Theorem pmtrdifellem2 19085

Description: Lemma 2 for pmtrdifel 19088. (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	⊢ (𝑄 ∈ 𝑇 → dom (𝑆 ∖ I ) = dom (𝑄 ∖ I ))

Proof of Theorem pmtrdifellem2

Step	Hyp	Ref	Expression
1		pmtrdifel.0	. . . 4 ⊢ 𝑆 = ((pmTrsp‘𝑁)‘dom (𝑄 ∖ I ))
2	1	difeq1i 4053	. . 3 ⊢ (𝑆 ∖ I ) = (((pmTrsp‘𝑁)‘dom (𝑄 ∖ I )) ∖ I )
3	2	dmeqi 5813	. 2 ⊢ dom (𝑆 ∖ I ) = dom (((pmTrsp‘𝑁)‘dom (𝑄 ∖ I )) ∖ I )
4		eqid 2738	. . . . 5 ⊢ (pmTrsp‘(𝑁 ∖ {𝐾})) = (pmTrsp‘(𝑁 ∖ {𝐾}))
5		pmtrdifel.t	. . . . 5 ⊢ 𝑇 = ran (pmTrsp‘(𝑁 ∖ {𝐾}))
6	4, 5	pmtrfb 19073	. . . 4 ⊢ (𝑄 ∈ 𝑇 ↔ ((𝑁 ∖ {𝐾}) ∈ V ∧ 𝑄:(𝑁 ∖ {𝐾})–1-1-onto→(𝑁 ∖ {𝐾}) ∧ dom (𝑄 ∖ I ) ≈ 2o))
7		difsnexi 7611	. . . . 5 ⊢ ((𝑁 ∖ {𝐾}) ∈ V → 𝑁 ∈ V)
8		f1of 6716	. . . . . 6 ⊢ (𝑄:(𝑁 ∖ {𝐾})–1-1-onto→(𝑁 ∖ {𝐾}) → 𝑄:(𝑁 ∖ {𝐾})⟶(𝑁 ∖ {𝐾}))
9		fdm 6609	. . . . . 6 ⊢ (𝑄:(𝑁 ∖ {𝐾})⟶(𝑁 ∖ {𝐾}) → dom 𝑄 = (𝑁 ∖ {𝐾}))
10		difssd 4067	. . . . . . . 8 ⊢ (dom 𝑄 = (𝑁 ∖ {𝐾}) → (𝑄 ∖ I ) ⊆ 𝑄)
11		dmss 5811	. . . . . . . 8 ⊢ ((𝑄 ∖ I ) ⊆ 𝑄 → dom (𝑄 ∖ I ) ⊆ dom 𝑄)
12	10, 11	syl 17	. . . . . . 7 ⊢ (dom 𝑄 = (𝑁 ∖ {𝐾}) → dom (𝑄 ∖ I ) ⊆ dom 𝑄)
13		difssd 4067	. . . . . . . 8 ⊢ (dom 𝑄 = (𝑁 ∖ {𝐾}) → (𝑁 ∖ {𝐾}) ⊆ 𝑁)
14		sseq1 3946	. . . . . . . 8 ⊢ (dom 𝑄 = (𝑁 ∖ {𝐾}) → (dom 𝑄 ⊆ 𝑁 ↔ (𝑁 ∖ {𝐾}) ⊆ 𝑁))
15	13, 14	mpbird 256	. . . . . . 7 ⊢ (dom 𝑄 = (𝑁 ∖ {𝐾}) → dom 𝑄 ⊆ 𝑁)
16	12, 15	sstrd 3931	. . . . . 6 ⊢ (dom 𝑄 = (𝑁 ∖ {𝐾}) → dom (𝑄 ∖ I ) ⊆ 𝑁)
17	8, 9, 16	3syl 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)
19	7, 17, 18	3anim123i 1150	. . . 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))
20	6, 19	sylbi 216	. . 3 ⊢ (𝑄 ∈ 𝑇 → (𝑁 ∈ V ∧ dom (𝑄 ∖ I ) ⊆ 𝑁 ∧ dom (𝑄 ∖ I ) ≈ 2o))
21		eqid 2738	. . . 4 ⊢ (pmTrsp‘𝑁) = (pmTrsp‘𝑁)
22	21	pmtrmvd 19064	. . 3 ⊢ ((𝑁 ∈ V ∧ dom (𝑄 ∖ I ) ⊆ 𝑁 ∧ dom (𝑄 ∖ I ) ≈ 2o) → dom (((pmTrsp‘𝑁)‘dom (𝑄 ∖ I )) ∖ I ) = dom (𝑄 ∖ I ))
23	20, 22	syl 17	. 2 ⊢ (𝑄 ∈ 𝑇 → dom (((pmTrsp‘𝑁)‘dom (𝑄 ∖ I )) ∖ I ) = dom (𝑄 ∖ I ))
24	3, 23	eqtrid 2790	1 ⊢ (𝑄 ∈ 𝑇 → dom (𝑆 ∖ I ) = dom (𝑄 ∖ I ))

Syntax hints:   → wi 4   ∧ w3a 1086   = wceq 1539   ∈ wcel 2106  Vcvv 3432   ∖ cdif 3884   ⊆ wss 3887  {csn 4561   class class class wbr 5074   I cid 5488  dom cdm 5589  ran crn 5590  ⟶wf 6429  –1-1-onto→wf1o 6432  ‘cfv 6433  2_oc2o 8291   ≈ cen 8730  pmTrspcpmtr 19049

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  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 2709  ax-rep 5209  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-pss 3906  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-tr 5192  df-id 5489  df-eprel 5495  df-po 5503  df-so 5504  df-fr 5544  df-we 5546  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-ord 6269  df-on 6270  df-lim 6271  df-suc 6272  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-om 7713  df-1o 8297  df-2o 8298  df-er 8498  df-en 8734  df-dom 8735  df-sdom 8736  df-fin 8737  df-pmtr 19050

This theorem is referenced by:  pmtrdifellem3  19086  pmtrdifellem4  19087