Theorem restval 17469

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 3478	. 2 ⊢ (𝐽 ∈ 𝑉 → 𝐽 ∈ V)
2		elex 3478	. 2 ⊢ (𝐴 ∈ 𝑊 → 𝐴 ∈ V)
3		mptexg 7209	. . . . 5 ⊢ (𝐽 ∈ V → (𝑥 ∈ 𝐽 ↦ (𝑥 ∩ 𝐴)) ∈ V)
4		rnexg 7887	. . . . 5 ⊢ ((𝑥 ∈ 𝐽 ↦ (𝑥 ∩ 𝐴)) ∈ V → ran (𝑥 ∈ 𝐽 ↦ (𝑥 ∩ 𝐴)) ∈ V)
5	3, 4	syl 18	. . . 4 ⊢ (𝐽 ∈ V → ran (𝑥 ∈ 𝐽 ↦ (𝑥 ∩ 𝐴)) ∈ V)
6	5	adantr 485	. . 3 ⊢ ((𝐽 ∈ V ∧ 𝐴 ∈ V) → ran (𝑥 ∈ 𝐽 ↦ (𝑥 ∩ 𝐴)) ∈ V)
7		simpl 487	. . . . . 6 ⊢ ((𝑗 = 𝐽 ∧ 𝑦 = 𝐴) → 𝑗 = 𝐽)
8		simpr 489	. . . . . . 7 ⊢ ((𝑗 = 𝐽 ∧ 𝑦 = 𝐴) → 𝑦 = 𝐴)
9	8	ineq2d 4175	. . . . . 6 ⊢ ((𝑗 = 𝐽 ∧ 𝑦 = 𝐴) → (𝑥 ∩ 𝑦) = (𝑥 ∩ 𝐴))
10	7, 9	mpteq12dv 5192	. . . . 5 ⊢ ((𝑗 = 𝐽 ∧ 𝑦 = 𝐴) → (𝑥 ∈ 𝑗 ↦ (𝑥 ∩ 𝑦)) = (𝑥 ∈ 𝐽 ↦ (𝑥 ∩ 𝐴)))
11	10	rneqd 5919	. . . 4 ⊢ ((𝑗 = 𝐽 ∧ 𝑦 = 𝐴) → ran (𝑥 ∈ 𝑗 ↦ (𝑥 ∩ 𝑦)) = ran (𝑥 ∈ 𝐽 ↦ (𝑥 ∩ 𝐴)))
12		df-rest 17465	. . . 4 ⊢ ↾t = (𝑗 ∈ V, 𝑦 ∈ V ↦ ran (𝑥 ∈ 𝑗 ↦ (𝑥 ∩ 𝑦)))
13	11, 12	ovmpoga 7554	. . 3 ⊢ ((𝐽 ∈ V ∧ 𝐴 ∈ V ∧ ran (𝑥 ∈ 𝐽 ↦ (𝑥 ∩ 𝐴)) ∈ V) → (𝐽 ↾t 𝐴) = ran (𝑥 ∈ 𝐽 ↦ (𝑥 ∩ 𝐴)))
14	6, 13	mpd3an3 1486	. 2 ⊢ ((𝐽 ∈ V ∧ 𝐴 ∈ V) → (𝐽 ↾t 𝐴) = ran (𝑥 ∈ 𝐽 ↦ (𝑥 ∩ 𝐴)))
15	1, 2, 14	syl2an 607	1 ⊢ ((𝐽 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊) → (𝐽 ↾t 𝐴) = ran (𝑥 ∈ 𝐽 ↦ (𝑥 ∩ 𝐴)))

Syntax hints:   → wi 4   ∧ wa 400   = wceq 1563   ∈ wcel 2145  Vcvv 3457   ∩ cin 3906   ↦ cmpt 5186  ran crn 5653  (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-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-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:  elrest  17470  0rest  17472  restid2  17473  tgrest  23277  resttopon  23279  restco  23282  rest0  23287  restfpw  23297  neitr  23298  ordtrest2  23322  1stcrest  23571  2ndcrest  23572  kgencmp  23663  xkoptsub  23772  trfilss  24007  trfg  24009  uzrest  24015  restmetu  24688  ellimc2  25997  limcflf  26001  ordtrest2NEW  34230  ptrest  38130