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Theorem pmtrfv 17918
 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 17917 . . . 4 ((𝐷𝑉𝑃𝐷𝑃 ≈ 2𝑜) → (𝑇𝑃) = (𝑧𝐷 ↦ if(𝑧𝑃, (𝑃 ∖ {𝑧}), 𝑧)))
32fveq1d 6231 . . 3 ((𝐷𝑉𝑃𝐷𝑃 ≈ 2𝑜) → ((𝑇𝑃)‘𝑍) = ((𝑧𝐷 ↦ if(𝑧𝑃, (𝑃 ∖ {𝑧}), 𝑧))‘𝑍))
43adantr 480 . 2 (((𝐷𝑉𝑃𝐷𝑃 ≈ 2𝑜) ∧ 𝑍𝐷) → ((𝑇𝑃)‘𝑍) = ((𝑧𝐷 ↦ if(𝑧𝑃, (𝑃 ∖ {𝑧}), 𝑧))‘𝑍))
5 simpr 476 . . 3 (((𝐷𝑉𝑃𝐷𝑃 ≈ 2𝑜) ∧ 𝑍𝐷) → 𝑍𝐷)
6 simpl3 1086 . . . . 5 (((𝐷𝑉𝑃𝐷𝑃 ≈ 2𝑜) ∧ 𝑍𝐷) → 𝑃 ≈ 2𝑜)
7 relen 8002 . . . . . 6 Rel ≈
87brrelexi 5192 . . . . 5 (𝑃 ≈ 2𝑜𝑃 ∈ V)
9 difexg 4841 . . . . 5 (𝑃 ∈ V → (𝑃 ∖ {𝑍}) ∈ V)
10 uniexg 6997 . . . . 5 ((𝑃 ∖ {𝑍}) ∈ V → (𝑃 ∖ {𝑍}) ∈ V)
116, 8, 9, 104syl 19 . . . 4 (((𝐷𝑉𝑃𝐷𝑃 ≈ 2𝑜) ∧ 𝑍𝐷) → (𝑃 ∖ {𝑍}) ∈ V)
12 ifexg 4190 . . . 4 (( (𝑃 ∖ {𝑍}) ∈ V ∧ 𝑍𝐷) → if(𝑍𝑃, (𝑃 ∖ {𝑍}), 𝑍) ∈ V)
1311, 5, 12syl2anc 694 . . 3 (((𝐷𝑉𝑃𝐷𝑃 ≈ 2𝑜) ∧ 𝑍𝐷) → if(𝑍𝑃, (𝑃 ∖ {𝑍}), 𝑍) ∈ V)
14 eleq1 2718 . . . . 5 (𝑧 = 𝑍 → (𝑧𝑃𝑍𝑃))
15 sneq 4220 . . . . . . 7 (𝑧 = 𝑍 → {𝑧} = {𝑍})
1615difeq2d 3761 . . . . . 6 (𝑧 = 𝑍 → (𝑃 ∖ {𝑧}) = (𝑃 ∖ {𝑍}))
1716unieqd 4478 . . . . 5 (𝑧 = 𝑍 (𝑃 ∖ {𝑧}) = (𝑃 ∖ {𝑍}))
18 id 22 . . . . 5 (𝑧 = 𝑍𝑧 = 𝑍)
1914, 17, 18ifbieq12d 4146 . . . 4 (𝑧 = 𝑍 → if(𝑧𝑃, (𝑃 ∖ {𝑧}), 𝑧) = if(𝑍𝑃, (𝑃 ∖ {𝑍}), 𝑍))
20 eqid 2651 . . . 4 (𝑧𝐷 ↦ if(𝑧𝑃, (𝑃 ∖ {𝑧}), 𝑧)) = (𝑧𝐷 ↦ if(𝑧𝑃, (𝑃 ∖ {𝑧}), 𝑧))
2119, 20fvmptg 6319 . . 3 ((𝑍𝐷 ∧ if(𝑍𝑃, (𝑃 ∖ {𝑍}), 𝑍) ∈ V) → ((𝑧𝐷 ↦ if(𝑧𝑃, (𝑃 ∖ {𝑧}), 𝑧))‘𝑍) = if(𝑍𝑃, (𝑃 ∖ {𝑍}), 𝑍))
225, 13, 21syl2anc 694 . 2 (((𝐷𝑉𝑃𝐷𝑃 ≈ 2𝑜) ∧ 𝑍𝐷) → ((𝑧𝐷 ↦ if(𝑧𝑃, (𝑃 ∖ {𝑧}), 𝑧))‘𝑍) = if(𝑍𝑃, (𝑃 ∖ {𝑍}), 𝑍))
234, 22eqtrd 2685 1 (((𝐷𝑉𝑃𝐷𝑃 ≈ 2𝑜) ∧ 𝑍𝐷) → ((𝑇𝑃)‘𝑍) = if(𝑍𝑃, (𝑃 ∖ {𝑍}), 𝑍))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   ∧ w3a 1054   = wceq 1523   ∈ wcel 2030  Vcvv 3231   ∖ cdif 3604   ⊆ wss 3607  ifcif 4119  {csn 4210  ∪ cuni 4468   class class class wbr 4685   ↦ cmpt 4762  ‘cfv 5926  2𝑜c2o 7599   ≈ cen 7994  pmTrspcpmtr 17907 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-en 7998  df-pmtr 17908 This theorem is referenced by:  pmtrprfv  17919  pmtrprfv3  17920  pmtrmvd  17922  pmtrffv  17925
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