Theorem pmtrdifellem2 19000

Description: Lemma 2 for pmtrdifel 19003. (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 4049	. . 3 ⊢ (𝑆 ∖ I ) = (((pmTrsp‘𝑁)‘dom (𝑄 ∖ I )) ∖ I )
3	2	dmeqi 5802	. 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 18988	. . . 4 ⊢ (𝑄 ∈ 𝑇 ↔ ((𝑁 ∖ {𝐾}) ∈ V ∧ 𝑄:(𝑁 ∖ {𝐾})–1-1-onto→(𝑁 ∖ {𝐾}) ∧ dom (𝑄 ∖ I ) ≈ 2o))
7		difsnexi 7589	. . . . 5 ⊢ ((𝑁 ∖ {𝐾}) ∈ V → 𝑁 ∈ V)
8		f1of 6700	. . . . . 6 ⊢ (𝑄:(𝑁 ∖ {𝐾})–1-1-onto→(𝑁 ∖ {𝐾}) → 𝑄:(𝑁 ∖ {𝐾})⟶(𝑁 ∖ {𝐾}))
9		fdm 6593	. . . . . 6 ⊢ (𝑄:(𝑁 ∖ {𝐾})⟶(𝑁 ∖ {𝐾}) → dom 𝑄 = (𝑁 ∖ {𝐾}))
10		difssd 4063	. . . . . . . 8 ⊢ (dom 𝑄 = (𝑁 ∖ {𝐾}) → (𝑄 ∖ I ) ⊆ 𝑄)
11		dmss 5800	. . . . . . . 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 4063	. . . . . . . 8 ⊢ (dom 𝑄 = (𝑁 ∖ {𝐾}) → (𝑁 ∖ {𝐾}) ⊆ 𝑁)
14		sseq1 3942	. . . . . . . 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 3927	. . . . . 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 1149	. . . 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 18979	. . 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 1085   = wceq 1539   ∈ wcel 2108  Vcvv 3422   ∖ cdif 3880   ⊆ wss 3883  {csn 4558   class class class wbr 5070   I cid 5479  dom cdm 5580  ran crn 5581  ⟶wf 6414  –1-1-onto→wf1o 6417  ‘cfv 6418  2_oc2o 8261   ≈ cen 8688  pmTrspcpmtr 18964

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-rep 5205  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347  ax-un 7566

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-ral 3068  df-rex 3069  df-reu 3070  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3902  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-tp 4563  df-op 4565  df-uni 4837  df-iun 4923  df-br 5071  df-opab 5133  df-mpt 5154  df-tr 5188  df-id 5480  df-eprel 5486  df-po 5494  df-so 5495  df-fr 5535  df-we 5537  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-ord 6254  df-on 6255  df-lim 6256  df-suc 6257  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-om 7688  df-1o 8267  df-2o 8268  df-er 8456  df-en 8692  df-dom 8693  df-sdom 8694  df-fin 8695  df-pmtr 18965

This theorem is referenced by:  pmtrdifellem3  19001  pmtrdifellem4  19002