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Theorem pmtrfv 19242
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 19241 . . . 4 ((𝐷𝑉𝑃𝐷𝑃 ≈ 2o) → (𝑇𝑃) = (𝑧𝐷 ↦ if(𝑧𝑃, (𝑃 ∖ {𝑧}), 𝑧)))
32fveq1d 6848 . . 3 ((𝐷𝑉𝑃𝐷𝑃 ≈ 2o) → ((𝑇𝑃)‘𝑍) = ((𝑧𝐷 ↦ if(𝑧𝑃, (𝑃 ∖ {𝑧}), 𝑧))‘𝑍))
43adantr 482 . 2 (((𝐷𝑉𝑃𝐷𝑃 ≈ 2o) ∧ 𝑍𝐷) → ((𝑇𝑃)‘𝑍) = ((𝑧𝐷 ↦ if(𝑧𝑃, (𝑃 ∖ {𝑧}), 𝑧))‘𝑍))
5 eqid 2733 . . 3 (𝑧𝐷 ↦ if(𝑧𝑃, (𝑃 ∖ {𝑧}), 𝑧)) = (𝑧𝐷 ↦ if(𝑧𝑃, (𝑃 ∖ {𝑧}), 𝑧))
6 eleq1 2822 . . . 4 (𝑧 = 𝑍 → (𝑧𝑃𝑍𝑃))
7 sneq 4600 . . . . . 6 (𝑧 = 𝑍 → {𝑧} = {𝑍})
87difeq2d 4086 . . . . 5 (𝑧 = 𝑍 → (𝑃 ∖ {𝑧}) = (𝑃 ∖ {𝑍}))
98unieqd 4883 . . . 4 (𝑧 = 𝑍 (𝑃 ∖ {𝑧}) = (𝑃 ∖ {𝑍}))
10 id 22 . . . 4 (𝑧 = 𝑍𝑧 = 𝑍)
116, 9, 10ifbieq12d 4518 . . 3 (𝑧 = 𝑍 → if(𝑧𝑃, (𝑃 ∖ {𝑧}), 𝑧) = if(𝑍𝑃, (𝑃 ∖ {𝑍}), 𝑍))
12 simpr 486 . . 3 (((𝐷𝑉𝑃𝐷𝑃 ≈ 2o) ∧ 𝑍𝐷) → 𝑍𝐷)
13 simpl3 1194 . . . . 5 (((𝐷𝑉𝑃𝐷𝑃 ≈ 2o) ∧ 𝑍𝐷) → 𝑃 ≈ 2o)
14 relen 8894 . . . . . 6 Rel ≈
1514brrelex1i 5692 . . . . 5 (𝑃 ≈ 2o𝑃 ∈ V)
16 difexg 5288 . . . . 5 (𝑃 ∈ V → (𝑃 ∖ {𝑍}) ∈ V)
17 uniexg 7681 . . . . 5 ((𝑃 ∖ {𝑍}) ∈ V → (𝑃 ∖ {𝑍}) ∈ V)
1813, 15, 16, 174syl 19 . . . 4 (((𝐷𝑉𝑃𝐷𝑃 ≈ 2o) ∧ 𝑍𝐷) → (𝑃 ∖ {𝑍}) ∈ V)
19 ifexg 4539 . . . 4 (( (𝑃 ∖ {𝑍}) ∈ V ∧ 𝑍𝐷) → if(𝑍𝑃, (𝑃 ∖ {𝑍}), 𝑍) ∈ V)
2018, 19sylancom 589 . . 3 (((𝐷𝑉𝑃𝐷𝑃 ≈ 2o) ∧ 𝑍𝐷) → if(𝑍𝑃, (𝑃 ∖ {𝑍}), 𝑍) ∈ V)
215, 11, 12, 20fvmptd3 6975 . 2 (((𝐷𝑉𝑃𝐷𝑃 ≈ 2o) ∧ 𝑍𝐷) → ((𝑧𝐷 ↦ if(𝑧𝑃, (𝑃 ∖ {𝑧}), 𝑧))‘𝑍) = if(𝑍𝑃, (𝑃 ∖ {𝑍}), 𝑍))
224, 21eqtrd 2773 1 (((𝐷𝑉𝑃𝐷𝑃 ≈ 2o) ∧ 𝑍𝐷) → ((𝑇𝑃)‘𝑍) = if(𝑍𝑃, (𝑃 ∖ {𝑍}), 𝑍))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397  w3a 1088   = wceq 1542  wcel 2107  Vcvv 3447  cdif 3911  wss 3914  ifcif 4490  {csn 4590   cuni 4869   class class class wbr 5109  cmpt 5192  cfv 6500  2oc2o 8410  cen 8886  pmTrspcpmtr 19231
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5246  ax-sep 5260  ax-nul 5267  ax-pow 5324  ax-pr 5388  ax-un 7676
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3353  df-rab 3407  df-v 3449  df-sbc 3744  df-csb 3860  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4287  df-if 4491  df-pw 4566  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4870  df-iun 4960  df-br 5110  df-opab 5172  df-mpt 5193  df-id 5535  df-xp 5643  df-rel 5644  df-cnv 5645  df-co 5646  df-dm 5647  df-rn 5648  df-res 5649  df-ima 5650  df-iota 6452  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-en 8890  df-pmtr 19232
This theorem is referenced by:  pmtrprfv  19243  pmtrprfv3  19244  pmtrmvd  19246  pmtrffv  19249
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