Theorem ressinbas 17290

Description: Restriction only cares about the part of the second set which intersects the base of the first. (Contributed by Stefan O'Rear, 29-Nov-2014.)

Hypothesis

Ref	Expression
ressid.1	⊢ 𝐵 = (Base‘𝑊)

Assertion

Ref	Expression
ressinbas	⊢ (𝐴 ∈ 𝑋 → (𝑊 ↾s 𝐴) = (𝑊 ↾s (𝐴 ∩ 𝐵)))

Proof of Theorem ressinbas

Step	Hyp	Ref	Expression
1		elex 3498	. 2 ⊢ (𝐴 ∈ 𝑋 → 𝐴 ∈ V)
2		eqid 2734	. . . . . . 7 ⊢ (𝑊 ↾s 𝐴) = (𝑊 ↾s 𝐴)
3		ressid.1	. . . . . . 7 ⊢ 𝐵 = (Base‘𝑊)
4	2, 3	ressid2 17277	. . . . . 6 ⊢ ((𝐵 ⊆ 𝐴 ∧ 𝑊 ∈ V ∧ 𝐴 ∈ V) → (𝑊 ↾s 𝐴) = 𝑊)
5		ssid 4017	. . . . . . . 8 ⊢ 𝐵 ⊆ 𝐵
6		incom 4216	. . . . . . . . 9 ⊢ (𝐴 ∩ 𝐵) = (𝐵 ∩ 𝐴)
7		dfss2 3980	. . . . . . . . . 10 ⊢ (𝐵 ⊆ 𝐴 ↔ (𝐵 ∩ 𝐴) = 𝐵)
8	7	biimpi 216	. . . . . . . . 9 ⊢ (𝐵 ⊆ 𝐴 → (𝐵 ∩ 𝐴) = 𝐵)
9	6, 8	eqtrid 2786	. . . . . . . 8 ⊢ (𝐵 ⊆ 𝐴 → (𝐴 ∩ 𝐵) = 𝐵)
10	5, 9	sseqtrrid 4048	. . . . . . 7 ⊢ (𝐵 ⊆ 𝐴 → 𝐵 ⊆ (𝐴 ∩ 𝐵))
11		elex 3498	. . . . . . 7 ⊢ (𝑊 ∈ V → 𝑊 ∈ V)
12		inex1g 5324	. . . . . . 7 ⊢ (𝐴 ∈ V → (𝐴 ∩ 𝐵) ∈ V)
13		eqid 2734	. . . . . . . 8 ⊢ (𝑊 ↾s (𝐴 ∩ 𝐵)) = (𝑊 ↾s (𝐴 ∩ 𝐵))
14	13, 3	ressid2 17277	. . . . . . 7 ⊢ ((𝐵 ⊆ (𝐴 ∩ 𝐵) ∧ 𝑊 ∈ V ∧ (𝐴 ∩ 𝐵) ∈ V) → (𝑊 ↾s (𝐴 ∩ 𝐵)) = 𝑊)
15	10, 11, 12, 14	syl3an 1159	. . . . . 6 ⊢ ((𝐵 ⊆ 𝐴 ∧ 𝑊 ∈ V ∧ 𝐴 ∈ V) → (𝑊 ↾s (𝐴 ∩ 𝐵)) = 𝑊)
16	4, 15	eqtr4d 2777	. . . . 5 ⊢ ((𝐵 ⊆ 𝐴 ∧ 𝑊 ∈ V ∧ 𝐴 ∈ V) → (𝑊 ↾s 𝐴) = (𝑊 ↾s (𝐴 ∩ 𝐵)))
17	16	3expb 1119	. . . 4 ⊢ ((𝐵 ⊆ 𝐴 ∧ (𝑊 ∈ V ∧ 𝐴 ∈ V)) → (𝑊 ↾s 𝐴) = (𝑊 ↾s (𝐴 ∩ 𝐵)))
18		inass 4235	. . . . . . . . 9 ⊢ ((𝐴 ∩ 𝐵) ∩ 𝐵) = (𝐴 ∩ (𝐵 ∩ 𝐵))
19		inidm 4234	. . . . . . . . . 10 ⊢ (𝐵 ∩ 𝐵) = 𝐵
20	19	ineq2i 4224	. . . . . . . . 9 ⊢ (𝐴 ∩ (𝐵 ∩ 𝐵)) = (𝐴 ∩ 𝐵)
21	18, 20	eqtr2i 2763	. . . . . . . 8 ⊢ (𝐴 ∩ 𝐵) = ((𝐴 ∩ 𝐵) ∩ 𝐵)
22	21	opeq2i 4881	. . . . . . 7 ⊢ ⟨(Base‘ndx), (𝐴 ∩ 𝐵)⟩ = ⟨(Base‘ndx), ((𝐴 ∩ 𝐵) ∩ 𝐵)⟩
23	22	oveq2i 7441	. . . . . 6 ⊢ (𝑊 sSet ⟨(Base‘ndx), (𝐴 ∩ 𝐵)⟩) = (𝑊 sSet ⟨(Base‘ndx), ((𝐴 ∩ 𝐵) ∩ 𝐵)⟩)
24	2, 3	ressval2 17278	. . . . . 6 ⊢ ((¬ 𝐵 ⊆ 𝐴 ∧ 𝑊 ∈ V ∧ 𝐴 ∈ V) → (𝑊 ↾s 𝐴) = (𝑊 sSet ⟨(Base‘ndx), (𝐴 ∩ 𝐵)⟩))
25		inss1 4244	. . . . . . . . 9 ⊢ (𝐴 ∩ 𝐵) ⊆ 𝐴
26		sstr 4003	. . . . . . . . 9 ⊢ ((𝐵 ⊆ (𝐴 ∩ 𝐵) ∧ (𝐴 ∩ 𝐵) ⊆ 𝐴) → 𝐵 ⊆ 𝐴)
27	25, 26	mpan2 691	. . . . . . . 8 ⊢ (𝐵 ⊆ (𝐴 ∩ 𝐵) → 𝐵 ⊆ 𝐴)
28	27	con3i 154	. . . . . . 7 ⊢ (¬ 𝐵 ⊆ 𝐴 → ¬ 𝐵 ⊆ (𝐴 ∩ 𝐵))
29	13, 3	ressval2 17278	. . . . . . 7 ⊢ ((¬ 𝐵 ⊆ (𝐴 ∩ 𝐵) ∧ 𝑊 ∈ V ∧ (𝐴 ∩ 𝐵) ∈ V) → (𝑊 ↾s (𝐴 ∩ 𝐵)) = (𝑊 sSet ⟨(Base‘ndx), ((𝐴 ∩ 𝐵) ∩ 𝐵)⟩))
30	28, 11, 12, 29	syl3an 1159	. . . . . 6 ⊢ ((¬ 𝐵 ⊆ 𝐴 ∧ 𝑊 ∈ V ∧ 𝐴 ∈ V) → (𝑊 ↾s (𝐴 ∩ 𝐵)) = (𝑊 sSet ⟨(Base‘ndx), ((𝐴 ∩ 𝐵) ∩ 𝐵)⟩))
31	23, 24, 30	3eqtr4a 2800	. . . . 5 ⊢ ((¬ 𝐵 ⊆ 𝐴 ∧ 𝑊 ∈ V ∧ 𝐴 ∈ V) → (𝑊 ↾s 𝐴) = (𝑊 ↾s (𝐴 ∩ 𝐵)))
32	31	3expb 1119	. . . 4 ⊢ ((¬ 𝐵 ⊆ 𝐴 ∧ (𝑊 ∈ V ∧ 𝐴 ∈ V)) → (𝑊 ↾s 𝐴) = (𝑊 ↾s (𝐴 ∩ 𝐵)))
33	17, 32	pm2.61ian 812	. . 3 ⊢ ((𝑊 ∈ V ∧ 𝐴 ∈ V) → (𝑊 ↾s 𝐴) = (𝑊 ↾s (𝐴 ∩ 𝐵)))
34		reldmress 17275	. . . . . 6 ⊢ Rel dom ↾s
35	34	ovprc1 7469	. . . . 5 ⊢ (¬ 𝑊 ∈ V → (𝑊 ↾s 𝐴) = ∅)
36	34	ovprc1 7469	. . . . 5 ⊢ (¬ 𝑊 ∈ V → (𝑊 ↾s (𝐴 ∩ 𝐵)) = ∅)
37	35, 36	eqtr4d 2777	. . . 4 ⊢ (¬ 𝑊 ∈ V → (𝑊 ↾s 𝐴) = (𝑊 ↾s (𝐴 ∩ 𝐵)))
38	37	adantr 480	. . 3 ⊢ ((¬ 𝑊 ∈ V ∧ 𝐴 ∈ V) → (𝑊 ↾s 𝐴) = (𝑊 ↾s (𝐴 ∩ 𝐵)))
39	33, 38	pm2.61ian 812	. 2 ⊢ (𝐴 ∈ V → (𝑊 ↾s 𝐴) = (𝑊 ↾s (𝐴 ∩ 𝐵)))
40	1, 39	syl 17	1 ⊢ (𝐴 ∈ 𝑋 → (𝑊 ↾s 𝐴) = (𝑊 ↾s (𝐴 ∩ 𝐵)))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   ∧ w3a 1086   = wceq 1536   ∈ wcel 2105  Vcvv 3477   ∩ cin 3961   ⊆ wss 3962  ∅c0 4338  ⟨cop 4636  ‘cfv 6562  (class class class)co 7430   sSet csts 17196  ndxcnx 17226  Basecbs 17244   ↾_s cress 17273

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-sep 5301  ax-nul 5311  ax-pr 5437

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3433  df-v 3479  df-sbc 3791  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-nul 4339  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-br 5148  df-opab 5210  df-id 5582  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-iota 6515  df-fun 6564  df-fv 6570  df-ov 7433  df-oprab 7434  df-mpo 7435  df-ress 17274

This theorem is referenced by:  ressress  17293  rescabs  17882  rescabsOLD  17883  resscat  17902  funcres2c  17954  ressffth  17991  cphsubrglem  25224  suborng  33324