Theorem pmtrdifellem2 19517

Description: Lemma 2 for pmtrdifel 19520. (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 4076	. . 3 ⊢ (𝑆 ∖ I ) = (((pmTrsp‘𝑁)‘dom (𝑄 ∖ I )) ∖ I )
3	2	dmeqi 5880	. 2 ⊢ dom (𝑆 ∖ I ) = dom (((pmTrsp‘𝑁)‘dom (𝑄 ∖ I )) ∖ I )
4		eqid 2762	. . . . 5 ⊢ (pmTrsp‘(𝑁 ∖ {𝐾})) = (pmTrsp‘(𝑁 ∖ {𝐾}))
5		pmtrdifel.t	. . . . 5 ⊢ 𝑇 = ran (pmTrsp‘(𝑁 ∖ {𝐾}))
6	4, 5	pmtrfb 19505	. . . 4 ⊢ (𝑄 ∈ 𝑇 ↔ ((𝑁 ∖ {𝐾}) ∈ V ∧ 𝑄:(𝑁 ∖ {𝐾})–1-1-onto→(𝑁 ∖ {𝐾}) ∧ dom (𝑄 ∖ I ) ≈ 2o))
7		difsnexi 7744	. . . . 5 ⊢ ((𝑁 ∖ {𝐾}) ∈ V → 𝑁 ∈ V)
8		f1of 6806	. . . . . 6 ⊢ (𝑄:(𝑁 ∖ {𝐾})–1-1-onto→(𝑁 ∖ {𝐾}) → 𝑄:(𝑁 ∖ {𝐾})⟶(𝑁 ∖ {𝐾}))
9		fdm 6701	. . . . . 6 ⊢ (𝑄:(𝑁 ∖ {𝐾})⟶(𝑁 ∖ {𝐾}) → dom 𝑄 = (𝑁 ∖ {𝐾}))
10		difssd 4090	. . . . . . . 8 ⊢ (dom 𝑄 = (𝑁 ∖ {𝐾}) → (𝑄 ∖ I ) ⊆ 𝑄)
11		dmss 5878	. . . . . . . 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 4090	. . . . . . . 8 ⊢ (dom 𝑄 = (𝑁 ∖ {𝐾}) → (𝑁 ∖ {𝐾}) ⊆ 𝑁)
14		sseq1 3961	. . . . . . . 8 ⊢ (dom 𝑄 = (𝑁 ∖ {𝐾}) → (dom 𝑄 ⊆ 𝑁 ↔ (𝑁 ∖ {𝐾}) ⊆ 𝑁))
15	13, 14	mpbird 259	. . . . . . 7 ⊢ (dom 𝑄 = (𝑁 ∖ {𝐾}) → dom 𝑄 ⊆ 𝑁)
16	12, 15	sstrd 3946	. . . . . 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 1164	. . . 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 219	. . 3 ⊢ (𝑄 ∈ 𝑇 → (𝑁 ∈ V ∧ dom (𝑄 ∖ I ) ⊆ 𝑁 ∧ dom (𝑄 ∖ I ) ≈ 2o))
21		eqid 2762	. . . 4 ⊢ (pmTrsp‘𝑁) = (pmTrsp‘𝑁)
22	21	pmtrmvd 19496	. . 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 2809	1 ⊢ (𝑄 ∈ 𝑇 → dom (𝑆 ∖ I ) = dom (𝑄 ∖ I ))

Syntax hints:   → wi 4   ∧ w3a 1098   = wceq 1560   ∈ wcel 2142  Vcvv 3454   ∖ cdif 3901   ⊆ wss 3904  {csn 4582   class class class wbr 5100   I cid 5541  dom cdm 5647  ran crn 5648  ⟶wf 6517  –1-1-onto→wf1o 6520  ‘cfv 6521  2_oc2o 8431   ≈ cen 8924  pmTrspcpmtr 19481

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829  ax-5 1930  ax-6 1987  ax-7 2028  ax-8 2144  ax-9 2152  ax-10 2175  ax-11 2191  ax-12 2212  ax-ext 2734  ax-rep 5227  ax-sep 5246  ax-nul 5256  ax-pow 5322  ax-pr 5390  ax-un 7718

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3or 1099  df-3an 1100  df-tru 1563  df-fal 1573  df-ex 1800  df-nf 1804  df-sb 2091  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3077  df-rex 3087  df-reu 3368  df-rab 3415  df-v 3456  df-sbc 3745  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-pss 3924  df-nul 4286  df-if 4481  df-pw 4557  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-iun 4951  df-br 5101  df-opab 5163  df-mpt 5182  df-tr 5208  df-id 5542  df-eprel 5547  df-po 5555  df-so 5556  df-fr 5600  df-we 5602  df-xp 5653  df-rel 5654  df-cnv 5655  df-co 5656  df-dm 5657  df-rn 5658  df-res 5659  df-ima 5660  df-ord 6349  df-on 6350  df-lim 6351  df-suc 6352  df-iota 6477  df-fun 6523  df-fn 6524  df-f 6525  df-f1 6526  df-fo 6527  df-f1o 6528  df-fv 6529  df-om 7847  df-1o 8437  df-2o 8438  df-er 8678  df-en 8928  df-dom 8929  df-sdom 8930  df-fin 8931  df-pmtr 19482

This theorem is referenced by:  pmtrdifellem3  19518  pmtrdifellem4  19519