Theorem ressressg 12537

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 2178	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝑊 ∈ 𝑍) → (𝑊 ↾s 𝐴) = (𝑊 ↾s 𝐴))
2		eqidd 2178	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝑊 ∈ 𝑍) → (Base‘𝑊) = (Base‘𝑊))
3		simp3 999	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝑊 ∈ 𝑍) → 𝑊 ∈ 𝑍)
4		simp1 997	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝑊 ∈ 𝑍) → 𝐴 ∈ 𝑋)
5	1, 2, 3, 4	ressbasd 12530	. . . . . 6 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝑊 ∈ 𝑍) → (𝐴 ∩ (Base‘𝑊)) = (Base‘(𝑊 ↾s 𝐴)))
6	5	ineq2d 3338	. . . . 5 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝑊 ∈ 𝑍) → (𝐵 ∩ (𝐴 ∩ (Base‘𝑊))) = (𝐵 ∩ (Base‘(𝑊 ↾s 𝐴))))
7		inass 3347	. . . . . 6 ⊢ ((𝐵 ∩ 𝐴) ∩ (Base‘𝑊)) = (𝐵 ∩ (𝐴 ∩ (Base‘𝑊)))
8		incom 3329	. . . . . . 7 ⊢ (𝐵 ∩ 𝐴) = (𝐴 ∩ 𝐵)
9	8	ineq1i 3334	. . . . . 6 ⊢ ((𝐵 ∩ 𝐴) ∩ (Base‘𝑊)) = ((𝐴 ∩ 𝐵) ∩ (Base‘𝑊))
10	7, 9	eqtr3i 2200	. . . . 5 ⊢ (𝐵 ∩ (𝐴 ∩ (Base‘𝑊))) = ((𝐴 ∩ 𝐵) ∩ (Base‘𝑊))
11	6, 10	eqtr3di 2225	. . . 4 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝑊 ∈ 𝑍) → (𝐵 ∩ (Base‘(𝑊 ↾s 𝐴))) = ((𝐴 ∩ 𝐵) ∩ (Base‘𝑊)))
12	11	opeq2d 3787	. . 3 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝑊 ∈ 𝑍) → ⟨(Base‘ndx), (𝐵 ∩ (Base‘(𝑊 ↾s 𝐴)))⟩ = ⟨(Base‘ndx), ((𝐴 ∩ 𝐵) ∩ (Base‘𝑊))⟩)
13	12	oveq2d 5894	. 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 12528	. . . . 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 998	. . . 4 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝑊 ∈ 𝑍) → 𝐵 ∈ 𝑌)
17		ressvalsets 12527	. . . 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 12527	. . . . 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 5893	. . 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 12521	. . . . 5 ⊢ (Base‘ndx) ∈ ℕ
23	22	a1i 9	. . . 4 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝑊 ∈ 𝑍) → (Base‘ndx) ∈ ℕ)
24		inex1g 4141	. . . . 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 4141	. . . . 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 12504	. . 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 2214	. 2 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝑊 ∈ 𝑍) → ((𝑊 ↾s 𝐴) ↾s 𝐵) = (𝑊 sSet ⟨(Base‘ndx), (𝐵 ∩ (Base‘(𝑊 ↾s 𝐴)))⟩))
30		inex1g 4141	. . . 4 ⊢ (𝐴 ∈ 𝑋 → (𝐴 ∩ 𝐵) ∈ V)
31	4, 30	syl 14	. . 3 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝑊 ∈ 𝑍) → (𝐴 ∩ 𝐵) ∈ V)
32		ressvalsets 12527	. . 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 2220	1 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝑊 ∈ 𝑍) → ((𝑊 ↾s 𝐴) ↾s 𝐵) = (𝑊 ↾s (𝐴 ∩ 𝐵)))

Syntax hints:   → wi 4   ∧ w3a 978   = wceq 1353   ∈ wcel 2148  Vcvv 2739   ∩ cin 3130  ⟨cop 3597  ‘cfv 5218  (class class class)co 5878  ℕcn 8922  ndxcnx 12462   sSet csts 12463  Basecbs 12465   ↾_s cress 12466

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4123  ax-pow 4176  ax-pr 4211  ax-un 4435  ax-setind 4538  ax-cnex 7905  ax-resscn 7906  ax-1re 7908  ax-addrcl 7911

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-rab 2464  df-v 2741  df-sbc 2965  df-dif 3133  df-un 3135  df-in 3137  df-ss 3144  df-nul 3425  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-int 3847  df-br 4006  df-opab 4067  df-mpt 4068  df-id 4295  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-rn 4639  df-res 4640  df-iota 5180  df-fun 5220  df-fv 5226  df-ov 5881  df-oprab 5882  df-mpo 5883  df-inn 8923  df-ndx 12468  df-slot 12469  df-base 12471  df-sets 12472  df-iress 12473

This theorem is referenced by:  ressabsg  12538