Theorem ssidcn 23303

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 23283	. . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋)) → (( I ↾ 𝑋) ∈ (𝐽 Cn 𝐾) ↔ (( I ↾ 𝑋):𝑋⟶𝑋 ∧ ∀𝑥 ∈ 𝐾 (◡( I ↾ 𝑋) “ 𝑥) ∈ 𝐽)))
2		f1oi 6840	. . . . 5 ⊢ ( I ↾ 𝑋):𝑋–1-1-onto→𝑋
3		f1of 6801	. . . . 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 538	. . 3 ⊢ (∀𝑥 ∈ 𝐾 (◡( I ↾ 𝑋) “ 𝑥) ∈ 𝐽 ↔ (( I ↾ 𝑋):𝑋⟶𝑋 ∧ ∀𝑥 ∈ 𝐾 (◡( I ↾ 𝑋) “ 𝑥) ∈ 𝐽))
6	1, 5	bitr4di 291	. 2 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋)) → (( I ↾ 𝑋) ∈ (𝐽 Cn 𝐾) ↔ ∀𝑥 ∈ 𝐾 (◡( I ↾ 𝑋) “ 𝑥) ∈ 𝐽))
7		cnvresid 6595	. . . . . . 7 ⊢ ◡( I ↾ 𝑋) = ( I ↾ 𝑋)
8	7	imaeq1i 6042	. . . . . 6 ⊢ (◡( I ↾ 𝑋) “ 𝑥) = (( I ↾ 𝑋) “ 𝑥)
9		elssuni 4894	. . . . . . . . 9 ⊢ (𝑥 ∈ 𝐾 → 𝑥 ⊆ ∪ 𝐾)
10	9	adantl 485	. . . . . . . 8 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋)) ∧ 𝑥 ∈ 𝐾) → 𝑥 ⊆ ∪ 𝐾)
11		toponuni 22962	. . . . . . . . 9 ⊢ (𝐾 ∈ (TopOn‘𝑋) → 𝑋 = ∪ 𝐾)
12	11	ad2antlr 737	. . . . . . . 8 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋)) ∧ 𝑥 ∈ 𝐾) → 𝑋 = ∪ 𝐾)
13	10, 12	sseqtrrd 3971	. . . . . . 7 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋)) ∧ 𝑥 ∈ 𝐾) → 𝑥 ⊆ 𝑋)
14		resiima 6061	. . . . . . 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 2808	. . . . 5 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋)) ∧ 𝑥 ∈ 𝐾) → (◡( I ↾ 𝑋) “ 𝑥) = 𝑥)
17	16	eleq1d 2846	. . . 4 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋)) ∧ 𝑥 ∈ 𝐾) → ((◡( I ↾ 𝑋) “ 𝑥) ∈ 𝐽 ↔ 𝑥 ∈ 𝐽))
18	17	ralbidva 3182	. . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋)) → (∀𝑥 ∈ 𝐾 (◡( I ↾ 𝑋) “ 𝑥) ∈ 𝐽 ↔ ∀𝑥 ∈ 𝐾 𝑥 ∈ 𝐽))
19		dfss3 3923	. . 3 ⊢ (𝐾 ⊆ 𝐽 ↔ ∀𝑥 ∈ 𝐾 𝑥 ∈ 𝐽)
20	18, 19	bitr4di 291	. 2 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋)) → (∀𝑥 ∈ 𝐾 (◡( I ↾ 𝑋) “ 𝑥) ∈ 𝐽 ↔ 𝐾 ⊆ 𝐽))
21	6, 20	bitrd 281	1 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋)) → (( I ↾ 𝑋) ∈ (𝐽 Cn 𝐾) ↔ 𝐾 ⊆ 𝐽))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 399   = wceq 1559   ∈ wcel 2141  ∀wral 3075   ⊆ wss 3902  ∪ cuni 4862   I cid 5537  ^◡ccnv 5642   ↾ cres 5645   “ cima 5646  ⟶wf 6512  –1-1-onto→wf1o 6515  ‘cfv 6516  (class class class)co 7391  TopOnctopon 22958   Cn ccn 23272

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-sep 5243  ax-nul 5253  ax-pow 5319  ax-pr 5387  ax-un 7713

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-nf 1803  df-sb 2090  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3076  df-rex 3086  df-rab 3414  df-v 3455  df-sbc 3743  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4284  df-if 4478  df-pw 4554  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4863  df-br 5098  df-opab 5160  df-mpt 5179  df-id 5538  df-xp 5649  df-rel 5650  df-cnv 5651  df-co 5652  df-dm 5653  df-rn 5654  df-res 5655  df-ima 5656  df-iota 6472  df-fun 6518  df-fn 6519  df-f 6520  df-f1 6521  df-fo 6522  df-f1o 6523  df-fv 6524  df-ov 7394  df-oprab 7395  df-mpo 7396  df-map 8804  df-top 22942  df-topon 22959  df-cn 23275

This theorem is referenced by:  idcn  23305  sshauslem  23420