Theorem pmtrdifellem2 17897

Description: Lemma 2 for pmtrdifel 17900. (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 3724	. . 3 ⊢ (𝑆 ∖ I ) = (((pmTrsp‘𝑁)‘dom (𝑄 ∖ I )) ∖ I )
3	2	dmeqi 5325	. 2 ⊢ dom (𝑆 ∖ I ) = dom (((pmTrsp‘𝑁)‘dom (𝑄 ∖ I )) ∖ I )
4		eqid 2622	. . . . 5 ⊢ (pmTrsp‘(𝑁 ∖ {𝐾})) = (pmTrsp‘(𝑁 ∖ {𝐾}))
5		pmtrdifel.t	. . . . 5 ⊢ 𝑇 = ran (pmTrsp‘(𝑁 ∖ {𝐾}))
6	4, 5	pmtrfb 17885	. . . 4 ⊢ (𝑄 ∈ 𝑇 ↔ ((𝑁 ∖ {𝐾}) ∈ V ∧ 𝑄:(𝑁 ∖ {𝐾})–1-1-onto→(𝑁 ∖ {𝐾}) ∧ dom (𝑄 ∖ I ) ≈ 2𝑜))
7		difsnexi 6970	. . . . 5 ⊢ ((𝑁 ∖ {𝐾}) ∈ V → 𝑁 ∈ V)
8		f1of 6137	. . . . . 6 ⊢ (𝑄:(𝑁 ∖ {𝐾})–1-1-onto→(𝑁 ∖ {𝐾}) → 𝑄:(𝑁 ∖ {𝐾})⟶(𝑁 ∖ {𝐾}))
9		fdm 6051	. . . . . 6 ⊢ (𝑄:(𝑁 ∖ {𝐾})⟶(𝑁 ∖ {𝐾}) → dom 𝑄 = (𝑁 ∖ {𝐾}))
10		difssd 3738	. . . . . . . 8 ⊢ (dom 𝑄 = (𝑁 ∖ {𝐾}) → (𝑄 ∖ I ) ⊆ 𝑄)
11		dmss 5323	. . . . . . . 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 3738	. . . . . . . 8 ⊢ (dom 𝑄 = (𝑁 ∖ {𝐾}) → (𝑁 ∖ {𝐾}) ⊆ 𝑁)
14		sseq1 3626	. . . . . . . 8 ⊢ (dom 𝑄 = (𝑁 ∖ {𝐾}) → (dom 𝑄 ⊆ 𝑁 ↔ (𝑁 ∖ {𝐾}) ⊆ 𝑁))
15	13, 14	mpbird 247	. . . . . . 7 ⊢ (dom 𝑄 = (𝑁 ∖ {𝐾}) → dom 𝑄 ⊆ 𝑁)
16	12, 15	sstrd 3613	. . . . . 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 1247	. . . 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 207	. . 3 ⊢ (𝑄 ∈ 𝑇 → (𝑁 ∈ V ∧ dom (𝑄 ∖ I ) ⊆ 𝑁 ∧ dom (𝑄 ∖ I ) ≈ 2𝑜))
21		eqid 2622	. . . 4 ⊢ (pmTrsp‘𝑁) = (pmTrsp‘𝑁)
22	21	pmtrmvd 17876	. . 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	syl5eq 2668	1 ⊢ (𝑄 ∈ 𝑇 → dom (𝑆 ∖ I ) = dom (𝑄 ∖ I ))

Syntax hints:   → wi 4   ∧ w3a 1037   = wceq 1483   ∈ wcel 1990  Vcvv 3200   ∖ cdif 3571   ⊆ wss 3574  {csn 4177   class class class wbr 4653   I cid 5023  dom cdm 5114  ran crn 5115  ⟶wf 5884  –1-1-onto→wf1o 5887  ‘cfv 5888  2_𝑜c2o 7554   ≈ cen 7952  pmTrspcpmtr 17861

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-om 7066  df-1o 7560  df-2o 7561  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-pmtr 17862

This theorem is referenced by:  pmtrdifellem3  17898  pmtrdifellem4  17899