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Theorem pmtrdifel 19449
Description: A transposition of elements of a set without a special element corresponds to a transposition of elements of the set. (Contributed by AV, 15-Jan-2019.)
Hypotheses
Ref Expression
pmtrdifel.t 𝑇 = ran (pmTrsp‘(𝑁 ∖ {𝐾}))
pmtrdifel.r 𝑅 = ran (pmTrsp‘𝑁)
Assertion
Ref Expression
pmtrdifel 𝑡𝑇𝑟𝑅𝑥 ∈ (𝑁 ∖ {𝐾})(𝑡𝑥) = (𝑟𝑥)
Distinct variable groups:   𝑡,𝑟,𝑥   𝐾,𝑟   𝑁,𝑟,𝑥   𝑅,𝑟   𝑥,𝑇
Allowed substitution hints:   𝑅(𝑥,𝑡)   𝑇(𝑡,𝑟)   𝐾(𝑥,𝑡)   𝑁(𝑡)

Proof of Theorem pmtrdifel
StepHypRef Expression
1 pmtrdifel.t . . . 4 𝑇 = ran (pmTrsp‘(𝑁 ∖ {𝐾}))
2 pmtrdifel.r . . . 4 𝑅 = ran (pmTrsp‘𝑁)
3 eqid 2728 . . . 4 ((pmTrsp‘𝑁)‘dom (𝑡 ∖ I )) = ((pmTrsp‘𝑁)‘dom (𝑡 ∖ I ))
41, 2, 3pmtrdifellem1 19445 . . 3 (𝑡𝑇 → ((pmTrsp‘𝑁)‘dom (𝑡 ∖ I )) ∈ 𝑅)
51, 2, 3pmtrdifellem3 19447 . . 3 (𝑡𝑇 → ∀𝑥 ∈ (𝑁 ∖ {𝐾})(𝑡𝑥) = (((pmTrsp‘𝑁)‘dom (𝑡 ∖ I ))‘𝑥))
6 fveq1 6901 . . . . . 6 (𝑟 = ((pmTrsp‘𝑁)‘dom (𝑡 ∖ I )) → (𝑟𝑥) = (((pmTrsp‘𝑁)‘dom (𝑡 ∖ I ))‘𝑥))
76eqeq2d 2739 . . . . 5 (𝑟 = ((pmTrsp‘𝑁)‘dom (𝑡 ∖ I )) → ((𝑡𝑥) = (𝑟𝑥) ↔ (𝑡𝑥) = (((pmTrsp‘𝑁)‘dom (𝑡 ∖ I ))‘𝑥)))
87ralbidv 3175 . . . 4 (𝑟 = ((pmTrsp‘𝑁)‘dom (𝑡 ∖ I )) → (∀𝑥 ∈ (𝑁 ∖ {𝐾})(𝑡𝑥) = (𝑟𝑥) ↔ ∀𝑥 ∈ (𝑁 ∖ {𝐾})(𝑡𝑥) = (((pmTrsp‘𝑁)‘dom (𝑡 ∖ I ))‘𝑥)))
98rspcev 3611 . . 3 ((((pmTrsp‘𝑁)‘dom (𝑡 ∖ I )) ∈ 𝑅 ∧ ∀𝑥 ∈ (𝑁 ∖ {𝐾})(𝑡𝑥) = (((pmTrsp‘𝑁)‘dom (𝑡 ∖ I ))‘𝑥)) → ∃𝑟𝑅𝑥 ∈ (𝑁 ∖ {𝐾})(𝑡𝑥) = (𝑟𝑥))
104, 5, 9syl2anc 582 . 2 (𝑡𝑇 → ∃𝑟𝑅𝑥 ∈ (𝑁 ∖ {𝐾})(𝑡𝑥) = (𝑟𝑥))
1110rgen 3060 1 𝑡𝑇𝑟𝑅𝑥 ∈ (𝑁 ∖ {𝐾})(𝑡𝑥) = (𝑟𝑥)
Colors of variables: wff setvar class
Syntax hints:   = wceq 1533  wcel 2098  wral 3058  wrex 3067  cdif 3946  {csn 4632   I cid 5579  dom cdm 5682  ran crn 5683  cfv 6553  pmTrspcpmtr 19410
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2699  ax-rep 5289  ax-sep 5303  ax-nul 5310  ax-pow 5369  ax-pr 5433  ax-un 7748
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-reu 3375  df-rab 3431  df-v 3475  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4327  df-if 4533  df-pw 4608  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-iun 5002  df-br 5153  df-opab 5215  df-mpt 5236  df-tr 5270  df-id 5580  df-eprel 5586  df-po 5594  df-so 5595  df-fr 5637  df-we 5639  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-ord 6377  df-on 6378  df-lim 6379  df-suc 6380  df-iota 6505  df-fun 6555  df-fn 6556  df-f 6557  df-f1 6558  df-fo 6559  df-f1o 6560  df-fv 6561  df-om 7879  df-1o 8495  df-2o 8496  df-er 8733  df-en 8973  df-dom 8974  df-sdom 8975  df-fin 8976  df-pmtr 19411
This theorem is referenced by: (None)
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