Theorem ressressg 13148

Description: Restriction composition law. (Contributed by Stefan O'Rear, 29-Nov-2014.) (Proof shortened by Mario Carneiro, 2-Dec-2014.)

Assertion

Ref	Expression
ressressg	⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝑊 ∈ 𝑍) → ((𝑊 ↾s 𝐴) ↾s 𝐵) = (𝑊 ↾s (𝐴 ∩ 𝐵)))

Proof of Theorem ressressg

Step	Hyp	Ref	Expression
1		eqidd 2230	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝑊 ∈ 𝑍) → (𝑊 ↾s 𝐴) = (𝑊 ↾s 𝐴))
2		eqidd 2230	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝑊 ∈ 𝑍) → (Base‘𝑊) = (Base‘𝑊))
3		simp3 1023	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝑊 ∈ 𝑍) → 𝑊 ∈ 𝑍)
4		simp1 1021	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝑊 ∈ 𝑍) → 𝐴 ∈ 𝑋)
5	1, 2, 3, 4	ressbasd 13140	. . . . . 6 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝑊 ∈ 𝑍) → (𝐴 ∩ (Base‘𝑊)) = (Base‘(𝑊 ↾s 𝐴)))
6	5	ineq2d 3406	. . . . 5 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝑊 ∈ 𝑍) → (𝐵 ∩ (𝐴 ∩ (Base‘𝑊))) = (𝐵 ∩ (Base‘(𝑊 ↾s 𝐴))))
7		inass 3415	. . . . . 6 ⊢ ((𝐵 ∩ 𝐴) ∩ (Base‘𝑊)) = (𝐵 ∩ (𝐴 ∩ (Base‘𝑊)))
8		incom 3397	. . . . . . 7 ⊢ (𝐵 ∩ 𝐴) = (𝐴 ∩ 𝐵)
9	8	ineq1i 3402	. . . . . 6 ⊢ ((𝐵 ∩ 𝐴) ∩ (Base‘𝑊)) = ((𝐴 ∩ 𝐵) ∩ (Base‘𝑊))
10	7, 9	eqtr3i 2252	. . . . 5 ⊢ (𝐵 ∩ (𝐴 ∩ (Base‘𝑊))) = ((𝐴 ∩ 𝐵) ∩ (Base‘𝑊))
11	6, 10	eqtr3di 2277	. . . 4 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝑊 ∈ 𝑍) → (𝐵 ∩ (Base‘(𝑊 ↾s 𝐴))) = ((𝐴 ∩ 𝐵) ∩ (Base‘𝑊)))
12	11	opeq2d 3867	. . 3 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝑊 ∈ 𝑍) → ⟨(Base‘ndx), (𝐵 ∩ (Base‘(𝑊 ↾s 𝐴)))⟩ = ⟨(Base‘ndx), ((𝐴 ∩ 𝐵) ∩ (Base‘𝑊))⟩)
13	12	oveq2d 6029	. 2 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝑊 ∈ 𝑍) → (𝑊 sSet ⟨(Base‘ndx), (𝐵 ∩ (Base‘(𝑊 ↾s 𝐴)))⟩) = (𝑊 sSet ⟨(Base‘ndx), ((𝐴 ∩ 𝐵) ∩ (Base‘𝑊))⟩))
14		ressex 13138	. . . . 5 ⊢ ((𝑊 ∈ 𝑍 ∧ 𝐴 ∈ 𝑋) → (𝑊 ↾s 𝐴) ∈ V)
15	3, 4, 14	syl2anc 411	. . . 4 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝑊 ∈ 𝑍) → (𝑊 ↾s 𝐴) ∈ V)
16		simp2 1022	. . . 4 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝑊 ∈ 𝑍) → 𝐵 ∈ 𝑌)
17		ressvalsets 13137	. . . 4 ⊢ (((𝑊 ↾s 𝐴) ∈ V ∧ 𝐵 ∈ 𝑌) → ((𝑊 ↾s 𝐴) ↾s 𝐵) = ((𝑊 ↾s 𝐴) sSet ⟨(Base‘ndx), (𝐵 ∩ (Base‘(𝑊 ↾s 𝐴)))⟩))
18	15, 16, 17	syl2anc 411	. . 3 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝑊 ∈ 𝑍) → ((𝑊 ↾s 𝐴) ↾s 𝐵) = ((𝑊 ↾s 𝐴) sSet ⟨(Base‘ndx), (𝐵 ∩ (Base‘(𝑊 ↾s 𝐴)))⟩))
19		ressvalsets 13137	. . . . 5 ⊢ ((𝑊 ∈ 𝑍 ∧ 𝐴 ∈ 𝑋) → (𝑊 ↾s 𝐴) = (𝑊 sSet ⟨(Base‘ndx), (𝐴 ∩ (Base‘𝑊))⟩))
20	3, 4, 19	syl2anc 411	. . . 4 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝑊 ∈ 𝑍) → (𝑊 ↾s 𝐴) = (𝑊 sSet ⟨(Base‘ndx), (𝐴 ∩ (Base‘𝑊))⟩))
21	20	oveq1d 6028	. . 3 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝑊 ∈ 𝑍) → ((𝑊 ↾s 𝐴) sSet ⟨(Base‘ndx), (𝐵 ∩ (Base‘(𝑊 ↾s 𝐴)))⟩) = ((𝑊 sSet ⟨(Base‘ndx), (𝐴 ∩ (Base‘𝑊))⟩) sSet ⟨(Base‘ndx), (𝐵 ∩ (Base‘(𝑊 ↾s 𝐴)))⟩))
22		basendxnn 13128	. . . . 5 ⊢ (Base‘ndx) ∈ ℕ
23	22	a1i 9	. . . 4 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝑊 ∈ 𝑍) → (Base‘ndx) ∈ ℕ)
24		inex1g 4223	. . . . 5 ⊢ (𝐴 ∈ 𝑋 → (𝐴 ∩ (Base‘𝑊)) ∈ V)
25	4, 24	syl 14	. . . 4 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝑊 ∈ 𝑍) → (𝐴 ∩ (Base‘𝑊)) ∈ V)
26		inex1g 4223	. . . . 5 ⊢ (𝐵 ∈ 𝑌 → (𝐵 ∩ (Base‘(𝑊 ↾s 𝐴))) ∈ V)
27	16, 26	syl 14	. . . 4 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝑊 ∈ 𝑍) → (𝐵 ∩ (Base‘(𝑊 ↾s 𝐴))) ∈ V)
28	3, 23, 25, 27	setsabsd 13111	. . 3 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝑊 ∈ 𝑍) → ((𝑊 sSet ⟨(Base‘ndx), (𝐴 ∩ (Base‘𝑊))⟩) sSet ⟨(Base‘ndx), (𝐵 ∩ (Base‘(𝑊 ↾s 𝐴)))⟩) = (𝑊 sSet ⟨(Base‘ndx), (𝐵 ∩ (Base‘(𝑊 ↾s 𝐴)))⟩))
29	18, 21, 28	3eqtrd 2266	. 2 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝑊 ∈ 𝑍) → ((𝑊 ↾s 𝐴) ↾s 𝐵) = (𝑊 sSet ⟨(Base‘ndx), (𝐵 ∩ (Base‘(𝑊 ↾s 𝐴)))⟩))
30		inex1g 4223	. . . 4 ⊢ (𝐴 ∈ 𝑋 → (𝐴 ∩ 𝐵) ∈ V)
31	4, 30	syl 14	. . 3 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝑊 ∈ 𝑍) → (𝐴 ∩ 𝐵) ∈ V)
32		ressvalsets 13137	. . 3 ⊢ ((𝑊 ∈ 𝑍 ∧ (𝐴 ∩ 𝐵) ∈ V) → (𝑊 ↾s (𝐴 ∩ 𝐵)) = (𝑊 sSet ⟨(Base‘ndx), ((𝐴 ∩ 𝐵) ∩ (Base‘𝑊))⟩))
33	3, 31, 32	syl2anc 411	. 2 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝑊 ∈ 𝑍) → (𝑊 ↾s (𝐴 ∩ 𝐵)) = (𝑊 sSet ⟨(Base‘ndx), ((𝐴 ∩ 𝐵) ∩ (Base‘𝑊))⟩))
34	13, 29, 33	3eqtr4d 2272	1 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝑊 ∈ 𝑍) → ((𝑊 ↾s 𝐴) ↾s 𝐵) = (𝑊 ↾s (𝐴 ∩ 𝐵)))

Syntax hints:   → wi 4   ∧ w3a 1002   = wceq 1395   ∈ wcel 2200  Vcvv 2800   ∩ cin 3197  ⟨cop 3670  ‘cfv 5324  (class class class)co 6013  ℕcn 9133  ndxcnx 13069   sSet csts 13070  Basecbs 13072   ↾_s cress 13073

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4205  ax-pow 4262  ax-pr 4297  ax-un 4528  ax-setind 4633  ax-cnex 8113  ax-resscn 8114  ax-1re 8116  ax-addrcl 8119

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-ral 2513  df-rex 2514  df-rab 2517  df-v 2802  df-sbc 3030  df-dif 3200  df-un 3202  df-in 3204  df-ss 3211  df-nul 3493  df-pw 3652  df-sn 3673  df-pr 3674  df-op 3676  df-uni 3892  df-int 3927  df-br 4087  df-opab 4149  df-mpt 4150  df-id 4388  df-xp 4729  df-rel 4730  df-cnv 4731  df-co 4732  df-dm 4733  df-rn 4734  df-res 4735  df-iota 5284  df-fun 5326  df-fv 5332  df-ov 6016  df-oprab 6017  df-mpo 6018  df-inn 9134  df-ndx 13075  df-slot 13076  df-base 13078  df-sets 13079  df-iress 13080

This theorem is referenced by:  ressabsg  13149