Theorem idqtop 23933

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 6616	. . . . . . 7 ⊢ ◡( I ↾ 𝑋) = ( I ↾ 𝑋)
2	1	imaeq1i 6057	. . . . . 6 ⊢ (◡( I ↾ 𝑋) “ 𝑥) = (( I ↾ 𝑋) “ 𝑥)
3		resiima 6076	. . . . . . 7 ⊢ (𝑥 ⊆ 𝑋 → (( I ↾ 𝑋) “ 𝑥) = 𝑥)
4	3	adantl 487	. . . . . 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 590	. . 3 ⊢ (𝐽 ∈ (TopOn‘𝑋) → ((𝑥 ⊆ 𝑋 ∧ (◡( I ↾ 𝑋) “ 𝑥) ∈ 𝐽) ↔ (𝑥 ⊆ 𝑋 ∧ 𝑥 ∈ 𝐽)))
8		f1oi 6860	. . . . 5 ⊢ ( I ↾ 𝑋):𝑋–1-1-onto→𝑋
9		f1ofo 6829	. . . . 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 23930	. . . 4 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ ( I ↾ 𝑋):𝑋–onto→𝑋) → (𝑥 ∈ (𝐽 qTop ( I ↾ 𝑋)) ↔ (𝑥 ⊆ 𝑋 ∧ (◡( I ↾ 𝑋) “ 𝑥) ∈ 𝐽)))
12	10, 11	mpdan 700	. . 3 ⊢ (𝐽 ∈ (TopOn‘𝑋) → (𝑥 ∈ (𝐽 qTop ( I ↾ 𝑋)) ↔ (𝑥 ⊆ 𝑋 ∧ (◡( I ↾ 𝑋) “ 𝑥) ∈ 𝐽)))
13		toponss 23153	. . . . 5 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑥 ∈ 𝐽) → 𝑥 ⊆ 𝑋)
14	13	ex 418	. . . 4 ⊢ (𝐽 ∈ (TopOn‘𝑋) → (𝑥 ∈ 𝐽 → 𝑥 ⊆ 𝑋))
15	14	pm4.71rd 572	. . 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 401   = wceq 1570   ∈ wcel 2145   ⊆ wss 3902   I cid 5553  ^◡ccnv 5658   ↾ cres 5661   “ cima 5662  –onto→wfo 6535  –1-1-onto→wf1o 6536  ‘cfv 6537  (class class class)co 7416   qTop cqtop 17593  TopOnctopon 23136

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2734  ax-rep 5236  ax-sep 5255  ax-nul 5267  ax-pow 5334  ax-pr 5402  ax-un 7739

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  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 3368  df-rab 3415  df-v 3455  df-sbc 3743  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4283  df-if 4486  df-pw 4562  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4871  df-iun 4956  df-br 5108  df-opab 5172  df-mpt 5191  df-id 5554  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-ov 7419  df-oprab 7420  df-mpo 7421  df-qtop 17597  df-topon 23137

This theorem is used by: (None)