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Theorem idqtop 21319
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 5882 . . . . . . 7 ( I ↾ 𝑋) = ( I ↾ 𝑋)
21imaeq1i 5382 . . . . . 6 (( I ↾ 𝑋) “ 𝑥) = (( I ↾ 𝑋) “ 𝑥)
3 resiima 5399 . . . . . . 7 (𝑥𝑋 → (( I ↾ 𝑋) “ 𝑥) = 𝑥)
43adantl 481 . . . . . 6 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑥𝑋) → (( I ↾ 𝑋) “ 𝑥) = 𝑥)
52, 4syl5eq 2656 . . . . 5 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑥𝑋) → (( I ↾ 𝑋) “ 𝑥) = 𝑥)
65eleq1d 2672 . . . 4 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑥𝑋) → ((( I ↾ 𝑋) “ 𝑥) ∈ 𝐽𝑥𝐽))
76pm5.32da 671 . . 3 (𝐽 ∈ (TopOn‘𝑋) → ((𝑥𝑋 ∧ (( I ↾ 𝑋) “ 𝑥) ∈ 𝐽) ↔ (𝑥𝑋𝑥𝐽)))
8 f1oi 6086 . . . . 5 ( I ↾ 𝑋):𝑋1-1-onto𝑋
9 f1ofo 6057 . . . . 5 (( I ↾ 𝑋):𝑋1-1-onto𝑋 → ( I ↾ 𝑋):𝑋onto𝑋)
108, 9mp1i 13 . . . 4 (𝐽 ∈ (TopOn‘𝑋) → ( I ↾ 𝑋):𝑋onto𝑋)
11 elqtop3 21316 . . . 4 ((𝐽 ∈ (TopOn‘𝑋) ∧ ( I ↾ 𝑋):𝑋onto𝑋) → (𝑥 ∈ (𝐽 qTop ( I ↾ 𝑋)) ↔ (𝑥𝑋 ∧ (( I ↾ 𝑋) “ 𝑥) ∈ 𝐽)))
1210, 11mpdan 699 . . 3 (𝐽 ∈ (TopOn‘𝑋) → (𝑥 ∈ (𝐽 qTop ( I ↾ 𝑋)) ↔ (𝑥𝑋 ∧ (( I ↾ 𝑋) “ 𝑥) ∈ 𝐽)))
13 toponss 20544 . . . . 5 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑥𝐽) → 𝑥𝑋)
1413ex 449 . . . 4 (𝐽 ∈ (TopOn‘𝑋) → (𝑥𝐽𝑥𝑋))
1514pm4.71rd 665 . . 3 (𝐽 ∈ (TopOn‘𝑋) → (𝑥𝐽 ↔ (𝑥𝑋𝑥𝐽)))
167, 12, 153bitr4d 299 . 2 (𝐽 ∈ (TopOn‘𝑋) → (𝑥 ∈ (𝐽 qTop ( I ↾ 𝑋)) ↔ 𝑥𝐽))
1716eqrdv 2608 1 (𝐽 ∈ (TopOn‘𝑋) → (𝐽 qTop ( I ↾ 𝑋)) = 𝐽)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 195  wa 383   = wceq 1475  wcel 1977  wss 3540   I cid 4948  ccnv 5037  cres 5040  cima 5041  ontowfo 5802  1-1-ontowf1o 5803  cfv 5804  (class class class)co 6549   qTop cqtop 15986  TopOnctopon 20518
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-qtop 15990  df-topon 20523
This theorem is referenced by: (None)
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