Theorem ssidcn 23481

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 23461	. . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋)) → (( I ↾ 𝑋) ∈ (𝐽 Cn 𝐾) ↔ (( I ↾ 𝑋):𝑋⟶𝑋 ∧ ∀𝑥 ∈ 𝐾 (◡( I ↾ 𝑋) “ 𝑥) ∈ 𝐽)))
2		f1oi 6857	. . . . 5 ⊢ ( I ↾ 𝑋):𝑋–1-1-onto→𝑋
3		f1of 6818	. . . . 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 540	. . 3 ⊢ (∀𝑥 ∈ 𝐾 (◡( I ↾ 𝑋) “ 𝑥) ∈ 𝐽 ↔ (( I ↾ 𝑋):𝑋⟶𝑋 ∧ ∀𝑥 ∈ 𝐾 (◡( I ↾ 𝑋) “ 𝑥) ∈ 𝐽))
6	1, 5	bitr4di 292	. 2 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋)) → (( I ↾ 𝑋) ∈ (𝐽 Cn 𝐾) ↔ ∀𝑥 ∈ 𝐾 (◡( I ↾ 𝑋) “ 𝑥) ∈ 𝐽))
7		cnvresid 6613	. . . . . . 7 ⊢ ◡( I ↾ 𝑋) = ( I ↾ 𝑋)
8	7	imaeq1i 6053	. . . . . 6 ⊢ (◡( I ↾ 𝑋) “ 𝑥) = (( I ↾ 𝑋) “ 𝑥)
9		elssuni 4899	. . . . . . . . 9 ⊢ (𝑥 ∈ 𝐾 → 𝑥 ⊆ ∪ 𝐾)
10	9	adantl 487	. . . . . . . 8 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋)) ∧ 𝑥 ∈ 𝐾) → 𝑥 ⊆ ∪ 𝐾)
11		toponuni 23140	. . . . . . . . 9 ⊢ (𝐾 ∈ (TopOn‘𝑋) → 𝑋 = ∪ 𝐾)
12	11	ad2antlr 740	. . . . . . . 8 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋)) ∧ 𝑥 ∈ 𝐾) → 𝑋 = ∪ 𝐾)
13	10, 12	sseqtrrd 3968	. . . . . . 7 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋)) ∧ 𝑥 ∈ 𝐾) → 𝑥 ⊆ 𝑋)
14		resiima 6072	. . . . . . 7 ⊢ (𝑥 ⊆ 𝑋 → (( I ↾ 𝑋) “ 𝑥) = 𝑥)
15	13, 14	syl 18	. . . . . 6 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋)) ∧ 𝑥 ∈ 𝐾) → (( I ↾ 𝑋) “ 𝑥) = 𝑥)
16	8, 15	eqtrid 2807	. . . . 5 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋)) ∧ 𝑥 ∈ 𝐾) → (◡( I ↾ 𝑋) “ 𝑥) = 𝑥)
17	16	eleq1d 2845	. . . 4 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋)) ∧ 𝑥 ∈ 𝐾) → ((◡( I ↾ 𝑋) “ 𝑥) ∈ 𝐽 ↔ 𝑥 ∈ 𝐽))
18	17	ralbidva 3183	. . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋)) → (∀𝑥 ∈ 𝐾 (◡( I ↾ 𝑋) “ 𝑥) ∈ 𝐽 ↔ ∀𝑥 ∈ 𝐾 𝑥 ∈ 𝐽))
19		dfss3 3920	. . 3 ⊢ (𝐾 ⊆ 𝐽 ↔ ∀𝑥 ∈ 𝐾 𝑥 ∈ 𝐽)
20	18, 19	bitr4di 292	. 2 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋)) → (∀𝑥 ∈ 𝐾 (◡( I ↾ 𝑋) “ 𝑥) ∈ 𝐽 ↔ 𝐾 ⊆ 𝐽))
21	6, 20	bitrd 282	1 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋)) → (( I ↾ 𝑋) ∈ (𝐽 Cn 𝐾) ↔ 𝐾 ⊆ 𝐽))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2145  ∀wral 3076   ⊆ wss 3899  ∪ cuni 4867   I cid 5549  ^◡ccnv 5654   ↾ cres 5657   “ cima 5658  ⟶wf 6529  –1-1-onto→wf1o 6532  ‘cfv 6533  (class class class)co 7414  TopOnctopon 23136   Cn ccn 23450

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 2213  ax-ext 2732  ax-sep 5251  ax-nul 5263  ax-pow 5330  ax-pr 5398  ax-un 7737

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 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-sbc 3740  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5104  df-opab 5168  df-mpt 5187  df-id 5550  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-iota 6489  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541  df-ov 7417  df-oprab 7418  df-mpo 7419  df-map 8829  df-top 23120  df-topon 23137  df-cn 23453

This theorem is used by:  idcn  23483  sshauslem  23598