Theorem idqtop 23874

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.

Step	Hyp	Ref	Expression
1		cnvresid 6615	. . . . . . 7 ⊢ ◡( I ↾ 𝑋) = ( I ↾ 𝑋)
2	1	imaeq1i 6058	. . . . . 6 ⊢ (◡( I ↾ 𝑋) “ 𝑥) = (( I ↾ 𝑋) “ 𝑥)
3		resiima 6077	. . . . . . 7 ⊢ (𝑥 ⊆ 𝑋 → (( I ↾ 𝑋) “ 𝑥) = 𝑥)
4	3	adantl 486	. . . . . 6 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑥 ⊆ 𝑋) → (( I ↾ 𝑋) “ 𝑥) = 𝑥)
5	2, 4	eqtrid 2809	. . . . 5 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑥 ⊆ 𝑋) → (◡( I ↾ 𝑋) “ 𝑥) = 𝑥)
6	5	eleq1d 2847	. . . 4 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑥 ⊆ 𝑋) → ((◡( I ↾ 𝑋) “ 𝑥) ∈ 𝐽 ↔ 𝑥 ∈ 𝐽))
7	6	pm5.32da 589	. . 3 ⊢ (𝐽 ∈ (TopOn‘𝑋) → ((𝑥 ⊆ 𝑋 ∧ (◡( I ↾ 𝑋) “ 𝑥) ∈ 𝐽) ↔ (𝑥 ⊆ 𝑋 ∧ 𝑥 ∈ 𝐽)))
8		f1oi 6859	. . . . 5 ⊢ ( I ↾ 𝑋):𝑋–1-1-onto→𝑋
9		f1ofo 6828	. . . . 5 ⊢ (( I ↾ 𝑋):𝑋–1-1-onto→𝑋 → ( I ↾ 𝑋):𝑋–onto→𝑋)
10	8, 9	mp1i 14	. . . 4 ⊢ (𝐽 ∈ (TopOn‘𝑋) → ( I ↾ 𝑋):𝑋–onto→𝑋)
11		elqtop3 23871	. . . 4 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ ( I ↾ 𝑋):𝑋–onto→𝑋) → (𝑥 ∈ (𝐽 qTop ( I ↾ 𝑋)) ↔ (𝑥 ⊆ 𝑋 ∧ (◡( I ↾ 𝑋) “ 𝑥) ∈ 𝐽)))
12	10, 11	mpdan 699	. . 3 ⊢ (𝐽 ∈ (TopOn‘𝑋) → (𝑥 ∈ (𝐽 qTop ( I ↾ 𝑋)) ↔ (𝑥 ⊆ 𝑋 ∧ (◡( I ↾ 𝑋) “ 𝑥) ∈ 𝐽)))
13		toponss 23095	. . . . 5 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑥 ∈ 𝐽) → 𝑥 ⊆ 𝑋)
14	13	ex 417	. . . 4 ⊢ (𝐽 ∈ (TopOn‘𝑋) → (𝑥 ∈ 𝐽 → 𝑥 ⊆ 𝑋))
15	14	pm4.71rd 571	. . 3 ⊢ (𝐽 ∈ (TopOn‘𝑋) → (𝑥 ∈ 𝐽 ↔ (𝑥 ⊆ 𝑋 ∧ 𝑥 ∈ 𝐽)))
16	7, 12, 15	3bitr4d 314	. 2 ⊢ (𝐽 ∈ (TopOn‘𝑋) → (𝑥 ∈ (𝐽 qTop ( I ↾ 𝑋)) ↔ 𝑥 ∈ 𝐽))
17	16	eqrdv 2760	1 ⊢ (𝐽 ∈ (TopOn‘𝑋) → (𝐽 qTop ( I ↾ 𝑋)) = 𝐽)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 400   = wceq 1569   ∈ wcel 2142   ⊆ wss 3904   I cid 5554  ^◡ccnv 5659   ↾ cres 5662   “ cima 5663  –onto→wfo 6534  –1-1-onto→wf1o 6535  ‘cfv 6536  (class class class)co 7412   qTop cqtop 17563  TopOnctopon 23078

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-8 2144  ax-9 2152  ax-10 2175  ax-11 2191  ax-12 2212  ax-ext 2734  ax-rep 5237  ax-sep 5256  ax-nul 5268  ax-pow 5335  ax-pr 5403  ax-un 7734

This proof depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1104  df-tru 1572  df-fal 1582  df-ex 1809  df-nf 1813  df-sb 2096  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3079  df-rex 3089  df-reu 3369  df-rab 3416  df-v 3456  df-sbc 3744  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4487  df-pw 4563  df-sn 4589  df-pr 4591  df-op 4595  df-uni 4872  df-iun 4957  df-br 5109  df-opab 5173  df-mpt 5192  df-id 5555  df-xp 5666  df-rel 5667  df-cnv 5668  df-co 5669  df-dm 5670  df-rn 5671  df-res 5672  df-ima 5673  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-ov 7415  df-oprab 7416  df-mpo 7417  df-qtop 17567  df-topon 23079

This theorem is used by: (None)