Theorem ssidcn 23263

Description: The identity function is a continuous function from one topology to another topology on the same set iff the domain is finer than the codomain. (Contributed by Mario Carneiro, 21-Mar-2015.) (Revised by Mario Carneiro, 21-Aug-2015.)

Assertion

Ref	Expression
ssidcn	⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋)) → (( I ↾ 𝑋) ∈ (𝐽 Cn 𝐾) ↔ 𝐾 ⊆ 𝐽))

Proof of Theorem ssidcn

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		iscn 23243	. . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋)) → (( I ↾ 𝑋) ∈ (𝐽 Cn 𝐾) ↔ (( I ↾ 𝑋):𝑋⟶𝑋 ∧ ∀𝑥 ∈ 𝐾 (◡( I ↾ 𝑋) “ 𝑥) ∈ 𝐽)))
2		f1oi 6886	. . . . 5 ⊢ ( I ↾ 𝑋):𝑋–1-1-onto→𝑋
3		f1of 6848	. . . . 5 ⊢ (( I ↾ 𝑋):𝑋–1-1-onto→𝑋 → ( I ↾ 𝑋):𝑋⟶𝑋)
4	2, 3	ax-mp 5	. . . 4 ⊢ ( I ↾ 𝑋):𝑋⟶𝑋
5	4	biantrur 530	. . 3 ⊢ (∀𝑥 ∈ 𝐾 (◡( I ↾ 𝑋) “ 𝑥) ∈ 𝐽 ↔ (( I ↾ 𝑋):𝑋⟶𝑋 ∧ ∀𝑥 ∈ 𝐾 (◡( I ↾ 𝑋) “ 𝑥) ∈ 𝐽))
6	1, 5	bitr4di 289	. 2 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋)) → (( I ↾ 𝑋) ∈ (𝐽 Cn 𝐾) ↔ ∀𝑥 ∈ 𝐾 (◡( I ↾ 𝑋) “ 𝑥) ∈ 𝐽))
7		cnvresid 6645	. . . . . . 7 ⊢ ◡( I ↾ 𝑋) = ( I ↾ 𝑋)
8	7	imaeq1i 6075	. . . . . 6 ⊢ (◡( I ↾ 𝑋) “ 𝑥) = (( I ↾ 𝑋) “ 𝑥)
9		elssuni 4937	. . . . . . . . 9 ⊢ (𝑥 ∈ 𝐾 → 𝑥 ⊆ ∪ 𝐾)
10	9	adantl 481	. . . . . . . 8 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋)) ∧ 𝑥 ∈ 𝐾) → 𝑥 ⊆ ∪ 𝐾)
11		toponuni 22920	. . . . . . . . 9 ⊢ (𝐾 ∈ (TopOn‘𝑋) → 𝑋 = ∪ 𝐾)
12	11	ad2antlr 727	. . . . . . . 8 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋)) ∧ 𝑥 ∈ 𝐾) → 𝑋 = ∪ 𝐾)
13	10, 12	sseqtrrd 4021	. . . . . . 7 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋)) ∧ 𝑥 ∈ 𝐾) → 𝑥 ⊆ 𝑋)
14		resiima 6094	. . . . . . 7 ⊢ (𝑥 ⊆ 𝑋 → (( I ↾ 𝑋) “ 𝑥) = 𝑥)
15	13, 14	syl 17	. . . . . 6 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋)) ∧ 𝑥 ∈ 𝐾) → (( I ↾ 𝑋) “ 𝑥) = 𝑥)
16	8, 15	eqtrid 2789	. . . . 5 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋)) ∧ 𝑥 ∈ 𝐾) → (◡( I ↾ 𝑋) “ 𝑥) = 𝑥)
17	16	eleq1d 2826	. . . 4 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋)) ∧ 𝑥 ∈ 𝐾) → ((◡( I ↾ 𝑋) “ 𝑥) ∈ 𝐽 ↔ 𝑥 ∈ 𝐽))
18	17	ralbidva 3176	. . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋)) → (∀𝑥 ∈ 𝐾 (◡( I ↾ 𝑋) “ 𝑥) ∈ 𝐽 ↔ ∀𝑥 ∈ 𝐾 𝑥 ∈ 𝐽))
19		dfss3 3972	. . 3 ⊢ (𝐾 ⊆ 𝐽 ↔ ∀𝑥 ∈ 𝐾 𝑥 ∈ 𝐽)
20	18, 19	bitr4di 289	. 2 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋)) → (∀𝑥 ∈ 𝐾 (◡( I ↾ 𝑋) “ 𝑥) ∈ 𝐽 ↔ 𝐾 ⊆ 𝐽))
21	6, 20	bitrd 279	1 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋)) → (( I ↾ 𝑋) ∈ (𝐽 Cn 𝐾) ↔ 𝐾 ⊆ 𝐽))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2108  ∀wral 3061   ⊆ wss 3951  ∪ cuni 4907   I cid 5577  ^◡ccnv 5684   ↾ cres 5687   “ cima 5688  ⟶wf 6557  –1-1-onto→wf1o 6560  ‘cfv 6561  (class class class)co 7431  TopOnctopon 22916   Cn ccn 23232

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 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-sbc 3789  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-map 8868  df-top 22900  df-topon 22917  df-cn 23235

This theorem is referenced by:  idcn  23265  sshauslem  23380