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Theorem idqtop 22311
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 6403 . . . . . . 7 ( I ↾ 𝑋) = ( I ↾ 𝑋)
21imaeq1i 5893 . . . . . 6 (( I ↾ 𝑋) “ 𝑥) = (( I ↾ 𝑋) “ 𝑥)
3 resiima 5911 . . . . . . 7 (𝑥𝑋 → (( I ↾ 𝑋) “ 𝑥) = 𝑥)
43adantl 485 . . . . . 6 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑥𝑋) → (( I ↾ 𝑋) “ 𝑥) = 𝑥)
52, 4syl5eq 2845 . . . . 5 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑥𝑋) → (( I ↾ 𝑋) “ 𝑥) = 𝑥)
65eleq1d 2874 . . . 4 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑥𝑋) → ((( I ↾ 𝑋) “ 𝑥) ∈ 𝐽𝑥𝐽))
76pm5.32da 582 . . 3 (𝐽 ∈ (TopOn‘𝑋) → ((𝑥𝑋 ∧ (( I ↾ 𝑋) “ 𝑥) ∈ 𝐽) ↔ (𝑥𝑋𝑥𝐽)))
8 f1oi 6627 . . . . 5 ( I ↾ 𝑋):𝑋1-1-onto𝑋
9 f1ofo 6597 . . . . 5 (( I ↾ 𝑋):𝑋1-1-onto𝑋 → ( I ↾ 𝑋):𝑋onto𝑋)
108, 9mp1i 13 . . . 4 (𝐽 ∈ (TopOn‘𝑋) → ( I ↾ 𝑋):𝑋onto𝑋)
11 elqtop3 22308 . . . 4 ((𝐽 ∈ (TopOn‘𝑋) ∧ ( I ↾ 𝑋):𝑋onto𝑋) → (𝑥 ∈ (𝐽 qTop ( I ↾ 𝑋)) ↔ (𝑥𝑋 ∧ (( I ↾ 𝑋) “ 𝑥) ∈ 𝐽)))
1210, 11mpdan 686 . . 3 (𝐽 ∈ (TopOn‘𝑋) → (𝑥 ∈ (𝐽 qTop ( I ↾ 𝑋)) ↔ (𝑥𝑋 ∧ (( I ↾ 𝑋) “ 𝑥) ∈ 𝐽)))
13 toponss 21532 . . . . 5 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑥𝐽) → 𝑥𝑋)
1413ex 416 . . . 4 (𝐽 ∈ (TopOn‘𝑋) → (𝑥𝐽𝑥𝑋))
1514pm4.71rd 566 . . 3 (𝐽 ∈ (TopOn‘𝑋) → (𝑥𝐽 ↔ (𝑥𝑋𝑥𝐽)))
167, 12, 153bitr4d 314 . 2 (𝐽 ∈ (TopOn‘𝑋) → (𝑥 ∈ (𝐽 qTop ( I ↾ 𝑋)) ↔ 𝑥𝐽))
1716eqrdv 2796 1 (𝐽 ∈ (TopOn‘𝑋) → (𝐽 qTop ( I ↾ 𝑋)) = 𝐽)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399   = wceq 1538  wcel 2111  wss 3881   I cid 5424  ccnv 5518  cres 5521  cima 5522  ontowfo 6322  1-1-ontowf1o 6323  cfv 6324  (class class class)co 7135   qTop cqtop 16768  TopOnctopon 21515
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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-rep 5154  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7441
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-reu 3113  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-id 5425  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-f1 6329  df-fo 6330  df-f1o 6331  df-fv 6332  df-ov 7138  df-oprab 7139  df-mpo 7140  df-qtop 16772  df-topon 21516
This theorem is referenced by: (None)
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