Theorem restin 15260

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 4583	. . . . 5 ⊢ (𝐽 ∈ 𝑉 → ∪ 𝐽 ∈ V)
3	1, 2	eqeltrid 2325	. . . 4 ⊢ (𝐽 ∈ 𝑉 → 𝑋 ∈ V)
4	3	adantr 276	. . 3 ⊢ ((𝐽 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊) → 𝑋 ∈ V)
5		restco 15258	. . . 4 ⊢ ((𝐽 ∈ 𝑉 ∧ 𝑋 ∈ V ∧ 𝐴 ∈ 𝑊) → ((𝐽 ↾t 𝑋) ↾t 𝐴) = (𝐽 ↾t (𝑋 ∩ 𝐴)))
6	5	3com23 1240	. . 3 ⊢ ((𝐽 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊 ∧ 𝑋 ∈ V) → ((𝐽 ↾t 𝑋) ↾t 𝐴) = (𝐽 ↾t (𝑋 ∩ 𝐴)))
7	4, 6	mpd3an3 1379	. 2 ⊢ ((𝐽 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊) → ((𝐽 ↾t 𝑋) ↾t 𝐴) = (𝐽 ↾t (𝑋 ∩ 𝐴)))
8	1	restid 13587	. . . 4 ⊢ (𝐽 ∈ 𝑉 → (𝐽 ↾t 𝑋) = 𝐽)
9	8	adantr 276	. . 3 ⊢ ((𝐽 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊) → (𝐽 ↾t 𝑋) = 𝐽)
10	9	oveq1d 6094	. 2 ⊢ ((𝐽 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊) → ((𝐽 ↾t 𝑋) ↾t 𝐴) = (𝐽 ↾t 𝐴))
11		incom 3421	. . . 4 ⊢ (𝑋 ∩ 𝐴) = (𝐴 ∩ 𝑋)
12	11	oveq2i 6090	. . 3 ⊢ (𝐽 ↾t (𝑋 ∩ 𝐴)) = (𝐽 ↾t (𝐴 ∩ 𝑋))
13	12	a1i 9	. 2 ⊢ ((𝐽 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊) → (𝐽 ↾t (𝑋 ∩ 𝐴)) = (𝐽 ↾t (𝐴 ∩ 𝑋)))
14	7, 10, 13	3eqtr3d 2279	1 ⊢ ((𝐽 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊) → (𝐽 ↾t 𝐴) = (𝐽 ↾t (𝐴 ∩ 𝑋)))

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1402   ∈ wcel 2209  Vcvv 2821   ∩ cin 3219  ∪ cuni 3933  (class class class)co 6079   ↾_t crest 13576

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4244  ax-sep 4247  ax-pow 4309  ax-pr 4344  ax-un 4576  ax-setind 4682

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-pw 3690  df-sn 3714  df-pr 3715  df-op 3717  df-uni 3934  df-iun 4012  df-br 4129  df-opab 4191  df-mpt 4192  df-id 4436  df-xp 4778  df-rel 4779  df-cnv 4780  df-co 4781  df-dm 4782  df-rn 4783  df-res 4784  df-ima 4785  df-iota 5335  df-fun 5377  df-fn 5378  df-f 5379  df-f1 5380  df-fo 5381  df-f1o 5382  df-fv 5383  df-ov 6082  df-oprab 6083  df-mpo 6084  df-1st 6368  df-2nd 6369  df-rest 13578

This theorem is referenced by:  restuni2  15261  cnrest2r  15321