MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  restval Structured version   Visualization version   GIF version

Theorem restval 15852
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 ((𝐽𝑉𝐴𝑊) → (𝐽t 𝐴) = ran (𝑥𝐽 ↦ (𝑥𝐴)))
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.
StepHypRef Expression
1 elex 3180 . 2 (𝐽𝑉𝐽 ∈ V)
2 elex 3180 . 2 (𝐴𝑊𝐴 ∈ V)
3 mptexg 6363 . . . . 5 (𝐽 ∈ V → (𝑥𝐽 ↦ (𝑥𝐴)) ∈ V)
4 rnexg 6963 . . . . 5 ((𝑥𝐽 ↦ (𝑥𝐴)) ∈ V → ran (𝑥𝐽 ↦ (𝑥𝐴)) ∈ V)
53, 4syl 17 . . . 4 (𝐽 ∈ V → ran (𝑥𝐽 ↦ (𝑥𝐴)) ∈ V)
65adantr 479 . . 3 ((𝐽 ∈ V ∧ 𝐴 ∈ V) → ran (𝑥𝐽 ↦ (𝑥𝐴)) ∈ V)
7 simpl 471 . . . . . 6 ((𝑗 = 𝐽𝑦 = 𝐴) → 𝑗 = 𝐽)
8 simpr 475 . . . . . . 7 ((𝑗 = 𝐽𝑦 = 𝐴) → 𝑦 = 𝐴)
98ineq2d 3771 . . . . . 6 ((𝑗 = 𝐽𝑦 = 𝐴) → (𝑥𝑦) = (𝑥𝐴))
107, 9mpteq12dv 4653 . . . . 5 ((𝑗 = 𝐽𝑦 = 𝐴) → (𝑥𝑗 ↦ (𝑥𝑦)) = (𝑥𝐽 ↦ (𝑥𝐴)))
1110rneqd 5257 . . . 4 ((𝑗 = 𝐽𝑦 = 𝐴) → ran (𝑥𝑗 ↦ (𝑥𝑦)) = ran (𝑥𝐽 ↦ (𝑥𝐴)))
12 df-rest 15848 . . . 4 t = (𝑗 ∈ V, 𝑦 ∈ V ↦ ran (𝑥𝑗 ↦ (𝑥𝑦)))
1311, 12ovmpt2ga 6662 . . 3 ((𝐽 ∈ V ∧ 𝐴 ∈ V ∧ ran (𝑥𝐽 ↦ (𝑥𝐴)) ∈ V) → (𝐽t 𝐴) = ran (𝑥𝐽 ↦ (𝑥𝐴)))
146, 13mpd3an3 1416 . 2 ((𝐽 ∈ V ∧ 𝐴 ∈ V) → (𝐽t 𝐴) = ran (𝑥𝐽 ↦ (𝑥𝐴)))
151, 2, 14syl2an 492 1 ((𝐽𝑉𝐴𝑊) → (𝐽t 𝐴) = ran (𝑥𝐽 ↦ (𝑥𝐴)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382   = wceq 1474  wcel 1975  Vcvv 3168  cin 3534  cmpt 4633  ran crn 5025  (class class class)co 6523  t crest 15846
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1711  ax-4 1726  ax-5 1825  ax-6 1873  ax-7 1920  ax-8 1977  ax-9 1984  ax-10 2004  ax-11 2019  ax-12 2031  ax-13 2228  ax-ext 2585  ax-rep 4689  ax-sep 4699  ax-nul 4708  ax-pr 4824  ax-un 6820
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1866  df-eu 2457  df-mo 2458  df-clab 2592  df-cleq 2598  df-clel 2601  df-nfc 2735  df-ne 2777  df-ral 2896  df-rex 2897  df-reu 2898  df-rab 2900  df-v 3170  df-sbc 3398  df-csb 3495  df-dif 3538  df-un 3540  df-in 3542  df-ss 3549  df-nul 3870  df-if 4032  df-sn 4121  df-pr 4123  df-op 4127  df-uni 4363  df-iun 4447  df-br 4574  df-opab 4634  df-mpt 4635  df-id 4939  df-xp 5030  df-rel 5031  df-cnv 5032  df-co 5033  df-dm 5034  df-rn 5035  df-res 5036  df-ima 5037  df-iota 5750  df-fun 5788  df-fn 5789  df-f 5790  df-f1 5791  df-fo 5792  df-f1o 5793  df-fv 5794  df-ov 6526  df-oprab 6527  df-mpt2 6528  df-rest 15848
This theorem is referenced by:  elrest  15853  0rest  15855  restid2  15856  tgrest  20711  resttopon  20713  restco  20716  rest0  20721  restfpw  20731  neitr  20732  ordtrest2  20756  1stcrest  21004  2ndcrest  21005  kgencmp  21096  xkoptsub  21205  trfilss  21441  trfg  21443  uzrest  21449  restmetu  22122  ellimc2  23360  limcflf  23364  ordtrest2NEW  29099  ptrest  32377
  Copyright terms: Public domain W3C validator