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Theorem idqtop 21419
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 5926 . . . . . . 7 ( I ↾ 𝑋) = ( I ↾ 𝑋)
21imaeq1i 5422 . . . . . 6 (( I ↾ 𝑋) “ 𝑥) = (( I ↾ 𝑋) “ 𝑥)
3 resiima 5439 . . . . . . 7 (𝑥𝑋 → (( I ↾ 𝑋) “ 𝑥) = 𝑥)
43adantl 482 . . . . . 6 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑥𝑋) → (( I ↾ 𝑋) “ 𝑥) = 𝑥)
52, 4syl5eq 2667 . . . . 5 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑥𝑋) → (( I ↾ 𝑋) “ 𝑥) = 𝑥)
65eleq1d 2683 . . . 4 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑥𝑋) → ((( I ↾ 𝑋) “ 𝑥) ∈ 𝐽𝑥𝐽))
76pm5.32da 672 . . 3 (𝐽 ∈ (TopOn‘𝑋) → ((𝑥𝑋 ∧ (( I ↾ 𝑋) “ 𝑥) ∈ 𝐽) ↔ (𝑥𝑋𝑥𝐽)))
8 f1oi 6131 . . . . 5 ( I ↾ 𝑋):𝑋1-1-onto𝑋
9 f1ofo 6101 . . . . 5 (( I ↾ 𝑋):𝑋1-1-onto𝑋 → ( I ↾ 𝑋):𝑋onto𝑋)
108, 9mp1i 13 . . . 4 (𝐽 ∈ (TopOn‘𝑋) → ( I ↾ 𝑋):𝑋onto𝑋)
11 elqtop3 21416 . . . 4 ((𝐽 ∈ (TopOn‘𝑋) ∧ ( I ↾ 𝑋):𝑋onto𝑋) → (𝑥 ∈ (𝐽 qTop ( I ↾ 𝑋)) ↔ (𝑥𝑋 ∧ (( I ↾ 𝑋) “ 𝑥) ∈ 𝐽)))
1210, 11mpdan 701 . . 3 (𝐽 ∈ (TopOn‘𝑋) → (𝑥 ∈ (𝐽 qTop ( I ↾ 𝑋)) ↔ (𝑥𝑋 ∧ (( I ↾ 𝑋) “ 𝑥) ∈ 𝐽)))
13 toponss 20644 . . . . 5 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑥𝐽) → 𝑥𝑋)
1413ex 450 . . . 4 (𝐽 ∈ (TopOn‘𝑋) → (𝑥𝐽𝑥𝑋))
1514pm4.71rd 666 . . 3 (𝐽 ∈ (TopOn‘𝑋) → (𝑥𝐽 ↔ (𝑥𝑋𝑥𝐽)))
167, 12, 153bitr4d 300 . 2 (𝐽 ∈ (TopOn‘𝑋) → (𝑥 ∈ (𝐽 qTop ( I ↾ 𝑋)) ↔ 𝑥𝐽))
1716eqrdv 2619 1 (𝐽 ∈ (TopOn‘𝑋) → (𝐽 qTop ( I ↾ 𝑋)) = 𝐽)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384   = wceq 1480  wcel 1987  wss 3555   I cid 4984  ccnv 5073  cres 5076  cima 5077  ontowfo 5845  1-1-ontowf1o 5846  cfv 5847  (class class class)co 6604   qTop cqtop 16084  TopOnctopon 20618
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4731  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-iun 4487  df-br 4614  df-opab 4674  df-mpt 4675  df-id 4989  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-qtop 16088  df-topon 20623
This theorem is referenced by: (None)
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