Theorem restval 17358

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 3463	. 2 ⊢ (𝐽 ∈ 𝑉 → 𝐽 ∈ V)
2		elex 3463	. 2 ⊢ (𝐴 ∈ 𝑊 → 𝐴 ∈ V)
3		mptexg 7177	. . . . 5 ⊢ (𝐽 ∈ V → (𝑥 ∈ 𝐽 ↦ (𝑥 ∩ 𝐴)) ∈ V)
4		rnexg 7854	. . . . 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 4174	. . . . . 6 ⊢ ((𝑗 = 𝐽 ∧ 𝑦 = 𝐴) → (𝑥 ∩ 𝑦) = (𝑥 ∩ 𝐴))
10	7, 9	mpteq12dv 5187	. . . . 5 ⊢ ((𝑗 = 𝐽 ∧ 𝑦 = 𝐴) → (𝑥 ∈ 𝑗 ↦ (𝑥 ∩ 𝑦)) = (𝑥 ∈ 𝐽 ↦ (𝑥 ∩ 𝐴)))
11	10	rneqd 5895	. . . 4 ⊢ ((𝑗 = 𝐽 ∧ 𝑦 = 𝐴) → ran (𝑥 ∈ 𝑗 ↦ (𝑥 ∩ 𝑦)) = ran (𝑥 ∈ 𝐽 ↦ (𝑥 ∩ 𝐴)))
12		df-rest 17354	. . . 4 ⊢ ↾t = (𝑗 ∈ V, 𝑦 ∈ V ↦ ran (𝑥 ∈ 𝑗 ↦ (𝑥 ∩ 𝑦)))
13	11, 12	ovmpoga 7522	. . 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 3442   ∩ cin 3902   ↦ cmpt 5181  ran crn 5633  (class class class)co 7368   ↾_t crest 17352

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 5226  ax-sep 5243  ax-nul 5253  ax-pr 5379  ax-un 7690

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 3063  df-reu 3353  df-rab 3402  df-v 3444  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-iun 4950  df-br 5101  df-opab 5163  df-mpt 5182  df-id 5527  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-res 5644  df-ima 5645  df-iota 6456  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-ov 7371  df-oprab 7372  df-mpo 7373  df-rest 17354

This theorem is referenced by:  elrest  17359  0rest  17361  restid2  17362  tgrest  23115  resttopon  23117  restco  23120  rest0  23125  restfpw  23135  neitr  23136  ordtrest2  23160  1stcrest  23409  2ndcrest  23410  kgencmp  23501  xkoptsub  23610  trfilss  23845  trfg  23847  uzrest  23853  restmetu  24526  ellimc2  25846  limcflf  25850  ordtrest2NEW  34100  ptrest  37859