Theorem ressinbas 16881

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 3440	. 2 ⊢ (𝐴 ∈ 𝑋 → 𝐴 ∈ V)
2		eqid 2738	. . . . . . 7 ⊢ (𝑊 ↾s 𝐴) = (𝑊 ↾s 𝐴)
3		ressid.1	. . . . . . 7 ⊢ 𝐵 = (Base‘𝑊)
4	2, 3	ressid2 16871	. . . . . 6 ⊢ ((𝐵 ⊆ 𝐴 ∧ 𝑊 ∈ V ∧ 𝐴 ∈ V) → (𝑊 ↾s 𝐴) = 𝑊)
5		ssid 3939	. . . . . . . 8 ⊢ 𝐵 ⊆ 𝐵
6		incom 4131	. . . . . . . . 9 ⊢ (𝐴 ∩ 𝐵) = (𝐵 ∩ 𝐴)
7		df-ss 3900	. . . . . . . . . 10 ⊢ (𝐵 ⊆ 𝐴 ↔ (𝐵 ∩ 𝐴) = 𝐵)
8	7	biimpi 215	. . . . . . . . 9 ⊢ (𝐵 ⊆ 𝐴 → (𝐵 ∩ 𝐴) = 𝐵)
9	6, 8	eqtrid 2790	. . . . . . . 8 ⊢ (𝐵 ⊆ 𝐴 → (𝐴 ∩ 𝐵) = 𝐵)
10	5, 9	sseqtrrid 3970	. . . . . . 7 ⊢ (𝐵 ⊆ 𝐴 → 𝐵 ⊆ (𝐴 ∩ 𝐵))
11		elex 3440	. . . . . . 7 ⊢ (𝑊 ∈ V → 𝑊 ∈ V)
12		inex1g 5238	. . . . . . 7 ⊢ (𝐴 ∈ V → (𝐴 ∩ 𝐵) ∈ V)
13		eqid 2738	. . . . . . . 8 ⊢ (𝑊 ↾s (𝐴 ∩ 𝐵)) = (𝑊 ↾s (𝐴 ∩ 𝐵))
14	13, 3	ressid2 16871	. . . . . . 7 ⊢ ((𝐵 ⊆ (𝐴 ∩ 𝐵) ∧ 𝑊 ∈ V ∧ (𝐴 ∩ 𝐵) ∈ V) → (𝑊 ↾s (𝐴 ∩ 𝐵)) = 𝑊)
15	10, 11, 12, 14	syl3an 1158	. . . . . 6 ⊢ ((𝐵 ⊆ 𝐴 ∧ 𝑊 ∈ V ∧ 𝐴 ∈ V) → (𝑊 ↾s (𝐴 ∩ 𝐵)) = 𝑊)
16	4, 15	eqtr4d 2781	. . . . 5 ⊢ ((𝐵 ⊆ 𝐴 ∧ 𝑊 ∈ V ∧ 𝐴 ∈ V) → (𝑊 ↾s 𝐴) = (𝑊 ↾s (𝐴 ∩ 𝐵)))
17	16	3expb 1118	. . . 4 ⊢ ((𝐵 ⊆ 𝐴 ∧ (𝑊 ∈ V ∧ 𝐴 ∈ V)) → (𝑊 ↾s 𝐴) = (𝑊 ↾s (𝐴 ∩ 𝐵)))
18		inass 4150	. . . . . . . . 9 ⊢ ((𝐴 ∩ 𝐵) ∩ 𝐵) = (𝐴 ∩ (𝐵 ∩ 𝐵))
19		inidm 4149	. . . . . . . . . 10 ⊢ (𝐵 ∩ 𝐵) = 𝐵
20	19	ineq2i 4140	. . . . . . . . 9 ⊢ (𝐴 ∩ (𝐵 ∩ 𝐵)) = (𝐴 ∩ 𝐵)
21	18, 20	eqtr2i 2767	. . . . . . . 8 ⊢ (𝐴 ∩ 𝐵) = ((𝐴 ∩ 𝐵) ∩ 𝐵)
22	21	opeq2i 4805	. . . . . . 7 ⊢ ⟨(Base‘ndx), (𝐴 ∩ 𝐵)⟩ = ⟨(Base‘ndx), ((𝐴 ∩ 𝐵) ∩ 𝐵)⟩
23	22	oveq2i 7266	. . . . . 6 ⊢ (𝑊 sSet ⟨(Base‘ndx), (𝐴 ∩ 𝐵)⟩) = (𝑊 sSet ⟨(Base‘ndx), ((𝐴 ∩ 𝐵) ∩ 𝐵)⟩)
24	2, 3	ressval2 16872	. . . . . 6 ⊢ ((¬ 𝐵 ⊆ 𝐴 ∧ 𝑊 ∈ V ∧ 𝐴 ∈ V) → (𝑊 ↾s 𝐴) = (𝑊 sSet ⟨(Base‘ndx), (𝐴 ∩ 𝐵)⟩))
25		inss1 4159	. . . . . . . . 9 ⊢ (𝐴 ∩ 𝐵) ⊆ 𝐴
26		sstr 3925	. . . . . . . . 9 ⊢ ((𝐵 ⊆ (𝐴 ∩ 𝐵) ∧ (𝐴 ∩ 𝐵) ⊆ 𝐴) → 𝐵 ⊆ 𝐴)
27	25, 26	mpan2 687	. . . . . . . 8 ⊢ (𝐵 ⊆ (𝐴 ∩ 𝐵) → 𝐵 ⊆ 𝐴)
28	27	con3i 154	. . . . . . 7 ⊢ (¬ 𝐵 ⊆ 𝐴 → ¬ 𝐵 ⊆ (𝐴 ∩ 𝐵))
29	13, 3	ressval2 16872	. . . . . . 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 1158	. . . . . 6 ⊢ ((¬ 𝐵 ⊆ 𝐴 ∧ 𝑊 ∈ V ∧ 𝐴 ∈ V) → (𝑊 ↾s (𝐴 ∩ 𝐵)) = (𝑊 sSet ⟨(Base‘ndx), ((𝐴 ∩ 𝐵) ∩ 𝐵)⟩))
31	23, 24, 30	3eqtr4a 2805	. . . . 5 ⊢ ((¬ 𝐵 ⊆ 𝐴 ∧ 𝑊 ∈ V ∧ 𝐴 ∈ V) → (𝑊 ↾s 𝐴) = (𝑊 ↾s (𝐴 ∩ 𝐵)))
32	31	3expb 1118	. . . 4 ⊢ ((¬ 𝐵 ⊆ 𝐴 ∧ (𝑊 ∈ V ∧ 𝐴 ∈ V)) → (𝑊 ↾s 𝐴) = (𝑊 ↾s (𝐴 ∩ 𝐵)))
33	17, 32	pm2.61ian 808	. . 3 ⊢ ((𝑊 ∈ V ∧ 𝐴 ∈ V) → (𝑊 ↾s 𝐴) = (𝑊 ↾s (𝐴 ∩ 𝐵)))
34		reldmress 16869	. . . . . 6 ⊢ Rel dom ↾s
35	34	ovprc1 7294	. . . . 5 ⊢ (¬ 𝑊 ∈ V → (𝑊 ↾s 𝐴) = ∅)
36	34	ovprc1 7294	. . . . 5 ⊢ (¬ 𝑊 ∈ V → (𝑊 ↾s (𝐴 ∩ 𝐵)) = ∅)
37	35, 36	eqtr4d 2781	. . . 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 808	. 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 1085   = wceq 1539   ∈ wcel 2108  Vcvv 3422   ∩ cin 3882   ⊆ wss 3883  ∅c0 4253  ⟨cop 4564  ‘cfv 6418  (class class class)co 7255   sSet csts 16792  ndxcnx 16822  Basecbs 16840   ↾_s cress 16867

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-sep 5218  ax-nul 5225  ax-pr 5347

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  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-rab 3072  df-v 3424  df-sbc 3712  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-br 5071  df-opab 5133  df-id 5480  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-iota 6376  df-fun 6420  df-fv 6426  df-ov 7258  df-oprab 7259  df-mpo 7260  df-ress 16868

This theorem is referenced by:  ressress  16884  rescabs  17464  resscat  17483  funcres2c  17533  ressffth  17570  cphsubrglem  24246  suborng  31416