Theorem ressressg 12778

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 2197	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝑊 ∈ 𝑍) → (𝑊 ↾s 𝐴) = (𝑊 ↾s 𝐴))
2		eqidd 2197	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝑊 ∈ 𝑍) → (Base‘𝑊) = (Base‘𝑊))
3		simp3 1001	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝑊 ∈ 𝑍) → 𝑊 ∈ 𝑍)
4		simp1 999	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝑊 ∈ 𝑍) → 𝐴 ∈ 𝑋)
5	1, 2, 3, 4	ressbasd 12770	. . . . . 6 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝑊 ∈ 𝑍) → (𝐴 ∩ (Base‘𝑊)) = (Base‘(𝑊 ↾s 𝐴)))
6	5	ineq2d 3365	. . . . 5 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝑊 ∈ 𝑍) → (𝐵 ∩ (𝐴 ∩ (Base‘𝑊))) = (𝐵 ∩ (Base‘(𝑊 ↾s 𝐴))))
7		inass 3374	. . . . . 6 ⊢ ((𝐵 ∩ 𝐴) ∩ (Base‘𝑊)) = (𝐵 ∩ (𝐴 ∩ (Base‘𝑊)))
8		incom 3356	. . . . . . 7 ⊢ (𝐵 ∩ 𝐴) = (𝐴 ∩ 𝐵)
9	8	ineq1i 3361	. . . . . 6 ⊢ ((𝐵 ∩ 𝐴) ∩ (Base‘𝑊)) = ((𝐴 ∩ 𝐵) ∩ (Base‘𝑊))
10	7, 9	eqtr3i 2219	. . . . 5 ⊢ (𝐵 ∩ (𝐴 ∩ (Base‘𝑊))) = ((𝐴 ∩ 𝐵) ∩ (Base‘𝑊))
11	6, 10	eqtr3di 2244	. . . 4 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝑊 ∈ 𝑍) → (𝐵 ∩ (Base‘(𝑊 ↾s 𝐴))) = ((𝐴 ∩ 𝐵) ∩ (Base‘𝑊)))
12	11	opeq2d 3816	. . 3 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝑊 ∈ 𝑍) → ⟨(Base‘ndx), (𝐵 ∩ (Base‘(𝑊 ↾s 𝐴)))⟩ = ⟨(Base‘ndx), ((𝐴 ∩ 𝐵) ∩ (Base‘𝑊))⟩)
13	12	oveq2d 5941	. 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 12768	. . . . 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 1000	. . . 4 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝑊 ∈ 𝑍) → 𝐵 ∈ 𝑌)
17		ressvalsets 12767	. . . 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 12767	. . . . 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 5940	. . 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 12759	. . . . 5 ⊢ (Base‘ndx) ∈ ℕ
23	22	a1i 9	. . . 4 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝑊 ∈ 𝑍) → (Base‘ndx) ∈ ℕ)
24		inex1g 4170	. . . . 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 4170	. . . . 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 12742	. . 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 2233	. 2 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝑊 ∈ 𝑍) → ((𝑊 ↾s 𝐴) ↾s 𝐵) = (𝑊 sSet ⟨(Base‘ndx), (𝐵 ∩ (Base‘(𝑊 ↾s 𝐴)))⟩))
30		inex1g 4170	. . . 4 ⊢ (𝐴 ∈ 𝑋 → (𝐴 ∩ 𝐵) ∈ V)
31	4, 30	syl 14	. . 3 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝑊 ∈ 𝑍) → (𝐴 ∩ 𝐵) ∈ V)
32		ressvalsets 12767	. . 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 2239	1 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝑊 ∈ 𝑍) → ((𝑊 ↾s 𝐴) ↾s 𝐵) = (𝑊 ↾s (𝐴 ∩ 𝐵)))

Syntax hints:   → wi 4   ∧ w3a 980   = wceq 1364   ∈ wcel 2167  Vcvv 2763   ∩ cin 3156  ⟨cop 3626  ‘cfv 5259  (class class class)co 5925  ℕcn 9007  ndxcnx 12700   sSet csts 12701  Basecbs 12703   ↾_s cress 12704

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 615  ax-in2 616  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-sep 4152  ax-pow 4208  ax-pr 4243  ax-un 4469  ax-setind 4574  ax-cnex 7987  ax-resscn 7988  ax-1re 7990  ax-addrcl 7993

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-ral 2480  df-rex 2481  df-rab 2484  df-v 2765  df-sbc 2990  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3452  df-pw 3608  df-sn 3629  df-pr 3630  df-op 3632  df-uni 3841  df-int 3876  df-br 4035  df-opab 4096  df-mpt 4097  df-id 4329  df-xp 4670  df-rel 4671  df-cnv 4672  df-co 4673  df-dm 4674  df-rn 4675  df-res 4676  df-iota 5220  df-fun 5261  df-fv 5267  df-ov 5928  df-oprab 5929  df-mpo 5930  df-inn 9008  df-ndx 12706  df-slot 12707  df-base 12709  df-sets 12710  df-iress 12711

This theorem is referenced by:  ressabsg  12779