Theorem hmeoimaf1o 23794

Description: The function mapping open sets to their images under a homeomorphism is a bijection of topologies. (Contributed by Mario Carneiro, 10-Sep-2015.)

Hypothesis

Ref	Expression
hmeoimaf1o.1	⊢ 𝐺 = (𝑥 ∈ 𝐽 ↦ (𝐹 “ 𝑥))

Assertion

Ref	Expression
hmeoimaf1o	⊢ (𝐹 ∈ (𝐽Homeo𝐾) → 𝐺:𝐽–1-1-onto→𝐾)

Distinct variable groups:   𝑥,𝐹   𝑥,𝐽   𝑥,𝐾

Allowed substitution hint:   𝐺(𝑥)

Proof of Theorem hmeoimaf1o

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		hmeoimaf1o.1	. 2 ⊢ 𝐺 = (𝑥 ∈ 𝐽 ↦ (𝐹 “ 𝑥))
2		hmeoima 23789	. 2 ⊢ ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝑥 ∈ 𝐽) → (𝐹 “ 𝑥) ∈ 𝐾)
3		hmeocn 23784	. . 3 ⊢ (𝐹 ∈ (𝐽Homeo𝐾) → 𝐹 ∈ (𝐽 Cn 𝐾))
4		cnima 23289	. . 3 ⊢ ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝑦 ∈ 𝐾) → (◡𝐹 “ 𝑦) ∈ 𝐽)
5	3, 4	sylan 580	. 2 ⊢ ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝑦 ∈ 𝐾) → (◡𝐹 “ 𝑦) ∈ 𝐽)
6		eqid 2735	. . . . . . 7 ⊢ ∪ 𝐽 = ∪ 𝐽
7		eqid 2735	. . . . . . 7 ⊢ ∪ 𝐾 = ∪ 𝐾
8	6, 7	hmeof1o 23788	. . . . . 6 ⊢ (𝐹 ∈ (𝐽Homeo𝐾) → 𝐹:∪ 𝐽–1-1-onto→∪ 𝐾)
9	8	adantr 480	. . . . 5 ⊢ ((𝐹 ∈ (𝐽Homeo𝐾) ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝐾)) → 𝐹:∪ 𝐽–1-1-onto→∪ 𝐾)
10		f1of1 6848	. . . . 5 ⊢ (𝐹:∪ 𝐽–1-1-onto→∪ 𝐾 → 𝐹:∪ 𝐽–1-1→∪ 𝐾)
11	9, 10	syl 17	. . . 4 ⊢ ((𝐹 ∈ (𝐽Homeo𝐾) ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝐾)) → 𝐹:∪ 𝐽–1-1→∪ 𝐾)
12		elssuni 4942	. . . . 5 ⊢ (𝑥 ∈ 𝐽 → 𝑥 ⊆ ∪ 𝐽)
13	12	ad2antrl 728	. . . 4 ⊢ ((𝐹 ∈ (𝐽Homeo𝐾) ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝐾)) → 𝑥 ⊆ ∪ 𝐽)
14		cnvimass 6102	. . . . 5 ⊢ (◡𝐹 “ 𝑦) ⊆ dom 𝐹
15		f1dm 6809	. . . . . 6 ⊢ (𝐹:∪ 𝐽–1-1→∪ 𝐾 → dom 𝐹 = ∪ 𝐽)
16	11, 15	syl 17	. . . . 5 ⊢ ((𝐹 ∈ (𝐽Homeo𝐾) ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝐾)) → dom 𝐹 = ∪ 𝐽)
17	14, 16	sseqtrid 4048	. . . 4 ⊢ ((𝐹 ∈ (𝐽Homeo𝐾) ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝐾)) → (◡𝐹 “ 𝑦) ⊆ ∪ 𝐽)
18		f1imaeq 7285	. . . 4 ⊢ ((𝐹:∪ 𝐽–1-1→∪ 𝐾 ∧ (𝑥 ⊆ ∪ 𝐽 ∧ (◡𝐹 “ 𝑦) ⊆ ∪ 𝐽)) → ((𝐹 “ 𝑥) = (𝐹 “ (◡𝐹 “ 𝑦)) ↔ 𝑥 = (◡𝐹 “ 𝑦)))
19	11, 13, 17, 18	syl12anc 837	. . 3 ⊢ ((𝐹 ∈ (𝐽Homeo𝐾) ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝐾)) → ((𝐹 “ 𝑥) = (𝐹 “ (◡𝐹 “ 𝑦)) ↔ 𝑥 = (◡𝐹 “ 𝑦)))
20		f1ofo 6856	. . . . . . 7 ⊢ (𝐹:∪ 𝐽–1-1-onto→∪ 𝐾 → 𝐹:∪ 𝐽–onto→∪ 𝐾)
21	9, 20	syl 17	. . . . . 6 ⊢ ((𝐹 ∈ (𝐽Homeo𝐾) ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝐾)) → 𝐹:∪ 𝐽–onto→∪ 𝐾)
22		elssuni 4942	. . . . . . 7 ⊢ (𝑦 ∈ 𝐾 → 𝑦 ⊆ ∪ 𝐾)
23	22	ad2antll 729	. . . . . 6 ⊢ ((𝐹 ∈ (𝐽Homeo𝐾) ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝐾)) → 𝑦 ⊆ ∪ 𝐾)
24		foimacnv 6866	. . . . . 6 ⊢ ((𝐹:∪ 𝐽–onto→∪ 𝐾 ∧ 𝑦 ⊆ ∪ 𝐾) → (𝐹 “ (◡𝐹 “ 𝑦)) = 𝑦)
25	21, 23, 24	syl2anc 584	. . . . 5 ⊢ ((𝐹 ∈ (𝐽Homeo𝐾) ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝐾)) → (𝐹 “ (◡𝐹 “ 𝑦)) = 𝑦)
26	25	eqeq2d 2746	. . . 4 ⊢ ((𝐹 ∈ (𝐽Homeo𝐾) ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝐾)) → ((𝐹 “ 𝑥) = (𝐹 “ (◡𝐹 “ 𝑦)) ↔ (𝐹 “ 𝑥) = 𝑦))
27		eqcom 2742	. . . 4 ⊢ ((𝐹 “ 𝑥) = 𝑦 ↔ 𝑦 = (𝐹 “ 𝑥))
28	26, 27	bitrdi 287	. . 3 ⊢ ((𝐹 ∈ (𝐽Homeo𝐾) ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝐾)) → ((𝐹 “ 𝑥) = (𝐹 “ (◡𝐹 “ 𝑦)) ↔ 𝑦 = (𝐹 “ 𝑥)))
29	19, 28	bitr3d 281	. 2 ⊢ ((𝐹 ∈ (𝐽Homeo𝐾) ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝐾)) → (𝑥 = (◡𝐹 “ 𝑦) ↔ 𝑦 = (𝐹 “ 𝑥)))
30	1, 2, 5, 29	f1o2d 7687	1 ⊢ (𝐹 ∈ (𝐽Homeo𝐾) → 𝐺:𝐽–1-1-onto→𝐾)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1537   ∈ wcel 2106   ⊆ wss 3963  ∪ cuni 4912   ↦ cmpt 5231  ^◡ccnv 5688  dom cdm 5689   “ cima 5692  –1-1→wf1 6560  –onto→wfo 6561  –1-1-onto→wf1o 6562  (class class class)co 7431   Cn ccn 23248  Homeochmeo 23777

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-sbc 3792  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-map 8867  df-top 22916  df-topon 22933  df-cn 23251  df-hmeo 23779

This theorem is referenced by:  hmphen  23809