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Theorem pmtrdifellem2 19259
Description: Lemma 2 for pmtrdifel 19262. (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 4078 . . 3 (𝑆 ∖ I ) = (((pmTrsp‘𝑁)‘dom (𝑄 ∖ I )) ∖ I )
32dmeqi 5860 . 2 dom (𝑆 ∖ I ) = dom (((pmTrsp‘𝑁)‘dom (𝑄 ∖ I )) ∖ I )
4 eqid 2736 . . . . 5 (pmTrsp‘(𝑁 ∖ {𝐾})) = (pmTrsp‘(𝑁 ∖ {𝐾}))
5 pmtrdifel.t . . . . 5 𝑇 = ran (pmTrsp‘(𝑁 ∖ {𝐾}))
64, 5pmtrfb 19247 . . . 4 (𝑄𝑇 ↔ ((𝑁 ∖ {𝐾}) ∈ V ∧ 𝑄:(𝑁 ∖ {𝐾})–1-1-onto→(𝑁 ∖ {𝐾}) ∧ dom (𝑄 ∖ I ) ≈ 2o))
7 difsnexi 7695 . . . . 5 ((𝑁 ∖ {𝐾}) ∈ V → 𝑁 ∈ V)
8 f1of 6784 . . . . . 6 (𝑄:(𝑁 ∖ {𝐾})–1-1-onto→(𝑁 ∖ {𝐾}) → 𝑄:(𝑁 ∖ {𝐾})⟶(𝑁 ∖ {𝐾}))
9 fdm 6677 . . . . . 6 (𝑄:(𝑁 ∖ {𝐾})⟶(𝑁 ∖ {𝐾}) → dom 𝑄 = (𝑁 ∖ {𝐾}))
10 difssd 4092 . . . . . . . 8 (dom 𝑄 = (𝑁 ∖ {𝐾}) → (𝑄 ∖ I ) ⊆ 𝑄)
11 dmss 5858 . . . . . . . 8 ((𝑄 ∖ I ) ⊆ 𝑄 → dom (𝑄 ∖ I ) ⊆ dom 𝑄)
1210, 11syl 17 . . . . . . 7 (dom 𝑄 = (𝑁 ∖ {𝐾}) → dom (𝑄 ∖ I ) ⊆ dom 𝑄)
13 difssd 4092 . . . . . . . 8 (dom 𝑄 = (𝑁 ∖ {𝐾}) → (𝑁 ∖ {𝐾}) ⊆ 𝑁)
14 sseq1 3969 . . . . . . . 8 (dom 𝑄 = (𝑁 ∖ {𝐾}) → (dom 𝑄𝑁 ↔ (𝑁 ∖ {𝐾}) ⊆ 𝑁))
1513, 14mpbird 256 . . . . . . 7 (dom 𝑄 = (𝑁 ∖ {𝐾}) → dom 𝑄𝑁)
1612, 15sstrd 3954 . . . . . 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 1151 . . . 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 2736 . . . 4 (pmTrsp‘𝑁) = (pmTrsp‘𝑁)
2221pmtrmvd 19238 . . 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 2788 1 (𝑄𝑇 → dom (𝑆 ∖ I ) = dom (𝑄 ∖ I ))
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1087   = wceq 1541  wcel 2106  Vcvv 3445  cdif 3907  wss 3910  {csn 4586   class class class wbr 5105   I cid 5530  dom cdm 5633  ran crn 5634  wf 6492  1-1-ontowf1o 6495  cfv 6496  2oc2o 8406  cen 8880  pmTrspcpmtr 19223
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  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 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-ral 3065  df-rex 3074  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-om 7803  df-1o 8412  df-2o 8413  df-er 8648  df-en 8884  df-dom 8885  df-sdom 8886  df-fin 8887  df-pmtr 19224
This theorem is referenced by:  pmtrdifellem3  19260  pmtrdifellem4  19261
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