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Theorem idqtop 23532
Description: The quotient topology induced by the identity function is the original topology. (Contributed by Mario Carneiro, 30-Aug-2015.)
Assertion
Ref Expression
idqtop (𝐽 ∈ (TopOn‘𝑋) → (𝐽 qTop ( I ↾ 𝑋)) = 𝐽)

Proof of Theorem idqtop
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 cnvresid 6617 . . . . . . 7 ( I ↾ 𝑋) = ( I ↾ 𝑋)
21imaeq1i 6046 . . . . . 6 (( I ↾ 𝑋) “ 𝑥) = (( I ↾ 𝑋) “ 𝑥)
3 resiima 6065 . . . . . . 7 (𝑥𝑋 → (( I ↾ 𝑋) “ 𝑥) = 𝑥)
43adantl 481 . . . . . 6 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑥𝑋) → (( I ↾ 𝑋) “ 𝑥) = 𝑥)
52, 4eqtrid 2776 . . . . 5 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑥𝑋) → (( I ↾ 𝑋) “ 𝑥) = 𝑥)
65eleq1d 2810 . . . 4 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑥𝑋) → ((( I ↾ 𝑋) “ 𝑥) ∈ 𝐽𝑥𝐽))
76pm5.32da 578 . . 3 (𝐽 ∈ (TopOn‘𝑋) → ((𝑥𝑋 ∧ (( I ↾ 𝑋) “ 𝑥) ∈ 𝐽) ↔ (𝑥𝑋𝑥𝐽)))
8 f1oi 6861 . . . . 5 ( I ↾ 𝑋):𝑋1-1-onto𝑋
9 f1ofo 6830 . . . . 5 (( I ↾ 𝑋):𝑋1-1-onto𝑋 → ( I ↾ 𝑋):𝑋onto𝑋)
108, 9mp1i 13 . . . 4 (𝐽 ∈ (TopOn‘𝑋) → ( I ↾ 𝑋):𝑋onto𝑋)
11 elqtop3 23529 . . . 4 ((𝐽 ∈ (TopOn‘𝑋) ∧ ( I ↾ 𝑋):𝑋onto𝑋) → (𝑥 ∈ (𝐽 qTop ( I ↾ 𝑋)) ↔ (𝑥𝑋 ∧ (( I ↾ 𝑋) “ 𝑥) ∈ 𝐽)))
1210, 11mpdan 684 . . 3 (𝐽 ∈ (TopOn‘𝑋) → (𝑥 ∈ (𝐽 qTop ( I ↾ 𝑋)) ↔ (𝑥𝑋 ∧ (( I ↾ 𝑋) “ 𝑥) ∈ 𝐽)))
13 toponss 22751 . . . . 5 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑥𝐽) → 𝑥𝑋)
1413ex 412 . . . 4 (𝐽 ∈ (TopOn‘𝑋) → (𝑥𝐽𝑥𝑋))
1514pm4.71rd 562 . . 3 (𝐽 ∈ (TopOn‘𝑋) → (𝑥𝐽 ↔ (𝑥𝑋𝑥𝐽)))
167, 12, 153bitr4d 311 . 2 (𝐽 ∈ (TopOn‘𝑋) → (𝑥 ∈ (𝐽 qTop ( I ↾ 𝑋)) ↔ 𝑥𝐽))
1716eqrdv 2722 1 (𝐽 ∈ (TopOn‘𝑋) → (𝐽 qTop ( I ↾ 𝑋)) = 𝐽)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 395   = wceq 1533  wcel 2098  wss 3940   I cid 5563  ccnv 5665  cres 5668  cima 5669  ontowfo 6531  1-1-ontowf1o 6532  cfv 6533  (class class class)co 7401   qTop cqtop 17448  TopOnctopon 22734
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-rep 5275  ax-sep 5289  ax-nul 5296  ax-pow 5353  ax-pr 5417  ax-un 7718
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-ral 3054  df-rex 3063  df-reu 3369  df-rab 3425  df-v 3468  df-sbc 3770  df-csb 3886  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-nul 4315  df-if 4521  df-pw 4596  df-sn 4621  df-pr 4623  df-op 4627  df-uni 4900  df-iun 4989  df-br 5139  df-opab 5201  df-mpt 5222  df-id 5564  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-rn 5677  df-res 5678  df-ima 5679  df-iota 6485  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541  df-ov 7404  df-oprab 7405  df-mpo 7406  df-qtop 17452  df-topon 22735
This theorem is referenced by: (None)
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