Theorem idqtop 23659

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 6566	. . . . . . 7 ⊢ ◡( I ↾ 𝑋) = ( I ↾ 𝑋)
2	1	imaeq1i 6011	. . . . . 6 ⊢ (◡( I ↾ 𝑋) “ 𝑥) = (( I ↾ 𝑋) “ 𝑥)
3		resiima 6030	. . . . . . 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 2782	. . . . 5 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑥 ⊆ 𝑋) → (◡( I ↾ 𝑋) “ 𝑥) = 𝑥)
6	5	eleq1d 2820	. . . 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 6807	. . . . 5 ⊢ ( I ↾ 𝑋):𝑋–1-1-onto→𝑋
9		f1ofo 6776	. . . . 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 23656	. . . 4 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ ( I ↾ 𝑋):𝑋–onto→𝑋) → (𝑥 ∈ (𝐽 qTop ( I ↾ 𝑋)) ↔ (𝑥 ⊆ 𝑋 ∧ (◡( I ↾ 𝑋) “ 𝑥) ∈ 𝐽)))
12	10, 11	mpdan 688	. . 3 ⊢ (𝐽 ∈ (TopOn‘𝑋) → (𝑥 ∈ (𝐽 qTop ( I ↾ 𝑋)) ↔ (𝑥 ⊆ 𝑋 ∧ (◡( I ↾ 𝑋) “ 𝑥) ∈ 𝐽)))
13		toponss 22880	. . . . 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 2733	1 ⊢ (𝐽 ∈ (TopOn‘𝑋) → (𝐽 qTop ( I ↾ 𝑋)) = 𝐽)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1542   ∈ wcel 2114   ⊆ wss 3885   I cid 5514  ^◡ccnv 5619   ↾ cres 5622   “ cima 5623  –onto→wfo 6485  –1-1-onto→wf1o 6486  ‘cfv 6487  (class class class)co 7356   qTop cqtop 17456  TopOnctopon 22863

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2184  ax-ext 2707  ax-rep 5201  ax-sep 5220  ax-nul 5230  ax-pow 5296  ax-pr 5364  ax-un 7678

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2538  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2810  df-nfc 2884  df-ne 2931  df-ral 3050  df-rex 3060  df-reu 3341  df-rab 3388  df-v 3429  df-sbc 3726  df-csb 3834  df-dif 3888  df-un 3890  df-in 3892  df-ss 3902  df-nul 4264  df-if 4457  df-pw 4533  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4841  df-iun 4925  df-br 5075  df-opab 5137  df-mpt 5156  df-id 5515  df-xp 5626  df-rel 5627  df-cnv 5628  df-co 5629  df-dm 5630  df-rn 5631  df-res 5632  df-ima 5633  df-iota 6443  df-fun 6489  df-fn 6490  df-f 6491  df-f1 6492  df-fo 6493  df-f1o 6494  df-fv 6495  df-ov 7359  df-oprab 7360  df-mpo 7361  df-qtop 17460  df-topon 22864

This theorem is referenced by: (None)