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Theorem pmtrfv 19319
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 19318 . . . 4 ((𝐷𝑉𝑃𝐷𝑃 ≈ 2o) → (𝑇𝑃) = (𝑧𝐷 ↦ if(𝑧𝑃, (𝑃 ∖ {𝑧}), 𝑧)))
32fveq1d 6893 . . 3 ((𝐷𝑉𝑃𝐷𝑃 ≈ 2o) → ((𝑇𝑃)‘𝑍) = ((𝑧𝐷 ↦ if(𝑧𝑃, (𝑃 ∖ {𝑧}), 𝑧))‘𝑍))
43adantr 481 . 2 (((𝐷𝑉𝑃𝐷𝑃 ≈ 2o) ∧ 𝑍𝐷) → ((𝑇𝑃)‘𝑍) = ((𝑧𝐷 ↦ if(𝑧𝑃, (𝑃 ∖ {𝑧}), 𝑧))‘𝑍))
5 eqid 2732 . . 3 (𝑧𝐷 ↦ if(𝑧𝑃, (𝑃 ∖ {𝑧}), 𝑧)) = (𝑧𝐷 ↦ if(𝑧𝑃, (𝑃 ∖ {𝑧}), 𝑧))
6 eleq1 2821 . . . 4 (𝑧 = 𝑍 → (𝑧𝑃𝑍𝑃))
7 sneq 4638 . . . . . 6 (𝑧 = 𝑍 → {𝑧} = {𝑍})
87difeq2d 4122 . . . . 5 (𝑧 = 𝑍 → (𝑃 ∖ {𝑧}) = (𝑃 ∖ {𝑍}))
98unieqd 4922 . . . 4 (𝑧 = 𝑍 (𝑃 ∖ {𝑧}) = (𝑃 ∖ {𝑍}))
10 id 22 . . . 4 (𝑧 = 𝑍𝑧 = 𝑍)
116, 9, 10ifbieq12d 4556 . . 3 (𝑧 = 𝑍 → if(𝑧𝑃, (𝑃 ∖ {𝑧}), 𝑧) = if(𝑍𝑃, (𝑃 ∖ {𝑍}), 𝑍))
12 simpr 485 . . 3 (((𝐷𝑉𝑃𝐷𝑃 ≈ 2o) ∧ 𝑍𝐷) → 𝑍𝐷)
13 simpl3 1193 . . . . 5 (((𝐷𝑉𝑃𝐷𝑃 ≈ 2o) ∧ 𝑍𝐷) → 𝑃 ≈ 2o)
14 relen 8943 . . . . . 6 Rel ≈
1514brrelex1i 5732 . . . . 5 (𝑃 ≈ 2o𝑃 ∈ V)
16 difexg 5327 . . . . 5 (𝑃 ∈ V → (𝑃 ∖ {𝑍}) ∈ V)
17 uniexg 7729 . . . . 5 ((𝑃 ∖ {𝑍}) ∈ V → (𝑃 ∖ {𝑍}) ∈ V)
1813, 15, 16, 174syl 19 . . . 4 (((𝐷𝑉𝑃𝐷𝑃 ≈ 2o) ∧ 𝑍𝐷) → (𝑃 ∖ {𝑍}) ∈ V)
19 ifexg 4577 . . . 4 (( (𝑃 ∖ {𝑍}) ∈ V ∧ 𝑍𝐷) → if(𝑍𝑃, (𝑃 ∖ {𝑍}), 𝑍) ∈ V)
2018, 19sylancom 588 . . 3 (((𝐷𝑉𝑃𝐷𝑃 ≈ 2o) ∧ 𝑍𝐷) → if(𝑍𝑃, (𝑃 ∖ {𝑍}), 𝑍) ∈ V)
215, 11, 12, 20fvmptd3 7021 . 2 (((𝐷𝑉𝑃𝐷𝑃 ≈ 2o) ∧ 𝑍𝐷) → ((𝑧𝐷 ↦ if(𝑧𝑃, (𝑃 ∖ {𝑧}), 𝑧))‘𝑍) = if(𝑍𝑃, (𝑃 ∖ {𝑍}), 𝑍))
224, 21eqtrd 2772 1 (((𝐷𝑉𝑃𝐷𝑃 ≈ 2o) ∧ 𝑍𝐷) → ((𝑇𝑃)‘𝑍) = if(𝑍𝑃, (𝑃 ∖ {𝑍}), 𝑍))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  w3a 1087   = wceq 1541  wcel 2106  Vcvv 3474  cdif 3945  wss 3948  ifcif 4528  {csn 4628   cuni 4908   class class class wbr 5148  cmpt 5231  cfv 6543  2oc2o 8459  cen 8935  pmTrspcpmtr 19308
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7724
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-en 8939  df-pmtr 19309
This theorem is referenced by:  pmtrprfv  19320  pmtrprfv3  19321  pmtrmvd  19323  pmtrffv  19326
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