Theorem restin 23397

Description: When the subspace region is not a subset of the base of the topology, the resulting set is the same as the subspace restricted to the base. (Contributed by Mario Carneiro, 15-Dec-2013.)

Hypothesis

Ref	Expression
restin.1	⊢ 𝑋 = ∪ 𝐽

Assertion

Ref	Expression
restin	⊢ ((𝐽 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊) → (𝐽 ↾t 𝐴) = (𝐽 ↾t (𝐴 ∩ 𝑋)))

Proof of Theorem restin

Step	Hyp	Ref	Expression
1		restin.1	. . . . 5 ⊢ 𝑋 = ∪ 𝐽
2		uniexg 7746	. . . . 5 ⊢ (𝐽 ∈ 𝑉 → ∪ 𝐽 ∈ V)
3	1, 2	eqeltrid 2866	. . . 4 ⊢ (𝐽 ∈ 𝑉 → 𝑋 ∈ V)
4	3	adantr 486	. . 3 ⊢ ((𝐽 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊) → 𝑋 ∈ V)
5		restco 23395	. . . 4 ⊢ ((𝐽 ∈ 𝑉 ∧ 𝑋 ∈ V ∧ 𝐴 ∈ 𝑊) → ((𝐽 ↾t 𝑋) ↾t 𝐴) = (𝐽 ↾t (𝑋 ∩ 𝐴)))
6	5	3com23 1144	. . 3 ⊢ ((𝐽 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊 ∧ 𝑋 ∈ V) → ((𝐽 ↾t 𝑋) ↾t 𝐴) = (𝐽 ↾t (𝑋 ∩ 𝐴)))
7	4, 6	mpd3an3 1491	. 2 ⊢ ((𝐽 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊) → ((𝐽 ↾t 𝑋) ↾t 𝐴) = (𝐽 ↾t (𝑋 ∩ 𝐴)))
8	1	restid 17524	. . . 4 ⊢ (𝐽 ∈ 𝑉 → (𝐽 ↾t 𝑋) = 𝐽)
9	8	adantr 486	. . 3 ⊢ ((𝐽 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊) → (𝐽 ↾t 𝑋) = 𝐽)
10	9	oveq1d 7432	. 2 ⊢ ((𝐽 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊) → ((𝐽 ↾t 𝑋) ↾t 𝐴) = (𝐽 ↾t 𝐴))
11		incom 4158	. . . 4 ⊢ (𝑋 ∩ 𝐴) = (𝐴 ∩ 𝑋)
12	11	oveq2i 7428	. . 3 ⊢ (𝐽 ↾t (𝑋 ∩ 𝐴)) = (𝐽 ↾t (𝐴 ∩ 𝑋))
13	12	a1i 11	. 2 ⊢ ((𝐽 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊) → (𝐽 ↾t (𝑋 ∩ 𝐴)) = (𝐽 ↾t (𝐴 ∩ 𝑋)))
14	7, 10, 13	3eqtr3d 2805	1 ⊢ ((𝐽 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊) → (𝐽 ↾t 𝐴) = (𝐽 ↾t (𝐴 ∩ 𝑋)))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2145  Vcvv 3453   ∩ cin 3901  ∪ cuni 4870  (class class class)co 7417   ↾_t crest 17511

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2734  ax-rep 5236  ax-sep 5255  ax-nul 5267  ax-pow 5334  ax-pr 5402  ax-un 7740

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3079  df-rex 3089  df-reu 3368  df-rab 3415  df-v 3455  df-sbc 3743  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4283  df-if 4486  df-pw 4562  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4871  df-iun 4956  df-br 5108  df-opab 5172  df-mpt 5191  df-id 5554  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-ov 7420  df-oprab 7421  df-mpo 7422  df-rest 17513

This theorem is used by:  restuni2  23398  cnrest2r  23518  cnrmi  23591  restcnrm  23593  resthauslem  23594  imacmp  23628  fiuncmp  23635  kgeni  23769  ressxms  24757  ptrest  38376  restuni6  45962