Theorem idqtop 23600

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 6598	. . . . . . 7 ⊢ ◡( I ↾ 𝑋) = ( I ↾ 𝑋)
2	1	imaeq1i 6031	. . . . . 6 ⊢ (◡( I ↾ 𝑋) “ 𝑥) = (( I ↾ 𝑋) “ 𝑥)
3		resiima 6050	. . . . . . 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 2777	. . . . 5 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑥 ⊆ 𝑋) → (◡( I ↾ 𝑋) “ 𝑥) = 𝑥)
6	5	eleq1d 2814	. . . 4 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑥 ⊆ 𝑋) → ((◡( I ↾ 𝑋) “ 𝑥) ∈ 𝐽 ↔ 𝑥 ∈ 𝐽))
7	6	pm5.32da 579	. . 3 ⊢ (𝐽 ∈ (TopOn‘𝑋) → ((𝑥 ⊆ 𝑋 ∧ (◡( I ↾ 𝑋) “ 𝑥) ∈ 𝐽) ↔ (𝑥 ⊆ 𝑋 ∧ 𝑥 ∈ 𝐽)))
8		f1oi 6841	. . . . 5 ⊢ ( I ↾ 𝑋):𝑋–1-1-onto→𝑋
9		f1ofo 6810	. . . . 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 23597	. . . 4 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ ( I ↾ 𝑋):𝑋–onto→𝑋) → (𝑥 ∈ (𝐽 qTop ( I ↾ 𝑋)) ↔ (𝑥 ⊆ 𝑋 ∧ (◡( I ↾ 𝑋) “ 𝑥) ∈ 𝐽)))
12	10, 11	mpdan 687	. . 3 ⊢ (𝐽 ∈ (TopOn‘𝑋) → (𝑥 ∈ (𝐽 qTop ( I ↾ 𝑋)) ↔ (𝑥 ⊆ 𝑋 ∧ (◡( I ↾ 𝑋) “ 𝑥) ∈ 𝐽)))
13		toponss 22821	. . . . 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 2728	1 ⊢ (𝐽 ∈ (TopOn‘𝑋) → (𝐽 qTop ( I ↾ 𝑋)) = 𝐽)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2109   ⊆ wss 3917   I cid 5535  ^◡ccnv 5640   ↾ cres 5643   “ cima 5644  –onto→wfo 6512  –1-1-onto→wf1o 6513  ‘cfv 6514  (class class class)co 7390   qTop cqtop 17473  TopOnctopon 22804

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-rep 5237  ax-sep 5254  ax-nul 5264  ax-pow 5323  ax-pr 5390  ax-un 7714

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-ral 3046  df-rex 3055  df-reu 3357  df-rab 3409  df-v 3452  df-sbc 3757  df-csb 3866  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-nul 4300  df-if 4492  df-pw 4568  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-iun 4960  df-br 5111  df-opab 5173  df-mpt 5192  df-id 5536  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-rn 5652  df-res 5653  df-ima 5654  df-iota 6467  df-fun 6516  df-fn 6517  df-f 6518  df-f1 6519  df-fo 6520  df-f1o 6521  df-fv 6522  df-ov 7393  df-oprab 7394  df-mpo 7395  df-qtop 17477  df-topon 22805

This theorem is referenced by: (None)