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Theorem pmtrdifellem2 19495
Description: Lemma 2 for pmtrdifel 19498. (Contributed by AV, 15-Jan-2019.)
Hypotheses
Ref Expression
pmtrdifel.t 𝑇 = ran (pmTrsp‘(𝑁 ∖ {𝐾}))
pmtrdifel.r 𝑅 = ran (pmTrsp‘𝑁)
pmtrdifel.0 𝑆 = ((pmTrsp‘𝑁)‘dom (𝑄 ∖ I ))
Assertion
Ref Expression
pmtrdifellem2 (𝑄𝑇 → dom (𝑆 ∖ I ) = dom (𝑄 ∖ I ))

Proof of Theorem pmtrdifellem2
StepHypRef Expression
1 pmtrdifel.0 . . . 4 𝑆 = ((pmTrsp‘𝑁)‘dom (𝑄 ∖ I ))
21difeq1i 4122 . . 3 (𝑆 ∖ I ) = (((pmTrsp‘𝑁)‘dom (𝑄 ∖ I )) ∖ I )
32dmeqi 5915 . 2 dom (𝑆 ∖ I ) = dom (((pmTrsp‘𝑁)‘dom (𝑄 ∖ I )) ∖ I )
4 eqid 2737 . . . . 5 (pmTrsp‘(𝑁 ∖ {𝐾})) = (pmTrsp‘(𝑁 ∖ {𝐾}))
5 pmtrdifel.t . . . . 5 𝑇 = ran (pmTrsp‘(𝑁 ∖ {𝐾}))
64, 5pmtrfb 19483 . . . 4 (𝑄𝑇 ↔ ((𝑁 ∖ {𝐾}) ∈ V ∧ 𝑄:(𝑁 ∖ {𝐾})–1-1-onto→(𝑁 ∖ {𝐾}) ∧ dom (𝑄 ∖ I ) ≈ 2o))
7 difsnexi 7781 . . . . 5 ((𝑁 ∖ {𝐾}) ∈ V → 𝑁 ∈ V)
8 f1of 6848 . . . . . 6 (𝑄:(𝑁 ∖ {𝐾})–1-1-onto→(𝑁 ∖ {𝐾}) → 𝑄:(𝑁 ∖ {𝐾})⟶(𝑁 ∖ {𝐾}))
9 fdm 6745 . . . . . 6 (𝑄:(𝑁 ∖ {𝐾})⟶(𝑁 ∖ {𝐾}) → dom 𝑄 = (𝑁 ∖ {𝐾}))
10 difssd 4137 . . . . . . . 8 (dom 𝑄 = (𝑁 ∖ {𝐾}) → (𝑄 ∖ I ) ⊆ 𝑄)
11 dmss 5913 . . . . . . . 8 ((𝑄 ∖ I ) ⊆ 𝑄 → dom (𝑄 ∖ I ) ⊆ dom 𝑄)
1210, 11syl 17 . . . . . . 7 (dom 𝑄 = (𝑁 ∖ {𝐾}) → dom (𝑄 ∖ I ) ⊆ dom 𝑄)
13 difssd 4137 . . . . . . . 8 (dom 𝑄 = (𝑁 ∖ {𝐾}) → (𝑁 ∖ {𝐾}) ⊆ 𝑁)
14 sseq1 4009 . . . . . . . 8 (dom 𝑄 = (𝑁 ∖ {𝐾}) → (dom 𝑄𝑁 ↔ (𝑁 ∖ {𝐾}) ⊆ 𝑁))
1513, 14mpbird 257 . . . . . . 7 (dom 𝑄 = (𝑁 ∖ {𝐾}) → dom 𝑄𝑁)
1612, 15sstrd 3994 . . . . . 6 (dom 𝑄 = (𝑁 ∖ {𝐾}) → dom (𝑄 ∖ I ) ⊆ 𝑁)
178, 9, 163syl 18 . . . . 5 (𝑄:(𝑁 ∖ {𝐾})–1-1-onto→(𝑁 ∖ {𝐾}) → dom (𝑄 ∖ I ) ⊆ 𝑁)
18 id 22 . . . . 5 (dom (𝑄 ∖ I ) ≈ 2o → dom (𝑄 ∖ I ) ≈ 2o)
197, 17, 183anim123i 1152 . . . 4 (((𝑁 ∖ {𝐾}) ∈ V ∧ 𝑄:(𝑁 ∖ {𝐾})–1-1-onto→(𝑁 ∖ {𝐾}) ∧ dom (𝑄 ∖ I ) ≈ 2o) → (𝑁 ∈ V ∧ dom (𝑄 ∖ I ) ⊆ 𝑁 ∧ dom (𝑄 ∖ I ) ≈ 2o))
206, 19sylbi 217 . . 3 (𝑄𝑇 → (𝑁 ∈ V ∧ dom (𝑄 ∖ I ) ⊆ 𝑁 ∧ dom (𝑄 ∖ I ) ≈ 2o))
21 eqid 2737 . . . 4 (pmTrsp‘𝑁) = (pmTrsp‘𝑁)
2221pmtrmvd 19474 . . 3 ((𝑁 ∈ V ∧ dom (𝑄 ∖ I ) ⊆ 𝑁 ∧ dom (𝑄 ∖ I ) ≈ 2o) → dom (((pmTrsp‘𝑁)‘dom (𝑄 ∖ I )) ∖ I ) = dom (𝑄 ∖ I ))
2320, 22syl 17 . 2 (𝑄𝑇 → dom (((pmTrsp‘𝑁)‘dom (𝑄 ∖ I )) ∖ I ) = dom (𝑄 ∖ I ))
243, 23eqtrid 2789 1 (𝑄𝑇 → dom (𝑆 ∖ I ) = dom (𝑄 ∖ I ))
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1087   = wceq 1540  wcel 2108  Vcvv 3480  cdif 3948  wss 3951  {csn 4626   class class class wbr 5143   I cid 5577  dom cdm 5685  ran crn 5686  wf 6557  1-1-ontowf1o 6560  cfv 6561  2oc2o 8500  cen 8982  pmTrspcpmtr 19459
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-om 7888  df-1o 8506  df-2o 8507  df-er 8745  df-en 8986  df-dom 8987  df-sdom 8988  df-fin 8989  df-pmtr 19460
This theorem is referenced by:  pmtrdifellem3  19496  pmtrdifellem4  19497
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