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Theorem pmtrfval 27370
 Description: The function generating transpositions on a set. (Contributed by Stefan O'Rear, 16-Aug-2015.)
Hypothesis
Ref Expression
pmtrfval.t pmTrsp
Assertion
Ref Expression
pmtrfval
Distinct variable groups:   ,,,   ,,,   ,
Allowed substitution hints:   (,)

Proof of Theorem pmtrfval
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 pmtrfval.t . 2 pmTrsp
2 elex 2964 . . 3
3 pweq 3802 . . . . . 6
4 rabeq 2950 . . . . . 6
53, 4syl 16 . . . . 5
6 mpteq1 4289 . . . . 5
75, 6mpteq12dv 4287 . . . 4
8 df-pmtr 27362 . . . 4 pmTrsp
9 vex 2959 . . . . . . 7
109pwex 4382 . . . . . 6
1110rabex 4354 . . . . 5
1211mptex 5966 . . . 4
137, 8, 12fvmpt3i 5809 . . 3 pmTrsp
142, 13syl 16 . 2 pmTrsp
151, 14syl5eq 2480 1
 Colors of variables: wff set class Syntax hints:   wi 4   wceq 1652   wcel 1725  crab 2709  cvv 2956   cdif 3317  cif 3739  cpw 3799  csn 3814  cuni 4015   class class class wbr 4212   cmpt 4266  cfv 5454  c2o 6718   cen 7106  pmTrspcpmtr 27361 This theorem is referenced by:  pmtrval  27371  pmtrrn  27376  pmtrfrn  27377 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403 This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-reu 2712  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-pmtr 27362
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