Theorem hmeoimaf1o 23927

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 23922	. 2 ⊢ ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝑥 ∈ 𝐽) → (𝐹 “ 𝑥) ∈ 𝐾)
3		hmeocn 23917	. . 3 ⊢ (𝐹 ∈ (𝐽Homeo𝐾) → 𝐹 ∈ (𝐽 Cn 𝐾))
4		cnima 23422	. . 3 ⊢ ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝑦 ∈ 𝐾) → (◡𝐹 “ 𝑦) ∈ 𝐽)
5	3, 4	sylan 591	. 2 ⊢ ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝑦 ∈ 𝐾) → (◡𝐹 “ 𝑦) ∈ 𝐽)
6		eqid 2763	. . . . . . 7 ⊢ ∪ 𝐽 = ∪ 𝐽
7		eqid 2763	. . . . . . 7 ⊢ ∪ 𝐾 = ∪ 𝐾
8	6, 7	hmeof1o 23921	. . . . . 6 ⊢ (𝐹 ∈ (𝐽Homeo𝐾) → 𝐹:∪ 𝐽–1-1-onto→∪ 𝐾)
9	8	adantr 485	. . . . 5 ⊢ ((𝐹 ∈ (𝐽Homeo𝐾) ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝐾)) → 𝐹:∪ 𝐽–1-1-onto→∪ 𝐾)
10		f1of1 6819	. . . . 5 ⊢ (𝐹:∪ 𝐽–1-1-onto→∪ 𝐾 → 𝐹:∪ 𝐽–1-1→∪ 𝐾)
11	9, 10	syl 18	. . . 4 ⊢ ((𝐹 ∈ (𝐽Homeo𝐾) ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝐾)) → 𝐹:∪ 𝐽–1-1→∪ 𝐾)
12		elssuni 4904	. . . . 5 ⊢ (𝑥 ∈ 𝐽 → 𝑥 ⊆ ∪ 𝐽)
13	12	ad2antrl 740	. . . 4 ⊢ ((𝐹 ∈ (𝐽Homeo𝐾) ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝐾)) → 𝑥 ⊆ ∪ 𝐽)
14		cnvimass 6084	. . . . 5 ⊢ (◡𝐹 “ 𝑦) ⊆ dom 𝐹
15		f1dm 6780	. . . . . 6 ⊢ (𝐹:∪ 𝐽–1-1→∪ 𝐾 → dom 𝐹 = ∪ 𝐽)
16	11, 15	syl 18	. . . . 5 ⊢ ((𝐹 ∈ (𝐽Homeo𝐾) ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝐾)) → dom 𝐹 = ∪ 𝐽)
17	14, 16	sseqtrid 3979	. . . 4 ⊢ ((𝐹 ∈ (𝐽Homeo𝐾) ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝐾)) → (◡𝐹 “ 𝑦) ⊆ ∪ 𝐽)
18		f1imaeq 7263	. . . 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 849	. . 3 ⊢ ((𝐹 ∈ (𝐽Homeo𝐾) ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝐾)) → ((𝐹 “ 𝑥) = (𝐹 “ (◡𝐹 “ 𝑦)) ↔ 𝑥 = (◡𝐹 “ 𝑦)))
20		f1ofo 6828	. . . . . . 7 ⊢ (𝐹:∪ 𝐽–1-1-onto→∪ 𝐾 → 𝐹:∪ 𝐽–onto→∪ 𝐾)
21	9, 20	syl 18	. . . . . 6 ⊢ ((𝐹 ∈ (𝐽Homeo𝐾) ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝐾)) → 𝐹:∪ 𝐽–onto→∪ 𝐾)
22		elssuni 4904	. . . . . . 7 ⊢ (𝑦 ∈ 𝐾 → 𝑦 ⊆ ∪ 𝐾)
23	22	ad2antll 741	. . . . . 6 ⊢ ((𝐹 ∈ (𝐽Homeo𝐾) ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝐾)) → 𝑦 ⊆ ∪ 𝐾)
24		foimacnv 6838	. . . . . 6 ⊢ ((𝐹:∪ 𝐽–onto→∪ 𝐾 ∧ 𝑦 ⊆ ∪ 𝐾) → (𝐹 “ (◡𝐹 “ 𝑦)) = 𝑦)
25	21, 23, 24	syl2anc 595	. . . . 5 ⊢ ((𝐹 ∈ (𝐽Homeo𝐾) ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝐾)) → (𝐹 “ (◡𝐹 “ 𝑦)) = 𝑦)
26	25	eqeq2d 2774	. . . 4 ⊢ ((𝐹 ∈ (𝐽Homeo𝐾) ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝐾)) → ((𝐹 “ 𝑥) = (𝐹 “ (◡𝐹 “ 𝑦)) ↔ (𝐹 “ 𝑥) = 𝑦))
27		eqcom 2770	. . . 4 ⊢ ((𝐹 “ 𝑥) = 𝑦 ↔ 𝑦 = (𝐹 “ 𝑥))
28	26, 27	bitrdi 290	. . 3 ⊢ ((𝐹 ∈ (𝐽Homeo𝐾) ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝐾)) → ((𝐹 “ 𝑥) = (𝐹 “ (◡𝐹 “ 𝑦)) ↔ 𝑦 = (𝐹 “ 𝑥)))
29	19, 28	bitr3d 284	. 2 ⊢ ((𝐹 ∈ (𝐽Homeo𝐾) ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝐾)) → (𝑥 = (◡𝐹 “ 𝑦) ↔ 𝑦 = (𝐹 “ 𝑥)))
30	1, 2, 5, 29	f1o2d 7664	1 ⊢ (𝐹 ∈ (𝐽Homeo𝐾) → 𝐺:𝐽–1-1-onto→𝐾)

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   = wceq 1570   ∈ wcel 2143   ⊆ wss 3905  ∪ cuni 4872   ↦ cmpt 5192  ^◡ccnv 5660  dom cdm 5661   “ cima 5664  –1-1→wf1 6533  –onto→wfo 6534  –1-1-onto→wf1o 6535  (class class class)co 7410   Cn ccn 23381  Homeochmeo 23910

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-sep 5257  ax-nul 5269  ax-pow 5336  ax-pr 5404  ax-un 7732

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-rab 3417  df-v 3457  df-sbc 3745  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4488  df-pw 4564  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4873  df-br 5110  df-opab 5174  df-mpt 5193  df-id 5556  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-rn 5672  df-res 5673  df-ima 5674  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-ov 7413  df-oprab 7414  df-mpo 7415  df-map 8822  df-top 23051  df-topon 23068  df-cn 23384  df-hmeo 23912

This theorem is referenced by:  hmphen  23942