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Theorem pmtrfv 18222
 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 18221 . . . 4 ((𝐷𝑉𝑃𝐷𝑃 ≈ 2o) → (𝑇𝑃) = (𝑧𝐷 ↦ if(𝑧𝑃, (𝑃 ∖ {𝑧}), 𝑧)))
32fveq1d 6435 . . 3 ((𝐷𝑉𝑃𝐷𝑃 ≈ 2o) → ((𝑇𝑃)‘𝑍) = ((𝑧𝐷 ↦ if(𝑧𝑃, (𝑃 ∖ {𝑧}), 𝑧))‘𝑍))
43adantr 474 . 2 (((𝐷𝑉𝑃𝐷𝑃 ≈ 2o) ∧ 𝑍𝐷) → ((𝑇𝑃)‘𝑍) = ((𝑧𝐷 ↦ if(𝑧𝑃, (𝑃 ∖ {𝑧}), 𝑧))‘𝑍))
5 eqid 2825 . . 3 (𝑧𝐷 ↦ if(𝑧𝑃, (𝑃 ∖ {𝑧}), 𝑧)) = (𝑧𝐷 ↦ if(𝑧𝑃, (𝑃 ∖ {𝑧}), 𝑧))
6 eleq1 2894 . . . 4 (𝑧 = 𝑍 → (𝑧𝑃𝑍𝑃))
7 sneq 4407 . . . . . 6 (𝑧 = 𝑍 → {𝑧} = {𝑍})
87difeq2d 3955 . . . . 5 (𝑧 = 𝑍 → (𝑃 ∖ {𝑧}) = (𝑃 ∖ {𝑍}))
98unieqd 4668 . . . 4 (𝑧 = 𝑍 (𝑃 ∖ {𝑧}) = (𝑃 ∖ {𝑍}))
10 id 22 . . . 4 (𝑧 = 𝑍𝑧 = 𝑍)
116, 9, 10ifbieq12d 4333 . . 3 (𝑧 = 𝑍 → if(𝑧𝑃, (𝑃 ∖ {𝑧}), 𝑧) = if(𝑍𝑃, (𝑃 ∖ {𝑍}), 𝑍))
12 simpr 479 . . 3 (((𝐷𝑉𝑃𝐷𝑃 ≈ 2o) ∧ 𝑍𝐷) → 𝑍𝐷)
13 simpl3 1252 . . . . 5 (((𝐷𝑉𝑃𝐷𝑃 ≈ 2o) ∧ 𝑍𝐷) → 𝑃 ≈ 2o)
14 relen 8227 . . . . . 6 Rel ≈
1514brrelex1i 5393 . . . . 5 (𝑃 ≈ 2o𝑃 ∈ V)
16 difexg 5033 . . . . 5 (𝑃 ∈ V → (𝑃 ∖ {𝑍}) ∈ V)
17 uniexg 7215 . . . . 5 ((𝑃 ∖ {𝑍}) ∈ V → (𝑃 ∖ {𝑍}) ∈ V)
1813, 15, 16, 174syl 19 . . . 4 (((𝐷𝑉𝑃𝐷𝑃 ≈ 2o) ∧ 𝑍𝐷) → (𝑃 ∖ {𝑍}) ∈ V)
19 ifexg 4353 . . . 4 (( (𝑃 ∖ {𝑍}) ∈ V ∧ 𝑍𝐷) → if(𝑍𝑃, (𝑃 ∖ {𝑍}), 𝑍) ∈ V)
2018, 12, 19syl2anc 581 . . 3 (((𝐷𝑉𝑃𝐷𝑃 ≈ 2o) ∧ 𝑍𝐷) → if(𝑍𝑃, (𝑃 ∖ {𝑍}), 𝑍) ∈ V)
215, 11, 12, 20fvmptd3 6550 . 2 (((𝐷𝑉𝑃𝐷𝑃 ≈ 2o) ∧ 𝑍𝐷) → ((𝑧𝐷 ↦ if(𝑧𝑃, (𝑃 ∖ {𝑧}), 𝑧))‘𝑍) = if(𝑍𝑃, (𝑃 ∖ {𝑍}), 𝑍))
224, 21eqtrd 2861 1 (((𝐷𝑉𝑃𝐷𝑃 ≈ 2o) ∧ 𝑍𝐷) → ((𝑇𝑃)‘𝑍) = if(𝑍𝑃, (𝑃 ∖ {𝑍}), 𝑍))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 386   ∧ w3a 1113   = wceq 1658   ∈ wcel 2166  Vcvv 3414   ∖ cdif 3795   ⊆ wss 3798  ifcif 4306  {csn 4397  ∪ cuni 4658   class class class wbr 4873   ↦ cmpt 4952  ‘cfv 6123  2oc2o 7820   ≈ cen 8219  pmTrspcpmtr 18211 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-8 2168  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-13 2391  ax-ext 2803  ax-rep 4994  ax-sep 5005  ax-nul 5013  ax-pow 5065  ax-pr 5127  ax-un 7209 This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-3an 1115  df-tru 1662  df-ex 1881  df-nf 1885  df-sb 2070  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ne 3000  df-ral 3122  df-rex 3123  df-reu 3124  df-rab 3126  df-v 3416  df-sbc 3663  df-csb 3758  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-nul 4145  df-if 4307  df-pw 4380  df-sn 4398  df-pr 4400  df-op 4404  df-uni 4659  df-iun 4742  df-br 4874  df-opab 4936  df-mpt 4953  df-id 5250  df-xp 5348  df-rel 5349  df-cnv 5350  df-co 5351  df-dm 5352  df-rn 5353  df-res 5354  df-ima 5355  df-iota 6086  df-fun 6125  df-fn 6126  df-f 6127  df-f1 6128  df-fo 6129  df-f1o 6130  df-fv 6131  df-en 8223  df-pmtr 18212 This theorem is referenced by:  pmtrprfv  18223  pmtrprfv3  18224  pmtrmvd  18226  pmtrffv  18229
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