Theorem ssidcn 23628

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 23608	. . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋)) → (( I ↾ 𝑋) ∈ (𝐽 Cn 𝐾) ↔ (( I ↾ 𝑋):𝑋⟶𝑋 ∧ ∀𝑥 ∈ 𝐾 (◡( I ↾ 𝑋) “ 𝑥) ∈ 𝐽)))
2		f1oi 6894	. . . . 5 ⊢ ( I ↾ 𝑋):𝑋⟶1-1onto𝑋
3		f1of 6854	. . . . 5 ⊢ (( I ↾ 𝑋):𝑋⟶1-1onto𝑋 → ( 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 6287	. . . . . . 7 ⊢ ◡( I ↾ 𝑋) = ( I ↾ 𝑋)
8	7	imaeq1i 6067	. . . . . 6 ⊢ (◡( I ↾ 𝑋) “ 𝑥) = (( I ↾ 𝑋) “ 𝑥)
9		elssuni 4904	. . . . . . . . 9 ⊢ (𝑥 ∈ 𝐾 → 𝑥 ⊆ ∪ 𝐾)
10	9	adantl 487	. . . . . . . 8 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋)) ∧ 𝑥 ∈ 𝐾) → 𝑥 ⊆ ∪ 𝐾)
11		toponuni 23287	. . . . . . . . 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 3950	. . . . . . 7 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋)) ∧ 𝑥 ∈ 𝐾) → 𝑥 ⊆ 𝑋)
14		resiima 6092	. . . . . . 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 2810	. . . . 5 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋)) ∧ 𝑥 ∈ 𝐾) → (◡( I ↾ 𝑋) “ 𝑥) = 𝑥)
17	16	eleq1d 2849	. . . 4 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋)) ∧ 𝑥 ∈ 𝐾) → ((◡( I ↾ 𝑋) “ 𝑥) ∈ 𝐽 ↔ 𝑥 ∈ 𝐽))
18	17	ralbidva 3187	. . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋)) → (∀𝑥 ∈ 𝐾 (◡( I ↾ 𝑋) “ 𝑥) ∈ 𝐽 ↔ ∀𝑥 ∈ 𝐾 𝑥 ∈ 𝐽))
19		dfss3 3902	. . 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 2147  ∀wral 3080   ⊆ wss 3900  ∪ cuni 4872   I cid 5552  ^◡ccnv 5657   ↾ cres 5660   “ cima 5661  ⟶wf 6562  ⟶1-1ontowf1o 6565  ‘cfv 6566  (class class class)co 7451  TopOnctopon 23283   Cn ccn 23597

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 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-sep 5255  ax-nul 5266  ax-pow 5334  ax-pr 5398  ax-un 7783

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 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-sbc 3743  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4841  df-uni 4873  df-br 5109  df-opab 5174  df-mpt 5193  df-id 5553  df-xp 5664  df-rel 5665  df-cnv 5666  df-co 5667  df-dm 5668  df-rn 5669  df-res 5670  df-ima 5671  df-iota 6520  df-fun 6568  df-fn 6569  df-f 6570  df-f1 6571  df-fo 6572  df-f1o 6573  df-fv 6574  df-ov 7454  df-oprab 7455  df-mpo 7456  df-map 8887  df-top 23267  df-topon 23284  df-cn 23600

This theorem is used by:  idcn  23630  sshauslem  23745