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Theorem pmtrdifel 19400
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 2726 . . . 4 ((pmTrsp‘𝑁)‘dom (𝑡 ∖ I )) = ((pmTrsp‘𝑁)‘dom (𝑡 ∖ I ))
41, 2, 3pmtrdifellem1 19396 . . 3 (𝑡𝑇 → ((pmTrsp‘𝑁)‘dom (𝑡 ∖ I )) ∈ 𝑅)
51, 2, 3pmtrdifellem3 19398 . . 3 (𝑡𝑇 → ∀𝑥 ∈ (𝑁 ∖ {𝐾})(𝑡𝑥) = (((pmTrsp‘𝑁)‘dom (𝑡 ∖ I ))‘𝑥))
6 fveq1 6884 . . . . . 6 (𝑟 = ((pmTrsp‘𝑁)‘dom (𝑡 ∖ I )) → (𝑟𝑥) = (((pmTrsp‘𝑁)‘dom (𝑡 ∖ I ))‘𝑥))
76eqeq2d 2737 . . . . 5 (𝑟 = ((pmTrsp‘𝑁)‘dom (𝑡 ∖ I )) → ((𝑡𝑥) = (𝑟𝑥) ↔ (𝑡𝑥) = (((pmTrsp‘𝑁)‘dom (𝑡 ∖ I ))‘𝑥)))
87ralbidv 3171 . . . 4 (𝑟 = ((pmTrsp‘𝑁)‘dom (𝑡 ∖ I )) → (∀𝑥 ∈ (𝑁 ∖ {𝐾})(𝑡𝑥) = (𝑟𝑥) ↔ ∀𝑥 ∈ (𝑁 ∖ {𝐾})(𝑡𝑥) = (((pmTrsp‘𝑁)‘dom (𝑡 ∖ I ))‘𝑥)))
98rspcev 3606 . . 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 583 . 2 (𝑡𝑇 → ∃𝑟𝑅𝑥 ∈ (𝑁 ∖ {𝐾})(𝑡𝑥) = (𝑟𝑥))
1110rgen 3057 1 𝑡𝑇𝑟𝑅𝑥 ∈ (𝑁 ∖ {𝐾})(𝑡𝑥) = (𝑟𝑥)
Colors of variables: wff setvar class
Syntax hints:   = wceq 1533  wcel 2098  wral 3055  wrex 3064  cdif 3940  {csn 4623   I cid 5566  dom cdm 5669  ran crn 5670  cfv 6537  pmTrspcpmtr 19361
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 2163  ax-ext 2697  ax-rep 5278  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7722
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-reu 3371  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-pss 3962  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-tr 5259  df-id 5567  df-eprel 5573  df-po 5581  df-so 5582  df-fr 5624  df-we 5626  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-ord 6361  df-on 6362  df-lim 6363  df-suc 6364  df-iota 6489  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-om 7853  df-1o 8467  df-2o 8468  df-er 8705  df-en 8942  df-dom 8943  df-sdom 8944  df-fin 8945  df-pmtr 19362
This theorem is referenced by: (None)
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