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Theorem pmtrdifel 18902
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 2739 . . . 4 ((pmTrsp‘𝑁)‘dom (𝑡 ∖ I )) = ((pmTrsp‘𝑁)‘dom (𝑡 ∖ I ))
41, 2, 3pmtrdifellem1 18898 . . 3 (𝑡𝑇 → ((pmTrsp‘𝑁)‘dom (𝑡 ∖ I )) ∈ 𝑅)
51, 2, 3pmtrdifellem3 18900 . . 3 (𝑡𝑇 → ∀𝑥 ∈ (𝑁 ∖ {𝐾})(𝑡𝑥) = (((pmTrsp‘𝑁)‘dom (𝑡 ∖ I ))‘𝑥))
6 fveq1 6737 . . . . . 6 (𝑟 = ((pmTrsp‘𝑁)‘dom (𝑡 ∖ I )) → (𝑟𝑥) = (((pmTrsp‘𝑁)‘dom (𝑡 ∖ I ))‘𝑥))
76eqeq2d 2750 . . . . 5 (𝑟 = ((pmTrsp‘𝑁)‘dom (𝑡 ∖ I )) → ((𝑡𝑥) = (𝑟𝑥) ↔ (𝑡𝑥) = (((pmTrsp‘𝑁)‘dom (𝑡 ∖ I ))‘𝑥)))
87ralbidv 3120 . . . 4 (𝑟 = ((pmTrsp‘𝑁)‘dom (𝑡 ∖ I )) → (∀𝑥 ∈ (𝑁 ∖ {𝐾})(𝑡𝑥) = (𝑟𝑥) ↔ ∀𝑥 ∈ (𝑁 ∖ {𝐾})(𝑡𝑥) = (((pmTrsp‘𝑁)‘dom (𝑡 ∖ I ))‘𝑥)))
98rspcev 3551 . . 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 587 . 2 (𝑡𝑇 → ∃𝑟𝑅𝑥 ∈ (𝑁 ∖ {𝐾})(𝑡𝑥) = (𝑟𝑥))
1110rgen 3073 1 𝑡𝑇𝑟𝑅𝑥 ∈ (𝑁 ∖ {𝐾})(𝑡𝑥) = (𝑟𝑥)
Colors of variables: wff setvar class
Syntax hints:   = wceq 1543  wcel 2112  wral 3063  wrex 3064  cdif 3879  {csn 4557   I cid 5470  dom cdm 5568  ran crn 5569  cfv 6400  pmTrspcpmtr 18863
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2160  ax-12 2177  ax-ext 2710  ax-rep 5195  ax-sep 5208  ax-nul 5215  ax-pow 5274  ax-pr 5338  ax-un 7544
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3or 1090  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2073  df-mo 2541  df-eu 2570  df-clab 2717  df-cleq 2731  df-clel 2818  df-nfc 2888  df-ne 2943  df-ral 3068  df-rex 3069  df-reu 3070  df-rab 3072  df-v 3424  df-sbc 3711  df-csb 3828  df-dif 3885  df-un 3887  df-in 3889  df-ss 3899  df-pss 3901  df-nul 4254  df-if 4456  df-pw 4531  df-sn 4558  df-pr 4560  df-tp 4562  df-op 4564  df-uni 4836  df-iun 4922  df-br 5070  df-opab 5132  df-mpt 5152  df-tr 5178  df-id 5471  df-eprel 5477  df-po 5485  df-so 5486  df-fr 5526  df-we 5528  df-xp 5574  df-rel 5575  df-cnv 5576  df-co 5577  df-dm 5578  df-rn 5579  df-res 5580  df-ima 5581  df-ord 6236  df-on 6237  df-lim 6238  df-suc 6239  df-iota 6358  df-fun 6402  df-fn 6403  df-f 6404  df-f1 6405  df-fo 6406  df-f1o 6407  df-fv 6408  df-om 7666  df-1o 8225  df-2o 8226  df-er 8414  df-en 8650  df-dom 8651  df-sdom 8652  df-fin 8653  df-pmtr 18864
This theorem is referenced by: (None)
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