Theorem ressressg 13277

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 2233	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝑊 ∈ 𝑍) → (𝑊 ↾s 𝐴) = (𝑊 ↾s 𝐴))
2		eqidd 2233	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝑊 ∈ 𝑍) → (Base‘𝑊) = (Base‘𝑊))
3		simp3 1026	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝑊 ∈ 𝑍) → 𝑊 ∈ 𝑍)
4		simp1 1024	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝑊 ∈ 𝑍) → 𝐴 ∈ 𝑋)
5	1, 2, 3, 4	ressbasd 13269	. . . . . 6 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝑊 ∈ 𝑍) → (𝐴 ∩ (Base‘𝑊)) = (Base‘(𝑊 ↾s 𝐴)))
6	5	ineq2d 3421	. . . . 5 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝑊 ∈ 𝑍) → (𝐵 ∩ (𝐴 ∩ (Base‘𝑊))) = (𝐵 ∩ (Base‘(𝑊 ↾s 𝐴))))
7		inass 3430	. . . . . 6 ⊢ ((𝐵 ∩ 𝐴) ∩ (Base‘𝑊)) = (𝐵 ∩ (𝐴 ∩ (Base‘𝑊)))
8		incom 3410	. . . . . . 7 ⊢ (𝐵 ∩ 𝐴) = (𝐴 ∩ 𝐵)
9	8	ineq1i 3417	. . . . . 6 ⊢ ((𝐵 ∩ 𝐴) ∩ (Base‘𝑊)) = ((𝐴 ∩ 𝐵) ∩ (Base‘𝑊))
10	7, 9	eqtr3i 2255	. . . . 5 ⊢ (𝐵 ∩ (𝐴 ∩ (Base‘𝑊))) = ((𝐴 ∩ 𝐵) ∩ (Base‘𝑊))
11	6, 10	eqtr3di 2280	. . . 4 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝑊 ∈ 𝑍) → (𝐵 ∩ (Base‘(𝑊 ↾s 𝐴))) = ((𝐴 ∩ 𝐵) ∩ (Base‘𝑊)))
12	11	opeq2d 3889	. . 3 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝑊 ∈ 𝑍) → ⟨(Base‘ndx), (𝐵 ∩ (Base‘(𝑊 ↾s 𝐴)))⟩ = ⟨(Base‘ndx), ((𝐴 ∩ 𝐵) ∩ (Base‘𝑊))⟩)
13	12	oveq2d 6065	. 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 13267	. . . . 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 1025	. . . 4 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝑊 ∈ 𝑍) → 𝐵 ∈ 𝑌)
17		ressvalsets 13266	. . . 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 13266	. . . . 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 6064	. . 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 13257	. . . . 5 ⊢ (Base‘ndx) ∈ ℕ
23	22	a1i 9	. . . 4 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝑊 ∈ 𝑍) → (Base‘ndx) ∈ ℕ)
24		inex1g 4245	. . . . 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 4245	. . . . 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 13240	. . 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 2269	. 2 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝑊 ∈ 𝑍) → ((𝑊 ↾s 𝐴) ↾s 𝐵) = (𝑊 sSet ⟨(Base‘ndx), (𝐵 ∩ (Base‘(𝑊 ↾s 𝐴)))⟩))
30		inex1g 4245	. . . 4 ⊢ (𝐴 ∈ 𝑋 → (𝐴 ∩ 𝐵) ∈ V)
31	4, 30	syl 14	. . 3 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝑊 ∈ 𝑍) → (𝐴 ∩ 𝐵) ∈ V)
32		ressvalsets 13266	. . 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 2275	1 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝑊 ∈ 𝑍) → ((𝑊 ↾s 𝐴) ↾s 𝐵) = (𝑊 ↾s (𝐴 ∩ 𝐵)))

Syntax hints:   → wi 4   ∧ w3a 1005   = wceq 1398   ∈ wcel 2203  Vcvv 2812   ∩ cin 3209  ⟨cop 3691  ‘cfv 5351  (class class class)co 6049  ℕcn 9233  ndxcnx 13198   sSet csts 13199  Basecbs 13201   ↾_s cress 13202

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2205  ax-14 2206  ax-ext 2214  ax-sep 4227  ax-pow 4286  ax-pr 4321  ax-un 4553  ax-setind 4658  ax-cnex 8214  ax-resscn 8215  ax-1re 8217  ax-addrcl 8220

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-ral 2525  df-rex 2526  df-rab 2529  df-v 2814  df-sbc 3042  df-dif 3212  df-un 3214  df-in 3216  df-ss 3223  df-nul 3508  df-pw 3670  df-sn 3694  df-pr 3695  df-op 3697  df-uni 3914  df-int 3949  df-br 4109  df-opab 4171  df-mpt 4172  df-id 4413  df-xp 4754  df-rel 4755  df-cnv 4756  df-co 4757  df-dm 4758  df-rn 4759  df-res 4760  df-iota 5311  df-fun 5353  df-fv 5359  df-ov 6052  df-oprab 6053  df-mpo 6054  df-inn 9234  df-ndx 13204  df-slot 13205  df-base 13207  df-sets 13208  df-iress 13209

This theorem is referenced by:  ressabsg  13278