Theorem ressinbas 17304

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 3509	. 2 ⊢ (𝐴 ∈ 𝑋 → 𝐴 ∈ V)
2		eqid 2740	. . . . . . 7 ⊢ (𝑊 ↾s 𝐴) = (𝑊 ↾s 𝐴)
3		ressid.1	. . . . . . 7 ⊢ 𝐵 = (Base‘𝑊)
4	2, 3	ressid2 17291	. . . . . 6 ⊢ ((𝐵 ⊆ 𝐴 ∧ 𝑊 ∈ V ∧ 𝐴 ∈ V) → (𝑊 ↾s 𝐴) = 𝑊)
5		ssid 4031	. . . . . . . 8 ⊢ 𝐵 ⊆ 𝐵
6		incom 4230	. . . . . . . . 9 ⊢ (𝐴 ∩ 𝐵) = (𝐵 ∩ 𝐴)
7		dfss2 3994	. . . . . . . . . 10 ⊢ (𝐵 ⊆ 𝐴 ↔ (𝐵 ∩ 𝐴) = 𝐵)
8	7	biimpi 216	. . . . . . . . 9 ⊢ (𝐵 ⊆ 𝐴 → (𝐵 ∩ 𝐴) = 𝐵)
9	6, 8	eqtrid 2792	. . . . . . . 8 ⊢ (𝐵 ⊆ 𝐴 → (𝐴 ∩ 𝐵) = 𝐵)
10	5, 9	sseqtrrid 4062	. . . . . . 7 ⊢ (𝐵 ⊆ 𝐴 → 𝐵 ⊆ (𝐴 ∩ 𝐵))
11		elex 3509	. . . . . . 7 ⊢ (𝑊 ∈ V → 𝑊 ∈ V)
12		inex1g 5337	. . . . . . 7 ⊢ (𝐴 ∈ V → (𝐴 ∩ 𝐵) ∈ V)
13		eqid 2740	. . . . . . . 8 ⊢ (𝑊 ↾s (𝐴 ∩ 𝐵)) = (𝑊 ↾s (𝐴 ∩ 𝐵))
14	13, 3	ressid2 17291	. . . . . . 7 ⊢ ((𝐵 ⊆ (𝐴 ∩ 𝐵) ∧ 𝑊 ∈ V ∧ (𝐴 ∩ 𝐵) ∈ V) → (𝑊 ↾s (𝐴 ∩ 𝐵)) = 𝑊)
15	10, 11, 12, 14	syl3an 1160	. . . . . 6 ⊢ ((𝐵 ⊆ 𝐴 ∧ 𝑊 ∈ V ∧ 𝐴 ∈ V) → (𝑊 ↾s (𝐴 ∩ 𝐵)) = 𝑊)
16	4, 15	eqtr4d 2783	. . . . 5 ⊢ ((𝐵 ⊆ 𝐴 ∧ 𝑊 ∈ V ∧ 𝐴 ∈ V) → (𝑊 ↾s 𝐴) = (𝑊 ↾s (𝐴 ∩ 𝐵)))
17	16	3expb 1120	. . . 4 ⊢ ((𝐵 ⊆ 𝐴 ∧ (𝑊 ∈ V ∧ 𝐴 ∈ V)) → (𝑊 ↾s 𝐴) = (𝑊 ↾s (𝐴 ∩ 𝐵)))
18		inass 4249	. . . . . . . . 9 ⊢ ((𝐴 ∩ 𝐵) ∩ 𝐵) = (𝐴 ∩ (𝐵 ∩ 𝐵))
19		inidm 4248	. . . . . . . . . 10 ⊢ (𝐵 ∩ 𝐵) = 𝐵
20	19	ineq2i 4238	. . . . . . . . 9 ⊢ (𝐴 ∩ (𝐵 ∩ 𝐵)) = (𝐴 ∩ 𝐵)
21	18, 20	eqtr2i 2769	. . . . . . . 8 ⊢ (𝐴 ∩ 𝐵) = ((𝐴 ∩ 𝐵) ∩ 𝐵)
22	21	opeq2i 4901	. . . . . . 7 ⊢ ⟨(Base‘ndx), (𝐴 ∩ 𝐵)⟩ = ⟨(Base‘ndx), ((𝐴 ∩ 𝐵) ∩ 𝐵)⟩
23	22	oveq2i 7459	. . . . . 6 ⊢ (𝑊 sSet ⟨(Base‘ndx), (𝐴 ∩ 𝐵)⟩) = (𝑊 sSet ⟨(Base‘ndx), ((𝐴 ∩ 𝐵) ∩ 𝐵)⟩)
24	2, 3	ressval2 17292	. . . . . 6 ⊢ ((¬ 𝐵 ⊆ 𝐴 ∧ 𝑊 ∈ V ∧ 𝐴 ∈ V) → (𝑊 ↾s 𝐴) = (𝑊 sSet ⟨(Base‘ndx), (𝐴 ∩ 𝐵)⟩))
25		inss1 4258	. . . . . . . . 9 ⊢ (𝐴 ∩ 𝐵) ⊆ 𝐴
26		sstr 4017	. . . . . . . . 9 ⊢ ((𝐵 ⊆ (𝐴 ∩ 𝐵) ∧ (𝐴 ∩ 𝐵) ⊆ 𝐴) → 𝐵 ⊆ 𝐴)
27	25, 26	mpan2 690	. . . . . . . 8 ⊢ (𝐵 ⊆ (𝐴 ∩ 𝐵) → 𝐵 ⊆ 𝐴)
28	27	con3i 154	. . . . . . 7 ⊢ (¬ 𝐵 ⊆ 𝐴 → ¬ 𝐵 ⊆ (𝐴 ∩ 𝐵))
29	13, 3	ressval2 17292	. . . . . . 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 1160	. . . . . 6 ⊢ ((¬ 𝐵 ⊆ 𝐴 ∧ 𝑊 ∈ V ∧ 𝐴 ∈ V) → (𝑊 ↾s (𝐴 ∩ 𝐵)) = (𝑊 sSet ⟨(Base‘ndx), ((𝐴 ∩ 𝐵) ∩ 𝐵)⟩))
31	23, 24, 30	3eqtr4a 2806	. . . . 5 ⊢ ((¬ 𝐵 ⊆ 𝐴 ∧ 𝑊 ∈ V ∧ 𝐴 ∈ V) → (𝑊 ↾s 𝐴) = (𝑊 ↾s (𝐴 ∩ 𝐵)))
32	31	3expb 1120	. . . 4 ⊢ ((¬ 𝐵 ⊆ 𝐴 ∧ (𝑊 ∈ V ∧ 𝐴 ∈ V)) → (𝑊 ↾s 𝐴) = (𝑊 ↾s (𝐴 ∩ 𝐵)))
33	17, 32	pm2.61ian 811	. . 3 ⊢ ((𝑊 ∈ V ∧ 𝐴 ∈ V) → (𝑊 ↾s 𝐴) = (𝑊 ↾s (𝐴 ∩ 𝐵)))
34		reldmress 17289	. . . . . 6 ⊢ Rel dom ↾s
35	34	ovprc1 7487	. . . . 5 ⊢ (¬ 𝑊 ∈ V → (𝑊 ↾s 𝐴) = ∅)
36	34	ovprc1 7487	. . . . 5 ⊢ (¬ 𝑊 ∈ V → (𝑊 ↾s (𝐴 ∩ 𝐵)) = ∅)
37	35, 36	eqtr4d 2783	. . . 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 811	. 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 1087   = wceq 1537   ∈ wcel 2108  Vcvv 3488   ∩ cin 3975   ⊆ wss 3976  ∅c0 4352  ⟨cop 4654  ‘cfv 6573  (class class class)co 7448   sSet csts 17210  ndxcnx 17240  Basecbs 17258   ↾_s cress 17287

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-sbc 3805  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5167  df-opab 5229  df-id 5593  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-iota 6525  df-fun 6575  df-fv 6581  df-ov 7451  df-oprab 7452  df-mpo 7453  df-ress 17288

This theorem is referenced by:  ressress  17307  rescabs  17896  rescabsOLD  17897  resscat  17916  funcres2c  17968  ressffth  18005  cphsubrglem  25230  suborng  33310