Theorem restin 23284

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 7727	. . . . 5 ⊢ (𝐽 ∈ 𝑉 → ∪ 𝐽 ∈ V)
3	1, 2	eqeltrid 2869	. . . 4 ⊢ (𝐽 ∈ 𝑉 → 𝑋 ∈ V)
4	3	adantr 485	. . 3 ⊢ ((𝐽 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊) → 𝑋 ∈ V)
5		restco 23282	. . . 4 ⊢ ((𝐽 ∈ 𝑉 ∧ 𝑋 ∈ V ∧ 𝐴 ∈ 𝑊) → ((𝐽 ↾t 𝑋) ↾t 𝐴) = (𝐽 ↾t (𝑋 ∩ 𝐴)))
6	5	3com23 1142	. . 3 ⊢ ((𝐽 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊 ∧ 𝑋 ∈ V) → ((𝐽 ↾t 𝑋) ↾t 𝐴) = (𝐽 ↾t (𝑋 ∩ 𝐴)))
7	4, 6	mpd3an3 1486	. 2 ⊢ ((𝐽 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊) → ((𝐽 ↾t 𝑋) ↾t 𝐴) = (𝐽 ↾t (𝑋 ∩ 𝐴)))
8	1	restid 17476	. . . 4 ⊢ (𝐽 ∈ 𝑉 → (𝐽 ↾t 𝑋) = 𝐽)
9	8	adantr 485	. . 3 ⊢ ((𝐽 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊) → (𝐽 ↾t 𝑋) = 𝐽)
10	9	oveq1d 7415	. 2 ⊢ ((𝐽 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊) → ((𝐽 ↾t 𝑋) ↾t 𝐴) = (𝐽 ↾t 𝐴))
11		incom 4164	. . . 4 ⊢ (𝑋 ∩ 𝐴) = (𝐴 ∩ 𝑋)
12	11	oveq2i 7411	. . 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 2808	1 ⊢ ((𝐽 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊) → (𝐽 ↾t 𝐴) = (𝐽 ↾t (𝐴 ∩ 𝑋)))

Syntax hints:   → wi 4   ∧ wa 400   = wceq 1563   ∈ wcel 2145  Vcvv 3457   ∩ cin 3906  ∪ cuni 4868  (class class class)co 7400   ↾_t crest 17463

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2737  ax-rep 5232  ax-sep 5251  ax-nul 5261  ax-pow 5327  ax-pr 5395  ax-un 7722

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-nf 1807  df-sb 2094  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3080  df-rex 3090  df-reu 3371  df-rab 3418  df-v 3459  df-sbc 3748  df-csb 3856  df-dif 3910  df-un 3912  df-in 3914  df-ss 3924  df-nul 4289  df-if 4484  df-pw 4560  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4869  df-iun 4954  df-br 5106  df-opab 5168  df-mpt 5187  df-id 5547  df-xp 5658  df-rel 5659  df-cnv 5660  df-co 5661  df-dm 5662  df-rn 5663  df-res 5664  df-ima 5665  df-iota 6481  df-fun 6527  df-fn 6528  df-f 6529  df-f1 6530  df-fo 6531  df-f1o 6532  df-fv 6533  df-ov 7403  df-oprab 7404  df-mpo 7405  df-rest 17465

This theorem is referenced by:  restuni2  23285  cnrest2r  23405  cnrmi  23478  restcnrm  23480  resthauslem  23481  imacmp  23515  fiuncmp  23522  kgeni  23655  ressxms  24643  ptrest  38130  restuni6  45698