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Theorem pmtrdifellem2 18534
Description: Lemma 2 for pmtrdifel 18537. (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 4092 . . 3 (𝑆 ∖ I ) = (((pmTrsp‘𝑁)‘dom (𝑄 ∖ I )) ∖ I )
32dmeqi 5766 . 2 dom (𝑆 ∖ I ) = dom (((pmTrsp‘𝑁)‘dom (𝑄 ∖ I )) ∖ I )
4 eqid 2818 . . . . 5 (pmTrsp‘(𝑁 ∖ {𝐾})) = (pmTrsp‘(𝑁 ∖ {𝐾}))
5 pmtrdifel.t . . . . 5 𝑇 = ran (pmTrsp‘(𝑁 ∖ {𝐾}))
64, 5pmtrfb 18522 . . . 4 (𝑄𝑇 ↔ ((𝑁 ∖ {𝐾}) ∈ V ∧ 𝑄:(𝑁 ∖ {𝐾})–1-1-onto→(𝑁 ∖ {𝐾}) ∧ dom (𝑄 ∖ I ) ≈ 2o))
7 difsnexi 7472 . . . . 5 ((𝑁 ∖ {𝐾}) ∈ V → 𝑁 ∈ V)
8 f1of 6608 . . . . . 6 (𝑄:(𝑁 ∖ {𝐾})–1-1-onto→(𝑁 ∖ {𝐾}) → 𝑄:(𝑁 ∖ {𝐾})⟶(𝑁 ∖ {𝐾}))
9 fdm 6515 . . . . . 6 (𝑄:(𝑁 ∖ {𝐾})⟶(𝑁 ∖ {𝐾}) → dom 𝑄 = (𝑁 ∖ {𝐾}))
10 difssd 4106 . . . . . . . 8 (dom 𝑄 = (𝑁 ∖ {𝐾}) → (𝑄 ∖ I ) ⊆ 𝑄)
11 dmss 5764 . . . . . . . 8 ((𝑄 ∖ I ) ⊆ 𝑄 → dom (𝑄 ∖ I ) ⊆ dom 𝑄)
1210, 11syl 17 . . . . . . 7 (dom 𝑄 = (𝑁 ∖ {𝐾}) → dom (𝑄 ∖ I ) ⊆ dom 𝑄)
13 difssd 4106 . . . . . . . 8 (dom 𝑄 = (𝑁 ∖ {𝐾}) → (𝑁 ∖ {𝐾}) ⊆ 𝑁)
14 sseq1 3989 . . . . . . . 8 (dom 𝑄 = (𝑁 ∖ {𝐾}) → (dom 𝑄𝑁 ↔ (𝑁 ∖ {𝐾}) ⊆ 𝑁))
1513, 14mpbird 258 . . . . . . 7 (dom 𝑄 = (𝑁 ∖ {𝐾}) → dom 𝑄𝑁)
1612, 15sstrd 3974 . . . . . 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 1143 . . . 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 218 . . 3 (𝑄𝑇 → (𝑁 ∈ V ∧ dom (𝑄 ∖ I ) ⊆ 𝑁 ∧ dom (𝑄 ∖ I ) ≈ 2o))
21 eqid 2818 . . . 4 (pmTrsp‘𝑁) = (pmTrsp‘𝑁)
2221pmtrmvd 18513 . . 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, 23syl5eq 2865 1 (𝑄𝑇 → dom (𝑆 ∖ I ) = dom (𝑄 ∖ I ))
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1079   = wceq 1528  wcel 2105  Vcvv 3492  cdif 3930  wss 3933  {csn 4557   class class class wbr 5057   I cid 5452  dom cdm 5548  ran crn 5549  wf 6344  1-1-ontowf1o 6347  cfv 6348  2oc2o 8085  cen 8494  pmTrspcpmtr 18498
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3or 1080  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-reu 3142  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-pss 3951  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-tp 4562  df-op 4564  df-uni 4831  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-tr 5164  df-id 5453  df-eprel 5458  df-po 5467  df-so 5468  df-fr 5507  df-we 5509  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-ord 6187  df-on 6188  df-lim 6189  df-suc 6190  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-om 7570  df-1o 8091  df-2o 8092  df-er 8278  df-en 8498  df-dom 8499  df-sdom 8500  df-fin 8501  df-pmtr 18499
This theorem is referenced by:  pmtrdifellem3  18535  pmtrdifellem4  18536
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