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Theorem restid2 15914
 Description: The subspace topology over a subset of the base set is the original topology. (Contributed by Mario Carneiro, 13-Aug-2015.)
Assertion
Ref Expression
restid2 ((𝐴𝑉𝐽 ⊆ 𝒫 𝐴) → (𝐽t 𝐴) = 𝐽)

Proof of Theorem restid2
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 pwexg 4776 . . . . 5 (𝐴𝑉 → 𝒫 𝐴 ∈ V)
21adantr 480 . . . 4 ((𝐴𝑉𝐽 ⊆ 𝒫 𝐴) → 𝒫 𝐴 ∈ V)
3 simpr 476 . . . 4 ((𝐴𝑉𝐽 ⊆ 𝒫 𝐴) → 𝐽 ⊆ 𝒫 𝐴)
42, 3ssexd 4733 . . 3 ((𝐴𝑉𝐽 ⊆ 𝒫 𝐴) → 𝐽 ∈ V)
5 simpl 472 . . 3 ((𝐴𝑉𝐽 ⊆ 𝒫 𝐴) → 𝐴𝑉)
6 restval 15910 . . 3 ((𝐽 ∈ V ∧ 𝐴𝑉) → (𝐽t 𝐴) = ran (𝑥𝐽 ↦ (𝑥𝐴)))
74, 5, 6syl2anc 691 . 2 ((𝐴𝑉𝐽 ⊆ 𝒫 𝐴) → (𝐽t 𝐴) = ran (𝑥𝐽 ↦ (𝑥𝐴)))
83sselda 3568 . . . . . . . 8 (((𝐴𝑉𝐽 ⊆ 𝒫 𝐴) ∧ 𝑥𝐽) → 𝑥 ∈ 𝒫 𝐴)
98elpwid 4118 . . . . . . 7 (((𝐴𝑉𝐽 ⊆ 𝒫 𝐴) ∧ 𝑥𝐽) → 𝑥𝐴)
10 df-ss 3554 . . . . . . 7 (𝑥𝐴 ↔ (𝑥𝐴) = 𝑥)
119, 10sylib 207 . . . . . 6 (((𝐴𝑉𝐽 ⊆ 𝒫 𝐴) ∧ 𝑥𝐽) → (𝑥𝐴) = 𝑥)
1211mpteq2dva 4672 . . . . 5 ((𝐴𝑉𝐽 ⊆ 𝒫 𝐴) → (𝑥𝐽 ↦ (𝑥𝐴)) = (𝑥𝐽𝑥))
13 mptresid 5375 . . . . 5 (𝑥𝐽𝑥) = ( I ↾ 𝐽)
1412, 13syl6eq 2660 . . . 4 ((𝐴𝑉𝐽 ⊆ 𝒫 𝐴) → (𝑥𝐽 ↦ (𝑥𝐴)) = ( I ↾ 𝐽))
1514rneqd 5274 . . 3 ((𝐴𝑉𝐽 ⊆ 𝒫 𝐴) → ran (𝑥𝐽 ↦ (𝑥𝐴)) = ran ( I ↾ 𝐽))
16 rnresi 5398 . . 3 ran ( I ↾ 𝐽) = 𝐽
1715, 16syl6eq 2660 . 2 ((𝐴𝑉𝐽 ⊆ 𝒫 𝐴) → ran (𝑥𝐽 ↦ (𝑥𝐴)) = 𝐽)
187, 17eqtrd 2644 1 ((𝐴𝑉𝐽 ⊆ 𝒫 𝐴) → (𝐽t 𝐴) = 𝐽)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   = wceq 1475   ∈ wcel 1977  Vcvv 3173   ∩ cin 3539   ⊆ wss 3540  𝒫 cpw 4108   ↦ cmpt 4643   I cid 4948  ran crn 5039   ↾ cres 5040  (class class class)co 6549   ↾t crest 15904 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-rest 15906 This theorem is referenced by:  restid  15917  topnid  15919  ssufl  21532
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