Theorem idqtop 23532

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 6617	. . . . . . 7 ⊢ ◡( I ↾ 𝑋) = ( I ↾ 𝑋)
2	1	imaeq1i 6046	. . . . . 6 ⊢ (◡( I ↾ 𝑋) “ 𝑥) = (( I ↾ 𝑋) “ 𝑥)
3		resiima 6065	. . . . . . 7 ⊢ (𝑥 ⊆ 𝑋 → (( I ↾ 𝑋) “ 𝑥) = 𝑥)
4	3	adantl 481	. . . . . 6 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑥 ⊆ 𝑋) → (( I ↾ 𝑋) “ 𝑥) = 𝑥)
5	2, 4	eqtrid 2776	. . . . 5 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑥 ⊆ 𝑋) → (◡( I ↾ 𝑋) “ 𝑥) = 𝑥)
6	5	eleq1d 2810	. . . 4 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑥 ⊆ 𝑋) → ((◡( I ↾ 𝑋) “ 𝑥) ∈ 𝐽 ↔ 𝑥 ∈ 𝐽))
7	6	pm5.32da 578	. . 3 ⊢ (𝐽 ∈ (TopOn‘𝑋) → ((𝑥 ⊆ 𝑋 ∧ (◡( I ↾ 𝑋) “ 𝑥) ∈ 𝐽) ↔ (𝑥 ⊆ 𝑋 ∧ 𝑥 ∈ 𝐽)))
8		f1oi 6861	. . . . 5 ⊢ ( I ↾ 𝑋):𝑋–1-1-onto→𝑋
9		f1ofo 6830	. . . . 5 ⊢ (( I ↾ 𝑋):𝑋–1-1-onto→𝑋 → ( I ↾ 𝑋):𝑋–onto→𝑋)
10	8, 9	mp1i 13	. . . 4 ⊢ (𝐽 ∈ (TopOn‘𝑋) → ( I ↾ 𝑋):𝑋–onto→𝑋)
11		elqtop3 23529	. . . 4 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ ( I ↾ 𝑋):𝑋–onto→𝑋) → (𝑥 ∈ (𝐽 qTop ( I ↾ 𝑋)) ↔ (𝑥 ⊆ 𝑋 ∧ (◡( I ↾ 𝑋) “ 𝑥) ∈ 𝐽)))
12	10, 11	mpdan 684	. . 3 ⊢ (𝐽 ∈ (TopOn‘𝑋) → (𝑥 ∈ (𝐽 qTop ( I ↾ 𝑋)) ↔ (𝑥 ⊆ 𝑋 ∧ (◡( I ↾ 𝑋) “ 𝑥) ∈ 𝐽)))
13		toponss 22751	. . . . 5 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑥 ∈ 𝐽) → 𝑥 ⊆ 𝑋)
14	13	ex 412	. . . 4 ⊢ (𝐽 ∈ (TopOn‘𝑋) → (𝑥 ∈ 𝐽 → 𝑥 ⊆ 𝑋))
15	14	pm4.71rd 562	. . 3 ⊢ (𝐽 ∈ (TopOn‘𝑋) → (𝑥 ∈ 𝐽 ↔ (𝑥 ⊆ 𝑋 ∧ 𝑥 ∈ 𝐽)))
16	7, 12, 15	3bitr4d 311	. 2 ⊢ (𝐽 ∈ (TopOn‘𝑋) → (𝑥 ∈ (𝐽 qTop ( I ↾ 𝑋)) ↔ 𝑥 ∈ 𝐽))
17	16	eqrdv 2722	1 ⊢ (𝐽 ∈ (TopOn‘𝑋) → (𝐽 qTop ( I ↾ 𝑋)) = 𝐽)

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395   = wceq 1533   ∈ wcel 2098   ⊆ wss 3940   I cid 5563  ^◡ccnv 5665   ↾ cres 5668   “ cima 5669  –onto→wfo 6531  –1-1-onto→wf1o 6532  ‘cfv 6533  (class class class)co 7401   qTop cqtop 17448  TopOnctopon 22734

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-rep 5275  ax-sep 5289  ax-nul 5296  ax-pow 5353  ax-pr 5417  ax-un 7718

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-ral 3054  df-rex 3063  df-reu 3369  df-rab 3425  df-v 3468  df-sbc 3770  df-csb 3886  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-nul 4315  df-if 4521  df-pw 4596  df-sn 4621  df-pr 4623  df-op 4627  df-uni 4900  df-iun 4989  df-br 5139  df-opab 5201  df-mpt 5222  df-id 5564  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-rn 5677  df-res 5678  df-ima 5679  df-iota 6485  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541  df-ov 7404  df-oprab 7405  df-mpo 7406  df-qtop 17452  df-topon 22735

This theorem is referenced by: (None)