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Theorem idqtop 21731
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 6129 . . . . . . 7 ( I ↾ 𝑋) = ( I ↾ 𝑋)
21imaeq1i 5621 . . . . . 6 (( I ↾ 𝑋) “ 𝑥) = (( I ↾ 𝑋) “ 𝑥)
3 resiima 5638 . . . . . . 7 (𝑥𝑋 → (( I ↾ 𝑋) “ 𝑥) = 𝑥)
43adantl 473 . . . . . 6 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑥𝑋) → (( I ↾ 𝑋) “ 𝑥) = 𝑥)
52, 4syl5eq 2806 . . . . 5 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑥𝑋) → (( I ↾ 𝑋) “ 𝑥) = 𝑥)
65eleq1d 2824 . . . 4 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑥𝑋) → ((( I ↾ 𝑋) “ 𝑥) ∈ 𝐽𝑥𝐽))
76pm5.32da 676 . . 3 (𝐽 ∈ (TopOn‘𝑋) → ((𝑥𝑋 ∧ (( I ↾ 𝑋) “ 𝑥) ∈ 𝐽) ↔ (𝑥𝑋𝑥𝐽)))
8 f1oi 6336 . . . . 5 ( I ↾ 𝑋):𝑋1-1-onto𝑋
9 f1ofo 6306 . . . . 5 (( I ↾ 𝑋):𝑋1-1-onto𝑋 → ( I ↾ 𝑋):𝑋onto𝑋)
108, 9mp1i 13 . . . 4 (𝐽 ∈ (TopOn‘𝑋) → ( I ↾ 𝑋):𝑋onto𝑋)
11 elqtop3 21728 . . . 4 ((𝐽 ∈ (TopOn‘𝑋) ∧ ( I ↾ 𝑋):𝑋onto𝑋) → (𝑥 ∈ (𝐽 qTop ( I ↾ 𝑋)) ↔ (𝑥𝑋 ∧ (( I ↾ 𝑋) “ 𝑥) ∈ 𝐽)))
1210, 11mpdan 705 . . 3 (𝐽 ∈ (TopOn‘𝑋) → (𝑥 ∈ (𝐽 qTop ( I ↾ 𝑋)) ↔ (𝑥𝑋 ∧ (( I ↾ 𝑋) “ 𝑥) ∈ 𝐽)))
13 toponss 20953 . . . . 5 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑥𝐽) → 𝑥𝑋)
1413ex 449 . . . 4 (𝐽 ∈ (TopOn‘𝑋) → (𝑥𝐽𝑥𝑋))
1514pm4.71rd 670 . . 3 (𝐽 ∈ (TopOn‘𝑋) → (𝑥𝐽 ↔ (𝑥𝑋𝑥𝐽)))
167, 12, 153bitr4d 300 . 2 (𝐽 ∈ (TopOn‘𝑋) → (𝑥 ∈ (𝐽 qTop ( I ↾ 𝑋)) ↔ 𝑥𝐽))
1716eqrdv 2758 1 (𝐽 ∈ (TopOn‘𝑋) → (𝐽 qTop ( I ↾ 𝑋)) = 𝐽)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383   = wceq 1632  wcel 2139  wss 3715   I cid 5173  ccnv 5265  cres 5268  cima 5269  ontowfo 6047  1-1-ontowf1o 6048  cfv 6049  (class class class)co 6814   qTop cqtop 16385  TopOnctopon 20937
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7115
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-reu 3057  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-ov 6817  df-oprab 6818  df-mpt2 6819  df-qtop 16389  df-topon 20938
This theorem is referenced by: (None)
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