Theorem ressinbas 17300

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 3476	. 2 ⊢ (𝐴 ∈ 𝑋 → 𝐴 ∈ V)
2		eqid 2763	. . . . . . 7 ⊢ (𝑊 ↾s 𝐴) = (𝑊 ↾s 𝐴)
3		ressid.1	. . . . . . 7 ⊢ 𝐵 = (Base‘𝑊)
4	2, 3	ressid2 17289	. . . . . 6 ⊢ ((𝐵 ⊆ 𝐴 ∧ 𝑊 ∈ V ∧ 𝐴 ∈ V) → (𝑊 ↾s 𝐴) = 𝑊)
5		ssid 3959	. . . . . . . 8 ⊢ 𝐵 ⊆ 𝐵
6		incom 4162	. . . . . . . . 9 ⊢ (𝐴 ∩ 𝐵) = (𝐵 ∩ 𝐴)
7		dfss2 3923	. . . . . . . . . 10 ⊢ (𝐵 ⊆ 𝐴 ↔ (𝐵 ∩ 𝐴) = 𝐵)
8	7	biimpi 219	. . . . . . . . 9 ⊢ (𝐵 ⊆ 𝐴 → (𝐵 ∩ 𝐴) = 𝐵)
9	6, 8	eqtrid 2810	. . . . . . . 8 ⊢ (𝐵 ⊆ 𝐴 → (𝐴 ∩ 𝐵) = 𝐵)
10	5, 9	sseqtrrid 3980	. . . . . . 7 ⊢ (𝐵 ⊆ 𝐴 → 𝐵 ⊆ (𝐴 ∩ 𝐵))
11		elex 3476	. . . . . . 7 ⊢ (𝑊 ∈ V → 𝑊 ∈ V)
12		inex1g 5288	. . . . . . 7 ⊢ (𝐴 ∈ V → (𝐴 ∩ 𝐵) ∈ V)
13		eqid 2763	. . . . . . . 8 ⊢ (𝑊 ↾s (𝐴 ∩ 𝐵)) = (𝑊 ↾s (𝐴 ∩ 𝐵))
14	13, 3	ressid2 17289	. . . . . . 7 ⊢ ((𝐵 ⊆ (𝐴 ∩ 𝐵) ∧ 𝑊 ∈ V ∧ (𝐴 ∩ 𝐵) ∈ V) → (𝑊 ↾s (𝐴 ∩ 𝐵)) = 𝑊)
15	10, 11, 12, 14	syl3an 1178	. . . . . 6 ⊢ ((𝐵 ⊆ 𝐴 ∧ 𝑊 ∈ V ∧ 𝐴 ∈ V) → (𝑊 ↾s (𝐴 ∩ 𝐵)) = 𝑊)
16	4, 15	eqtr4d 2801	. . . . 5 ⊢ ((𝐵 ⊆ 𝐴 ∧ 𝑊 ∈ V ∧ 𝐴 ∈ V) → (𝑊 ↾s 𝐴) = (𝑊 ↾s (𝐴 ∩ 𝐵)))
17	16	3expb 1138	. . . 4 ⊢ ((𝐵 ⊆ 𝐴 ∧ (𝑊 ∈ V ∧ 𝐴 ∈ V)) → (𝑊 ↾s 𝐴) = (𝑊 ↾s (𝐴 ∩ 𝐵)))
18		inass 4180	. . . . . . . . 9 ⊢ ((𝐴 ∩ 𝐵) ∩ 𝐵) = (𝐴 ∩ (𝐵 ∩ 𝐵))
19		inidm 4179	. . . . . . . . . 10 ⊢ (𝐵 ∩ 𝐵) = 𝐵
20	19	ineq2i 4170	. . . . . . . . 9 ⊢ (𝐴 ∩ (𝐵 ∩ 𝐵)) = (𝐴 ∩ 𝐵)
21	18, 20	eqtr2i 2787	. . . . . . . 8 ⊢ (𝐴 ∩ 𝐵) = ((𝐴 ∩ 𝐵) ∩ 𝐵)
22	21	opeq2i 4842	. . . . . . 7 ⊢ ⟨(Base‘ndx), (𝐴 ∩ 𝐵)⟩ = ⟨(Base‘ndx), ((𝐴 ∩ 𝐵) ∩ 𝐵)⟩
23	22	oveq2i 7421	. . . . . 6 ⊢ (𝑊 sSet ⟨(Base‘ndx), (𝐴 ∩ 𝐵)⟩) = (𝑊 sSet ⟨(Base‘ndx), ((𝐴 ∩ 𝐵) ∩ 𝐵)⟩)
24	2, 3	ressval2 17290	. . . . . 6 ⊢ ((¬ 𝐵 ⊆ 𝐴 ∧ 𝑊 ∈ V ∧ 𝐴 ∈ V) → (𝑊 ↾s 𝐴) = (𝑊 sSet ⟨(Base‘ndx), (𝐴 ∩ 𝐵)⟩))
25		inss1 4189	. . . . . . . . 9 ⊢ (𝐴 ∩ 𝐵) ⊆ 𝐴
26		sstr 3945	. . . . . . . . 9 ⊢ ((𝐵 ⊆ (𝐴 ∩ 𝐵) ∧ (𝐴 ∩ 𝐵) ⊆ 𝐴) → 𝐵 ⊆ 𝐴)
27	25, 26	mpan2 703	. . . . . . . 8 ⊢ (𝐵 ⊆ (𝐴 ∩ 𝐵) → 𝐵 ⊆ 𝐴)
28	27	con3i 155	. . . . . . 7 ⊢ (¬ 𝐵 ⊆ 𝐴 → ¬ 𝐵 ⊆ (𝐴 ∩ 𝐵))
29	13, 3	ressval2 17290	. . . . . . 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 1178	. . . . . 6 ⊢ ((¬ 𝐵 ⊆ 𝐴 ∧ 𝑊 ∈ V ∧ 𝐴 ∈ V) → (𝑊 ↾s (𝐴 ∩ 𝐵)) = (𝑊 sSet ⟨(Base‘ndx), ((𝐴 ∩ 𝐵) ∩ 𝐵)⟩))
31	23, 24, 30	3eqtr4a 2824	. . . . 5 ⊢ ((¬ 𝐵 ⊆ 𝐴 ∧ 𝑊 ∈ V ∧ 𝐴 ∈ V) → (𝑊 ↾s 𝐴) = (𝑊 ↾s (𝐴 ∩ 𝐵)))
32	31	3expb 1138	. . . 4 ⊢ ((¬ 𝐵 ⊆ 𝐴 ∧ (𝑊 ∈ V ∧ 𝐴 ∈ V)) → (𝑊 ↾s 𝐴) = (𝑊 ↾s (𝐴 ∩ 𝐵)))
33	17, 32	pm2.61ian 823	. . 3 ⊢ ((𝑊 ∈ V ∧ 𝐴 ∈ V) → (𝑊 ↾s 𝐴) = (𝑊 ↾s (𝐴 ∩ 𝐵)))
34		reldmress 17287	. . . . . 6 ⊢ Rel dom ↾s
35	34	ovprc1 7449	. . . . 5 ⊢ (¬ 𝑊 ∈ V → (𝑊 ↾s 𝐴) = ∅)
36	34	ovprc1 7449	. . . . 5 ⊢ (¬ 𝑊 ∈ V → (𝑊 ↾s (𝐴 ∩ 𝐵)) = ∅)
37	35, 36	eqtr4d 2801	. . . 4 ⊢ (¬ 𝑊 ∈ V → (𝑊 ↾s 𝐴) = (𝑊 ↾s (𝐴 ∩ 𝐵)))
38	37	adantr 485	. . 3 ⊢ ((¬ 𝑊 ∈ V ∧ 𝐴 ∈ V) → (𝑊 ↾s 𝐴) = (𝑊 ↾s (𝐴 ∩ 𝐵)))
39	33, 38	pm2.61ian 823	. 2 ⊢ (𝐴 ∈ V → (𝑊 ↾s 𝐴) = (𝑊 ↾s (𝐴 ∩ 𝐵)))
40	1, 39	syl 18	1 ⊢ (𝐴 ∈ 𝑋 → (𝑊 ↾s 𝐴) = (𝑊 ↾s (𝐴 ∩ 𝐵)))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 400   ∧ w3a 1103   = wceq 1570   ∈ wcel 2143  Vcvv 3455   ∩ cin 3904   ⊆ wss 3905  ∅c0 4286  ⟨cop 4595  ‘cfv 6536  (class class class)co 7410   sSet csts 17218  ndxcnx 17248  Basecbs 17264   ↾_s cress 17285

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-sep 5257  ax-nul 5269  ax-pr 5404

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-rab 3417  df-v 3457  df-sbc 3745  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4488  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4873  df-br 5110  df-opab 5174  df-id 5556  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-iota 6492  df-fun 6538  df-fv 6544  df-ov 7413  df-oprab 7414  df-mpo 7415  df-ress 17286

This theorem is referenced by:  ressress  17302  rescabs  17885  resscat  17904  funcres2c  17955  ressffth  17992  suborng  20979  cphsubrglem  25336