Theorem pmtrdifel 19088

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	⊢ ∀𝑡 ∈ 𝑇 ∃𝑟 ∈ 𝑅 ∀𝑥 ∈ (𝑁 ∖ {𝐾})(𝑡‘𝑥) = (𝑟‘𝑥)

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

Step	Hyp	Ref	Expression
1		pmtrdifel.t	. . . 4 ⊢ 𝑇 = ran (pmTrsp‘(𝑁 ∖ {𝐾}))
2		pmtrdifel.r	. . . 4 ⊢ 𝑅 = ran (pmTrsp‘𝑁)
3		eqid 2738	. . . 4 ⊢ ((pmTrsp‘𝑁)‘dom (𝑡 ∖ I )) = ((pmTrsp‘𝑁)‘dom (𝑡 ∖ I ))
4	1, 2, 3	pmtrdifellem1 19084	. . 3 ⊢ (𝑡 ∈ 𝑇 → ((pmTrsp‘𝑁)‘dom (𝑡 ∖ I )) ∈ 𝑅)
5	1, 2, 3	pmtrdifellem3 19086	. . 3 ⊢ (𝑡 ∈ 𝑇 → ∀𝑥 ∈ (𝑁 ∖ {𝐾})(𝑡‘𝑥) = (((pmTrsp‘𝑁)‘dom (𝑡 ∖ I ))‘𝑥))
6		fveq1 6773	. . . . . 6 ⊢ (𝑟 = ((pmTrsp‘𝑁)‘dom (𝑡 ∖ I )) → (𝑟‘𝑥) = (((pmTrsp‘𝑁)‘dom (𝑡 ∖ I ))‘𝑥))
7	6	eqeq2d 2749	. . . . 5 ⊢ (𝑟 = ((pmTrsp‘𝑁)‘dom (𝑡 ∖ I )) → ((𝑡‘𝑥) = (𝑟‘𝑥) ↔ (𝑡‘𝑥) = (((pmTrsp‘𝑁)‘dom (𝑡 ∖ I ))‘𝑥)))
8	7	ralbidv 3112	. . . 4 ⊢ (𝑟 = ((pmTrsp‘𝑁)‘dom (𝑡 ∖ I )) → (∀𝑥 ∈ (𝑁 ∖ {𝐾})(𝑡‘𝑥) = (𝑟‘𝑥) ↔ ∀𝑥 ∈ (𝑁 ∖ {𝐾})(𝑡‘𝑥) = (((pmTrsp‘𝑁)‘dom (𝑡 ∖ I ))‘𝑥)))
9	8	rspcev 3561	. . 3 ⊢ ((((pmTrsp‘𝑁)‘dom (𝑡 ∖ I )) ∈ 𝑅 ∧ ∀𝑥 ∈ (𝑁 ∖ {𝐾})(𝑡‘𝑥) = (((pmTrsp‘𝑁)‘dom (𝑡 ∖ I ))‘𝑥)) → ∃𝑟 ∈ 𝑅 ∀𝑥 ∈ (𝑁 ∖ {𝐾})(𝑡‘𝑥) = (𝑟‘𝑥))
10	4, 5, 9	syl2anc 584	. 2 ⊢ (𝑡 ∈ 𝑇 → ∃𝑟 ∈ 𝑅 ∀𝑥 ∈ (𝑁 ∖ {𝐾})(𝑡‘𝑥) = (𝑟‘𝑥))
11	10	rgen 3074	1 ⊢ ∀𝑡 ∈ 𝑇 ∃𝑟 ∈ 𝑅 ∀𝑥 ∈ (𝑁 ∖ {𝐾})(𝑡‘𝑥) = (𝑟‘𝑥)

Syntax hints:   = wceq 1539   ∈ wcel 2106  ∀wral 3064  ∃wrex 3065   ∖ cdif 3884  {csn 4561   I cid 5488  dom cdm 5589  ran crn 5590  ‘cfv 6433  pmTrspcpmtr 19049

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-rep 5209  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-pss 3906  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-tr 5192  df-id 5489  df-eprel 5495  df-po 5503  df-so 5504  df-fr 5544  df-we 5546  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-ord 6269  df-on 6270  df-lim 6271  df-suc 6272  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-om 7713  df-1o 8297  df-2o 8298  df-er 8498  df-en 8734  df-dom 8735  df-sdom 8736  df-fin 8737  df-pmtr 19050

This theorem is referenced by: (None)