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Theorem pmtrdifellem2 19471
Description: Lemma 2 for pmtrdifel 19474. (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 4114 . . 3 (𝑆 ∖ I ) = (((pmTrsp‘𝑁)‘dom (𝑄 ∖ I )) ∖ I )
32dmeqi 5903 . 2 dom (𝑆 ∖ I ) = dom (((pmTrsp‘𝑁)‘dom (𝑄 ∖ I )) ∖ I )
4 eqid 2726 . . . . 5 (pmTrsp‘(𝑁 ∖ {𝐾})) = (pmTrsp‘(𝑁 ∖ {𝐾}))
5 pmtrdifel.t . . . . 5 𝑇 = ran (pmTrsp‘(𝑁 ∖ {𝐾}))
64, 5pmtrfb 19459 . . . 4 (𝑄𝑇 ↔ ((𝑁 ∖ {𝐾}) ∈ V ∧ 𝑄:(𝑁 ∖ {𝐾})–1-1-onto→(𝑁 ∖ {𝐾}) ∧ dom (𝑄 ∖ I ) ≈ 2o))
7 difsnexi 7761 . . . . 5 ((𝑁 ∖ {𝐾}) ∈ V → 𝑁 ∈ V)
8 f1of 6835 . . . . . 6 (𝑄:(𝑁 ∖ {𝐾})–1-1-onto→(𝑁 ∖ {𝐾}) → 𝑄:(𝑁 ∖ {𝐾})⟶(𝑁 ∖ {𝐾}))
9 fdm 6729 . . . . . 6 (𝑄:(𝑁 ∖ {𝐾})⟶(𝑁 ∖ {𝐾}) → dom 𝑄 = (𝑁 ∖ {𝐾}))
10 difssd 4129 . . . . . . . 8 (dom 𝑄 = (𝑁 ∖ {𝐾}) → (𝑄 ∖ I ) ⊆ 𝑄)
11 dmss 5901 . . . . . . . 8 ((𝑄 ∖ I ) ⊆ 𝑄 → dom (𝑄 ∖ I ) ⊆ dom 𝑄)
1210, 11syl 17 . . . . . . 7 (dom 𝑄 = (𝑁 ∖ {𝐾}) → dom (𝑄 ∖ I ) ⊆ dom 𝑄)
13 difssd 4129 . . . . . . . 8 (dom 𝑄 = (𝑁 ∖ {𝐾}) → (𝑁 ∖ {𝐾}) ⊆ 𝑁)
14 sseq1 4004 . . . . . . . 8 (dom 𝑄 = (𝑁 ∖ {𝐾}) → (dom 𝑄𝑁 ↔ (𝑁 ∖ {𝐾}) ⊆ 𝑁))
1513, 14mpbird 256 . . . . . . 7 (dom 𝑄 = (𝑁 ∖ {𝐾}) → dom 𝑄𝑁)
1612, 15sstrd 3989 . . . . . 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 1148 . . . 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 216 . . 3 (𝑄𝑇 → (𝑁 ∈ V ∧ dom (𝑄 ∖ I ) ⊆ 𝑁 ∧ dom (𝑄 ∖ I ) ≈ 2o))
21 eqid 2726 . . . 4 (pmTrsp‘𝑁) = (pmTrsp‘𝑁)
2221pmtrmvd 19450 . . 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 2778 1 (𝑄𝑇 → dom (𝑆 ∖ I ) = dom (𝑄 ∖ I ))
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1084   = wceq 1534  wcel 2099  Vcvv 3462  cdif 3943  wss 3946  {csn 4623   class class class wbr 5145   I cid 5571  dom cdm 5674  ran crn 5675  wf 6542  1-1-ontowf1o 6545  cfv 6546  2oc2o 8482  cen 8963  pmTrspcpmtr 19435
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2697  ax-rep 5282  ax-sep 5296  ax-nul 5303  ax-pow 5361  ax-pr 5425  ax-un 7738
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2704  df-cleq 2718  df-clel 2803  df-nfc 2878  df-ne 2931  df-ral 3052  df-rex 3061  df-reu 3365  df-rab 3420  df-v 3464  df-sbc 3776  df-csb 3892  df-dif 3949  df-un 3951  df-in 3953  df-ss 3963  df-pss 3966  df-nul 4323  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4906  df-iun 4995  df-br 5146  df-opab 5208  df-mpt 5229  df-tr 5263  df-id 5572  df-eprel 5578  df-po 5586  df-so 5587  df-fr 5629  df-we 5631  df-xp 5680  df-rel 5681  df-cnv 5682  df-co 5683  df-dm 5684  df-rn 5685  df-res 5686  df-ima 5687  df-ord 6371  df-on 6372  df-lim 6373  df-suc 6374  df-iota 6498  df-fun 6548  df-fn 6549  df-f 6550  df-f1 6551  df-fo 6552  df-f1o 6553  df-fv 6554  df-om 7869  df-1o 8488  df-2o 8489  df-er 8726  df-en 8967  df-dom 8968  df-sdom 8969  df-fin 8970  df-pmtr 19436
This theorem is referenced by:  pmtrdifellem3  19472  pmtrdifellem4  19473
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