contributed by Stefan O'Rear, 16-Aug-2015.)
Hypothesis
Ref Expression
pmtrfval.t 𝑇 = (pmTrsp‘𝐷)
Assertion
Ref Expression
pmtrfv (((𝐷𝑉𝑃𝐷𝑃 ≈ 2o) ∧ 𝑍𝐷) → ((𝑇𝑃)‘𝑍) = if(𝑍𝑃, (𝑃 ∖ {𝑍}), 𝑍))

Proof of Theorem pmtrfv
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 pmtrfval.t . . . . 5 𝑇 = (pmTrsp‘𝐷)
21pmtrval 19444 . . . 4 ((𝐷𝑉𝑃𝐷𝑃 ≈ 2o) → (𝑇𝑃) = (𝑧𝐷 ↦ if(𝑧𝑃, (𝑃 ∖ {𝑧}), 𝑧)))
32fveq1d 6894 . . 3 ((𝐷𝑉𝑃𝐷𝑃 ≈ 2o) → ((𝑇𝑃)‘𝑍) = ((𝑧𝐷 ↦ if(𝑧𝑃, (𝑃 ∖ {𝑧}), 𝑧))‘𝑍))
43adantr 479 . 2 (((𝐷𝑉𝑃𝐷𝑃 ≈ 2o) ∧ 𝑍𝐷) → ((𝑇𝑃)‘𝑍) = ((𝑧𝐷 ↦ if(𝑧𝑃, (𝑃 ∖ {𝑧}), 𝑧))‘𝑍))
5 eqid 2726 . . 3 (𝑧𝐷 ↦ if(𝑧𝑃, (𝑃 ∖ {𝑧}), 𝑧)) = (𝑧𝐷 ↦ if(𝑧𝑃, (𝑃 ∖ {𝑧}), 𝑧))
6 eleq1 2814 . . . 4 (𝑧 = 𝑍 → (𝑧𝑃𝑍𝑃))
7 sneq 4635 . . . . . 6 (𝑧 = 𝑍 → {𝑧} = {𝑍})
87difeq2d 4120 . . . . 5 (𝑧 = 𝑍 → (𝑃 ∖ {𝑧}) = (𝑃 ∖ {𝑍}))
98unieqd 4920 . . . 4 (𝑧 = 𝑍 (𝑃 ∖ {𝑧}) = (𝑃 ∖ {𝑍}))
10 id 22 . . . 4 (𝑧 = 𝑍𝑧 = 𝑍)
116, 9, 10ifbieq12d 4553 . . 3 (𝑧 = 𝑍 → if(𝑧𝑃, (𝑃 ∖ {𝑧}), 𝑧) = if(𝑍𝑃, (𝑃 ∖ {𝑍}), 𝑍))
12 simpr 483 . . 3 (((𝐷𝑉𝑃𝐷𝑃 ≈ 2o) ∧ 𝑍𝐷) → 𝑍𝐷)
13 simpl3 1190 . . . . 5 (((𝐷𝑉𝑃𝐷𝑃 ≈ 2o) ∧ 𝑍𝐷) → 𝑃 ≈ 2o)
14 relen 8970 . . . . . 6 Rel ≈
1514brrelex1i 5730 . . . . 5 (𝑃 ≈ 2o𝑃 ∈ V)
16 difexg 5326 . . . . 5 (𝑃 ∈ V → (𝑃 ∖ {𝑍}) ∈ V)
17 uniexg 7742 . . . . 5 ((𝑃 ∖ {𝑍}) ∈ V → (𝑃 ∖ {𝑍}) ∈ V)
1813, 15, 16, 174syl 19 . . . 4 (((𝐷𝑉𝑃𝐷𝑃 ≈ 2o) ∧ 𝑍𝐷) → (𝑃 ∖ {𝑍}) ∈ V)
19 ifexg 4574 . . . 4 (( (𝑃 ∖ {𝑍}) ∈ V ∧ 𝑍𝐷) → if(𝑍𝑃, (𝑃 ∖ {𝑍}), 𝑍) ∈ V)
2018, 19sylancom 586 . . 3 (((𝐷𝑉𝑃𝐷𝑃 ≈ 2o) ∧ 𝑍𝐷) → if(𝑍𝑃, (𝑃 ∖ {𝑍}), 𝑍) ∈ V)
215, 11, 12, 20fvmptd3 7023 . 2 (((𝐷𝑉𝑃𝐷𝑃 ≈ 2o) ∧ 𝑍𝐷) → ((𝑧𝐷 ↦ if(𝑧𝑃, (𝑃 ∖ {𝑧}), 𝑧))‘𝑍) = if(𝑍𝑃, (𝑃 ∖ {𝑍}), 𝑍))
224, 21eqtrd 2766 1 (((𝐷𝑉𝑃𝐷𝑃 ≈ 2o) ∧ 𝑍𝐷) → ((𝑇𝑃)‘𝑍) = if(𝑍𝑃, (𝑃 ∖ {𝑍}), 𝑍))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 394  w3a 1084   = wceq 1534  wcel 2099  Vcvv 3464  cdif 3945  wss 3948  ifcif 4525  {csn 4625   cuni 4907   class class class wbr 5145  cmpt 5228  cfv 6545  2oc2o 8481  cen 8962  pmTrspcpmtr 19434
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2697  ax-rep 5282  ax-sep 5296  ax-nul 5303  ax-pow 5361  ax-pr 5425  ax-un 7737
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2704  df-cleq 2718  df-clel 2803  df-nfc 2878  df-ne 2931  df-ral 3052  df-rex 3061  df-reu 3366  df-rab 3421  df-v 3466  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4325  df-if 4526  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4908  df-iun 4997  df-br 5146  df-opab 5208  df-mpt 5229  df-id 5572  df-xp 5680  df-rel 5681  df-cnv 5682  df-co 5683  df-dm 5684  df-rn 5685  df-res 5686  df-ima 5687  df-iota 6497  df-fun 6547  df-fn 6548  df-f 6549  df-f1 6550  df-fo 6551  df-f1o 6552  df-fv 6553  df-en 8966  df-pmtr 19435
This theorem is referenced by:  pmtrprfv  19446  pmtrprfv3  19447  pmtrmvd  19449  pmtrffv  19452
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