Theorem restval 17478

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 3474	. 2 ⊢ (𝐽 ∈ 𝑉 → 𝐽 ∈ V)
2		elex 3474	. 2 ⊢ (𝐴 ∈ 𝑊 → 𝐴 ∈ V)
3		mptexg 7219	. . . . 5 ⊢ (𝐽 ∈ V → (𝑥 ∈ 𝐽 ↦ (𝑥 ∩ 𝐴)) ∈ V)
4		rnexg 7898	. . . . 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 4172	. . . . . 6 ⊢ ((𝑗 = 𝐽 ∧ 𝑦 = 𝐴) → (𝑥 ∩ 𝑦) = (𝑥 ∩ 𝐴))
10	7, 9	mpteq12dv 5197	. . . . 5 ⊢ ((𝑗 = 𝐽 ∧ 𝑦 = 𝐴) → (𝑥 ∈ 𝑗 ↦ (𝑥 ∩ 𝑦)) = (𝑥 ∈ 𝐽 ↦ (𝑥 ∩ 𝐴)))
11	10	rneqd 5928	. . . 4 ⊢ ((𝑗 = 𝐽 ∧ 𝑦 = 𝐴) → ran (𝑥 ∈ 𝑗 ↦ (𝑥 ∩ 𝑦)) = ran (𝑥 ∈ 𝐽 ↦ (𝑥 ∩ 𝐴)))
12		df-rest 17474	. . . 4 ⊢ ↾t = (𝑗 ∈ V, 𝑦 ∈ V ↦ ran (𝑥 ∈ 𝑗 ↦ (𝑥 ∩ 𝑦)))
13	11, 12	ovmpoga 7564	. . 3 ⊢ ((𝐽 ∈ V ∧ 𝐴 ∈ V ∧ ran (𝑥 ∈ 𝐽 ↦ (𝑥 ∩ 𝐴)) ∈ V) → (𝐽 ↾t 𝐴) = ran (𝑥 ∈ 𝐽 ↦ (𝑥 ∩ 𝐴)))
14	6, 13	mpd3an3 1489	. 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 1568   ∈ wcel 2141  Vcvv 3453   ∩ cin 3903   ↦ cmpt 5191  ran crn 5662  (class class class)co 7410   ↾_t crest 17472

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1823  ax-4 1837  ax-5 1938  ax-6 1995  ax-7 2036  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-rep 5237  ax-sep 5256  ax-nul 5268  ax-pr 5404  ax-un 7732

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1571  df-fal 1581  df-ex 1808  df-nf 1812  df-sb 2095  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3078  df-rex 3088  df-reu 3368  df-rab 3415  df-v 3455  df-sbc 3744  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4487  df-sn 4589  df-pr 4591  df-op 4595  df-uni 4872  df-iun 4957  df-br 5109  df-opab 5173  df-mpt 5192  df-id 5556  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-rn 5672  df-res 5673  df-ima 5674  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-ov 7413  df-oprab 7414  df-mpo 7415  df-rest 17474

This theorem is referenced by:  elrest  17479  0rest  17481  restid2  17482  tgrest  23295  resttopon  23297  restco  23300  rest0  23305  restfpw  23315  neitr  23316  ordtrest2  23340  1stcrest  23589  2ndcrest  23590  kgencmp  23681  xkoptsub  23790  trfilss  24025  trfg  24027  uzrest  24033  restmetu  24706  ellimc2  26015  limcflf  26019  ordtrest2NEW  34279  ptrest  38236