Theorem pmtrdifel 19387

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 2731	. . . 4 ⊢ ((pmTrsp‘𝑁)‘dom (𝑡 ∖ I )) = ((pmTrsp‘𝑁)‘dom (𝑡 ∖ I ))
4	1, 2, 3	pmtrdifellem1 19383	. . 3 ⊢ (𝑡 ∈ 𝑇 → ((pmTrsp‘𝑁)‘dom (𝑡 ∖ I )) ∈ 𝑅)
5	1, 2, 3	pmtrdifellem3 19385	. . 3 ⊢ (𝑡 ∈ 𝑇 → ∀𝑥 ∈ (𝑁 ∖ {𝐾})(𝑡‘𝑥) = (((pmTrsp‘𝑁)‘dom (𝑡 ∖ I ))‘𝑥))
6		fveq1 6816	. . . . . 6 ⊢ (𝑟 = ((pmTrsp‘𝑁)‘dom (𝑡 ∖ I )) → (𝑟‘𝑥) = (((pmTrsp‘𝑁)‘dom (𝑡 ∖ I ))‘𝑥))
7	6	eqeq2d 2742	. . . . 5 ⊢ (𝑟 = ((pmTrsp‘𝑁)‘dom (𝑡 ∖ I )) → ((𝑡‘𝑥) = (𝑟‘𝑥) ↔ (𝑡‘𝑥) = (((pmTrsp‘𝑁)‘dom (𝑡 ∖ I ))‘𝑥)))
8	7	ralbidv 3155	. . . 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 3572	. . 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 3049	1 ⊢ ∀𝑡 ∈ 𝑇 ∃𝑟 ∈ 𝑅 ∀𝑥 ∈ (𝑁 ∖ {𝐾})(𝑡‘𝑥) = (𝑟‘𝑥)

Syntax hints:   = wceq 1541   ∈ wcel 2111  ∀wral 3047  ∃wrex 3056   ∖ cdif 3894  {csn 4571   I cid 5505  dom cdm 5611  ran crn 5612  ‘cfv 6476  pmTrspcpmtr 19348

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-rep 5212  ax-sep 5229  ax-nul 5239  ax-pow 5298  ax-pr 5365  ax-un 7663

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-reu 3347  df-rab 3396  df-v 3438  df-sbc 3737  df-csb 3846  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-pss 3917  df-nul 4279  df-if 4471  df-pw 4547  df-sn 4572  df-pr 4574  df-op 4578  df-uni 4855  df-iun 4938  df-br 5087  df-opab 5149  df-mpt 5168  df-tr 5194  df-id 5506  df-eprel 5511  df-po 5519  df-so 5520  df-fr 5564  df-we 5566  df-xp 5617  df-rel 5618  df-cnv 5619  df-co 5620  df-dm 5621  df-rn 5622  df-res 5623  df-ima 5624  df-ord 6304  df-on 6305  df-lim 6306  df-suc 6307  df-iota 6432  df-fun 6478  df-fn 6479  df-f 6480  df-f1 6481  df-fo 6482  df-f1o 6483  df-fv 6484  df-om 7792  df-1o 8380  df-2o 8381  df-er 8617  df-en 8865  df-dom 8866  df-sdom 8867  df-fin 8868  df-pmtr 19349

This theorem is referenced by: (None)