Theorem restval 17350

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 3462	. 2 ⊢ (𝐽 ∈ 𝑉 → 𝐽 ∈ V)
2		elex 3462	. 2 ⊢ (𝐴 ∈ 𝑊 → 𝐴 ∈ V)
3		mptexg 7169	. . . . 5 ⊢ (𝐽 ∈ V → (𝑥 ∈ 𝐽 ↦ (𝑥 ∩ 𝐴)) ∈ V)
4		rnexg 7846	. . . . 5 ⊢ ((𝑥 ∈ 𝐽 ↦ (𝑥 ∩ 𝐴)) ∈ V → ran (𝑥 ∈ 𝐽 ↦ (𝑥 ∩ 𝐴)) ∈ V)
5	3, 4	syl 17	. . . 4 ⊢ (𝐽 ∈ V → ran (𝑥 ∈ 𝐽 ↦ (𝑥 ∩ 𝐴)) ∈ V)
6	5	adantr 480	. . 3 ⊢ ((𝐽 ∈ V ∧ 𝐴 ∈ V) → ran (𝑥 ∈ 𝐽 ↦ (𝑥 ∩ 𝐴)) ∈ V)
7		simpl 482	. . . . . 6 ⊢ ((𝑗 = 𝐽 ∧ 𝑦 = 𝐴) → 𝑗 = 𝐽)
8		simpr 484	. . . . . . 7 ⊢ ((𝑗 = 𝐽 ∧ 𝑦 = 𝐴) → 𝑦 = 𝐴)
9	8	ineq2d 4173	. . . . . 6 ⊢ ((𝑗 = 𝐽 ∧ 𝑦 = 𝐴) → (𝑥 ∩ 𝑦) = (𝑥 ∩ 𝐴))
10	7, 9	mpteq12dv 5186	. . . . 5 ⊢ ((𝑗 = 𝐽 ∧ 𝑦 = 𝐴) → (𝑥 ∈ 𝑗 ↦ (𝑥 ∩ 𝑦)) = (𝑥 ∈ 𝐽 ↦ (𝑥 ∩ 𝐴)))
11	10	rneqd 5888	. . . 4 ⊢ ((𝑗 = 𝐽 ∧ 𝑦 = 𝐴) → ran (𝑥 ∈ 𝑗 ↦ (𝑥 ∩ 𝑦)) = ran (𝑥 ∈ 𝐽 ↦ (𝑥 ∩ 𝐴)))
12		df-rest 17346	. . . 4 ⊢ ↾t = (𝑗 ∈ V, 𝑦 ∈ V ↦ ran (𝑥 ∈ 𝑗 ↦ (𝑥 ∩ 𝑦)))
13	11, 12	ovmpoga 7514	. . 3 ⊢ ((𝐽 ∈ V ∧ 𝐴 ∈ V ∧ ran (𝑥 ∈ 𝐽 ↦ (𝑥 ∩ 𝐴)) ∈ V) → (𝐽 ↾t 𝐴) = ran (𝑥 ∈ 𝐽 ↦ (𝑥 ∩ 𝐴)))
14	6, 13	mpd3an3 1465	. 2 ⊢ ((𝐽 ∈ V ∧ 𝐴 ∈ V) → (𝐽 ↾t 𝐴) = ran (𝑥 ∈ 𝐽 ↦ (𝑥 ∩ 𝐴)))
15	1, 2, 14	syl2an 597	1 ⊢ ((𝐽 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊) → (𝐽 ↾t 𝐴) = ran (𝑥 ∈ 𝐽 ↦ (𝑥 ∩ 𝐴)))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1542   ∈ wcel 2114  Vcvv 3441   ∩ cin 3901   ↦ cmpt 5180  ran crn 5626  (class class class)co 7360   ↾_t crest 17344

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5225  ax-sep 5242  ax-nul 5252  ax-pr 5378  ax-un 7682

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3062  df-reu 3352  df-rab 3401  df-v 3443  df-sbc 3742  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4287  df-if 4481  df-sn 4582  df-pr 4584  df-op 4588  df-uni 4865  df-iun 4949  df-br 5100  df-opab 5162  df-mpt 5181  df-id 5520  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-iota 6449  df-fun 6495  df-fn 6496  df-f 6497  df-f1 6498  df-fo 6499  df-f1o 6500  df-fv 6501  df-ov 7363  df-oprab 7364  df-mpo 7365  df-rest 17346

This theorem is referenced by:  elrest  17351  0rest  17353  restid2  17354  tgrest  23107  resttopon  23109  restco  23112  rest0  23117  restfpw  23127  neitr  23128  ordtrest2  23152  1stcrest  23401  2ndcrest  23402  kgencmp  23493  xkoptsub  23602  trfilss  23837  trfg  23839  uzrest  23845  restmetu  24518  ellimc2  25838  limcflf  25842  ordtrest2NEW  34061  ptrest  37791