Theorem restval 12859

Description: The subspace topology induced by the topology 𝐽 on the set 𝐴. (Contributed by FL, 20-Sep-2010.) (Revised by Mario Carneiro, 1-May-2015.)

Assertion

Ref	Expression
restval	⊢ ((𝐽 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊) → (𝐽 ↾t 𝐴) = ran (𝑥 ∈ 𝐽 ↦ (𝑥 ∩ 𝐴)))

Distinct variable groups:   𝑥,𝐴   𝑥,𝐽

Allowed substitution hints:   𝑉(𝑥)   𝑊(𝑥)

Proof of Theorem restval

Dummy variables 𝑗 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		elex 2771	. 2 ⊢ (𝐽 ∈ 𝑉 → 𝐽 ∈ V)
2		elex 2771	. 2 ⊢ (𝐴 ∈ 𝑊 → 𝐴 ∈ V)
3		mptexg 5784	. . . . 5 ⊢ (𝐽 ∈ V → (𝑥 ∈ 𝐽 ↦ (𝑥 ∩ 𝐴)) ∈ V)
4		rnexg 4928	. . . . 5 ⊢ ((𝑥 ∈ 𝐽 ↦ (𝑥 ∩ 𝐴)) ∈ V → ran (𝑥 ∈ 𝐽 ↦ (𝑥 ∩ 𝐴)) ∈ V)
5	3, 4	syl 14	. . . 4 ⊢ (𝐽 ∈ V → ran (𝑥 ∈ 𝐽 ↦ (𝑥 ∩ 𝐴)) ∈ V)
6	5	adantr 276	. . 3 ⊢ ((𝐽 ∈ V ∧ 𝐴 ∈ V) → ran (𝑥 ∈ 𝐽 ↦ (𝑥 ∩ 𝐴)) ∈ V)
7		simpl 109	. . . . . 6 ⊢ ((𝑗 = 𝐽 ∧ 𝑦 = 𝐴) → 𝑗 = 𝐽)
8		simpr 110	. . . . . . 7 ⊢ ((𝑗 = 𝐽 ∧ 𝑦 = 𝐴) → 𝑦 = 𝐴)
9	8	ineq2d 3361	. . . . . 6 ⊢ ((𝑗 = 𝐽 ∧ 𝑦 = 𝐴) → (𝑥 ∩ 𝑦) = (𝑥 ∩ 𝐴))
10	7, 9	mpteq12dv 4112	. . . . 5 ⊢ ((𝑗 = 𝐽 ∧ 𝑦 = 𝐴) → (𝑥 ∈ 𝑗 ↦ (𝑥 ∩ 𝑦)) = (𝑥 ∈ 𝐽 ↦ (𝑥 ∩ 𝐴)))
11	10	rneqd 4892	. . . 4 ⊢ ((𝑗 = 𝐽 ∧ 𝑦 = 𝐴) → ran (𝑥 ∈ 𝑗 ↦ (𝑥 ∩ 𝑦)) = ran (𝑥 ∈ 𝐽 ↦ (𝑥 ∩ 𝐴)))
12		df-rest 12855	. . . 4 ⊢ ↾t = (𝑗 ∈ V, 𝑦 ∈ V ↦ ran (𝑥 ∈ 𝑗 ↦ (𝑥 ∩ 𝑦)))
13	11, 12	ovmpoga 6049	. . 3 ⊢ ((𝐽 ∈ V ∧ 𝐴 ∈ V ∧ ran (𝑥 ∈ 𝐽 ↦ (𝑥 ∩ 𝐴)) ∈ V) → (𝐽 ↾t 𝐴) = ran (𝑥 ∈ 𝐽 ↦ (𝑥 ∩ 𝐴)))
14	6, 13	mpd3an3 1349	. 2 ⊢ ((𝐽 ∈ V ∧ 𝐴 ∈ V) → (𝐽 ↾t 𝐴) = ran (𝑥 ∈ 𝐽 ↦ (𝑥 ∩ 𝐴)))
15	1, 2, 14	syl2an 289	1 ⊢ ((𝐽 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊) → (𝐽 ↾t 𝐴) = ran (𝑥 ∈ 𝐽 ↦ (𝑥 ∩ 𝐴)))

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1364   ∈ wcel 2164  Vcvv 2760   ∩ cin 3153   ↦ cmpt 4091  ran crn 4661  (class class class)co 5919   ↾_t crest 12853

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 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-coll 4145  ax-sep 4148  ax-pow 4204  ax-pr 4239  ax-un 4465  ax-setind 4570

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-ral 2477  df-rex 2478  df-reu 2479  df-rab 2481  df-v 2762  df-sbc 2987  df-csb 3082  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-pw 3604  df-sn 3625  df-pr 3626  df-op 3628  df-uni 3837  df-iun 3915  df-br 4031  df-opab 4092  df-mpt 4093  df-id 4325  df-xp 4666  df-rel 4667  df-cnv 4668  df-co 4669  df-dm 4670  df-rn 4671  df-res 4672  df-ima 4673  df-iota 5216  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5922  df-oprab 5923  df-mpo 5924  df-rest 12855

This theorem is referenced by:  elrest  12860  restid2  12862  tgrest  14348  resttopon  14350  restco  14353  rest0  14358