Theorem ressinbas 15936

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 3212	. 2 ⊢ (𝐴 ∈ 𝑋 → 𝐴 ∈ V)
2		eqid 2622	. . . . . . 7 ⊢ (𝑊 ↾s 𝐴) = (𝑊 ↾s 𝐴)
3		ressid.1	. . . . . . 7 ⊢ 𝐵 = (Base‘𝑊)
4	2, 3	ressid2 15928	. . . . . 6 ⊢ ((𝐵 ⊆ 𝐴 ∧ 𝑊 ∈ V ∧ 𝐴 ∈ V) → (𝑊 ↾s 𝐴) = 𝑊)
5		ssid 3624	. . . . . . . 8 ⊢ 𝐵 ⊆ 𝐵
6		incom 3805	. . . . . . . . 9 ⊢ (𝐴 ∩ 𝐵) = (𝐵 ∩ 𝐴)
7		df-ss 3588	. . . . . . . . . 10 ⊢ (𝐵 ⊆ 𝐴 ↔ (𝐵 ∩ 𝐴) = 𝐵)
8	7	biimpi 206	. . . . . . . . 9 ⊢ (𝐵 ⊆ 𝐴 → (𝐵 ∩ 𝐴) = 𝐵)
9	6, 8	syl5eq 2668	. . . . . . . 8 ⊢ (𝐵 ⊆ 𝐴 → (𝐴 ∩ 𝐵) = 𝐵)
10	5, 9	syl5sseqr 3654	. . . . . . 7 ⊢ (𝐵 ⊆ 𝐴 → 𝐵 ⊆ (𝐴 ∩ 𝐵))
11		elex 3212	. . . . . . 7 ⊢ (𝑊 ∈ V → 𝑊 ∈ V)
12		inex1g 4801	. . . . . . 7 ⊢ (𝐴 ∈ V → (𝐴 ∩ 𝐵) ∈ V)
13		eqid 2622	. . . . . . . 8 ⊢ (𝑊 ↾s (𝐴 ∩ 𝐵)) = (𝑊 ↾s (𝐴 ∩ 𝐵))
14	13, 3	ressid2 15928	. . . . . . 7 ⊢ ((𝐵 ⊆ (𝐴 ∩ 𝐵) ∧ 𝑊 ∈ V ∧ (𝐴 ∩ 𝐵) ∈ V) → (𝑊 ↾s (𝐴 ∩ 𝐵)) = 𝑊)
15	10, 11, 12, 14	syl3an 1368	. . . . . 6 ⊢ ((𝐵 ⊆ 𝐴 ∧ 𝑊 ∈ V ∧ 𝐴 ∈ V) → (𝑊 ↾s (𝐴 ∩ 𝐵)) = 𝑊)
16	4, 15	eqtr4d 2659	. . . . 5 ⊢ ((𝐵 ⊆ 𝐴 ∧ 𝑊 ∈ V ∧ 𝐴 ∈ V) → (𝑊 ↾s 𝐴) = (𝑊 ↾s (𝐴 ∩ 𝐵)))
17	16	3expb 1266	. . . 4 ⊢ ((𝐵 ⊆ 𝐴 ∧ (𝑊 ∈ V ∧ 𝐴 ∈ V)) → (𝑊 ↾s 𝐴) = (𝑊 ↾s (𝐴 ∩ 𝐵)))
18		inass 3823	. . . . . . . . 9 ⊢ ((𝐴 ∩ 𝐵) ∩ 𝐵) = (𝐴 ∩ (𝐵 ∩ 𝐵))
19		inidm 3822	. . . . . . . . . 10 ⊢ (𝐵 ∩ 𝐵) = 𝐵
20	19	ineq2i 3811	. . . . . . . . 9 ⊢ (𝐴 ∩ (𝐵 ∩ 𝐵)) = (𝐴 ∩ 𝐵)
21	18, 20	eqtr2i 2645	. . . . . . . 8 ⊢ (𝐴 ∩ 𝐵) = ((𝐴 ∩ 𝐵) ∩ 𝐵)
22	21	opeq2i 4406	. . . . . . 7 ⊢ ⟨(Base‘ndx), (𝐴 ∩ 𝐵)⟩ = ⟨(Base‘ndx), ((𝐴 ∩ 𝐵) ∩ 𝐵)⟩
23	22	oveq2i 6661	. . . . . 6 ⊢ (𝑊 sSet ⟨(Base‘ndx), (𝐴 ∩ 𝐵)⟩) = (𝑊 sSet ⟨(Base‘ndx), ((𝐴 ∩ 𝐵) ∩ 𝐵)⟩)
24	2, 3	ressval2 15929	. . . . . 6 ⊢ ((¬ 𝐵 ⊆ 𝐴 ∧ 𝑊 ∈ V ∧ 𝐴 ∈ V) → (𝑊 ↾s 𝐴) = (𝑊 sSet ⟨(Base‘ndx), (𝐴 ∩ 𝐵)⟩))
25		inss1 3833	. . . . . . . . 9 ⊢ (𝐴 ∩ 𝐵) ⊆ 𝐴
26		sstr 3611	. . . . . . . . 9 ⊢ ((𝐵 ⊆ (𝐴 ∩ 𝐵) ∧ (𝐴 ∩ 𝐵) ⊆ 𝐴) → 𝐵 ⊆ 𝐴)
27	25, 26	mpan2 707	. . . . . . . 8 ⊢ (𝐵 ⊆ (𝐴 ∩ 𝐵) → 𝐵 ⊆ 𝐴)
28	27	con3i 150	. . . . . . 7 ⊢ (¬ 𝐵 ⊆ 𝐴 → ¬ 𝐵 ⊆ (𝐴 ∩ 𝐵))
29	13, 3	ressval2 15929	. . . . . . 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 1368	. . . . . 6 ⊢ ((¬ 𝐵 ⊆ 𝐴 ∧ 𝑊 ∈ V ∧ 𝐴 ∈ V) → (𝑊 ↾s (𝐴 ∩ 𝐵)) = (𝑊 sSet ⟨(Base‘ndx), ((𝐴 ∩ 𝐵) ∩ 𝐵)⟩))
31	23, 24, 30	3eqtr4a 2682	. . . . 5 ⊢ ((¬ 𝐵 ⊆ 𝐴 ∧ 𝑊 ∈ V ∧ 𝐴 ∈ V) → (𝑊 ↾s 𝐴) = (𝑊 ↾s (𝐴 ∩ 𝐵)))
32	31	3expb 1266	. . . 4 ⊢ ((¬ 𝐵 ⊆ 𝐴 ∧ (𝑊 ∈ V ∧ 𝐴 ∈ V)) → (𝑊 ↾s 𝐴) = (𝑊 ↾s (𝐴 ∩ 𝐵)))
33	17, 32	pm2.61ian 831	. . 3 ⊢ ((𝑊 ∈ V ∧ 𝐴 ∈ V) → (𝑊 ↾s 𝐴) = (𝑊 ↾s (𝐴 ∩ 𝐵)))
34		reldmress 15926	. . . . . 6 ⊢ Rel dom ↾s
35	34	ovprc1 6684	. . . . 5 ⊢ (¬ 𝑊 ∈ V → (𝑊 ↾s 𝐴) = ∅)
36	34	ovprc1 6684	. . . . 5 ⊢ (¬ 𝑊 ∈ V → (𝑊 ↾s (𝐴 ∩ 𝐵)) = ∅)
37	35, 36	eqtr4d 2659	. . . 4 ⊢ (¬ 𝑊 ∈ V → (𝑊 ↾s 𝐴) = (𝑊 ↾s (𝐴 ∩ 𝐵)))
38	37	adantr 481	. . 3 ⊢ ((¬ 𝑊 ∈ V ∧ 𝐴 ∈ V) → (𝑊 ↾s 𝐴) = (𝑊 ↾s (𝐴 ∩ 𝐵)))
39	33, 38	pm2.61ian 831	. 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 384   ∧ w3a 1037   = wceq 1483   ∈ wcel 1990  Vcvv 3200   ∩ cin 3573   ⊆ wss 3574  ∅c0 3915  ⟨cop 4183  ‘cfv 5888  (class class class)co 6650  ndxcnx 15854   sSet csts 15855  Basecbs 15857   ↾_s cress 15858

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-iota 5851  df-fun 5890  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-ress 15865

This theorem is referenced by:  ressress  15938  rescabs  16493  resscat  16512  funcres2c  16561  ressffth  16598  cphsubrglem  22977  suborng  29815