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Theorem hmeoimaf1o 21902
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 = (𝑥𝐽 ↦ (𝐹𝑥))
Assertion
Ref Expression
hmeoimaf1o (𝐹 ∈ (𝐽Homeo𝐾) → 𝐺:𝐽1-1-onto𝐾)
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.
StepHypRef Expression
1 hmeoimaf1o.1 . 2 𝐺 = (𝑥𝐽 ↦ (𝐹𝑥))
2 hmeoima 21897 . 2 ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝑥𝐽) → (𝐹𝑥) ∈ 𝐾)
3 hmeocn 21892 . . 3 (𝐹 ∈ (𝐽Homeo𝐾) → 𝐹 ∈ (𝐽 Cn 𝐾))
4 cnima 21398 . . 3 ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝑦𝐾) → (𝐹𝑦) ∈ 𝐽)
53, 4sylan 576 . 2 ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝑦𝐾) → (𝐹𝑦) ∈ 𝐽)
6 eqid 2799 . . . . . . 7 𝐽 = 𝐽
7 eqid 2799 . . . . . . 7 𝐾 = 𝐾
86, 7hmeof1o 21896 . . . . . 6 (𝐹 ∈ (𝐽Homeo𝐾) → 𝐹: 𝐽1-1-onto 𝐾)
98adantr 473 . . . . 5 ((𝐹 ∈ (𝐽Homeo𝐾) ∧ (𝑥𝐽𝑦𝐾)) → 𝐹: 𝐽1-1-onto 𝐾)
10 f1of1 6355 . . . . 5 (𝐹: 𝐽1-1-onto 𝐾𝐹: 𝐽1-1 𝐾)
119, 10syl 17 . . . 4 ((𝐹 ∈ (𝐽Homeo𝐾) ∧ (𝑥𝐽𝑦𝐾)) → 𝐹: 𝐽1-1 𝐾)
12 elssuni 4659 . . . . 5 (𝑥𝐽𝑥 𝐽)
1312ad2antrl 720 . . . 4 ((𝐹 ∈ (𝐽Homeo𝐾) ∧ (𝑥𝐽𝑦𝐾)) → 𝑥 𝐽)
14 cnvimass 5702 . . . . 5 (𝐹𝑦) ⊆ dom 𝐹
15 f1dm 6320 . . . . . 6 (𝐹: 𝐽1-1 𝐾 → dom 𝐹 = 𝐽)
1611, 15syl 17 . . . . 5 ((𝐹 ∈ (𝐽Homeo𝐾) ∧ (𝑥𝐽𝑦𝐾)) → dom 𝐹 = 𝐽)
1714, 16syl5sseq 3849 . . . 4 ((𝐹 ∈ (𝐽Homeo𝐾) ∧ (𝑥𝐽𝑦𝐾)) → (𝐹𝑦) ⊆ 𝐽)
18 f1imaeq 6750 . . . 4 ((𝐹: 𝐽1-1 𝐾 ∧ (𝑥 𝐽 ∧ (𝐹𝑦) ⊆ 𝐽)) → ((𝐹𝑥) = (𝐹 “ (𝐹𝑦)) ↔ 𝑥 = (𝐹𝑦)))
1911, 13, 17, 18syl12anc 866 . . 3 ((𝐹 ∈ (𝐽Homeo𝐾) ∧ (𝑥𝐽𝑦𝐾)) → ((𝐹𝑥) = (𝐹 “ (𝐹𝑦)) ↔ 𝑥 = (𝐹𝑦)))
20 f1ofo 6363 . . . . . . 7 (𝐹: 𝐽1-1-onto 𝐾𝐹: 𝐽onto 𝐾)
219, 20syl 17 . . . . . 6 ((𝐹 ∈ (𝐽Homeo𝐾) ∧ (𝑥𝐽𝑦𝐾)) → 𝐹: 𝐽onto 𝐾)
22 elssuni 4659 . . . . . . 7 (𝑦𝐾𝑦 𝐾)
2322ad2antll 721 . . . . . 6 ((𝐹 ∈ (𝐽Homeo𝐾) ∧ (𝑥𝐽𝑦𝐾)) → 𝑦 𝐾)
24 foimacnv 6373 . . . . . 6 ((𝐹: 𝐽onto 𝐾𝑦 𝐾) → (𝐹 “ (𝐹𝑦)) = 𝑦)
2521, 23, 24syl2anc 580 . . . . 5 ((𝐹 ∈ (𝐽Homeo𝐾) ∧ (𝑥𝐽𝑦𝐾)) → (𝐹 “ (𝐹𝑦)) = 𝑦)
2625eqeq2d 2809 . . . 4 ((𝐹 ∈ (𝐽Homeo𝐾) ∧ (𝑥𝐽𝑦𝐾)) → ((𝐹𝑥) = (𝐹 “ (𝐹𝑦)) ↔ (𝐹𝑥) = 𝑦))
27 eqcom 2806 . . . 4 ((𝐹𝑥) = 𝑦𝑦 = (𝐹𝑥))
2826, 27syl6bb 279 . . 3 ((𝐹 ∈ (𝐽Homeo𝐾) ∧ (𝑥𝐽𝑦𝐾)) → ((𝐹𝑥) = (𝐹 “ (𝐹𝑦)) ↔ 𝑦 = (𝐹𝑥)))
2919, 28bitr3d 273 . 2 ((𝐹 ∈ (𝐽Homeo𝐾) ∧ (𝑥𝐽𝑦𝐾)) → (𝑥 = (𝐹𝑦) ↔ 𝑦 = (𝐹𝑥)))
301, 2, 5, 29f1o2d 7121 1 (𝐹 ∈ (𝐽Homeo𝐾) → 𝐺:𝐽1-1-onto𝐾)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wa 385   = wceq 1653  wcel 2157  wss 3769   cuni 4628  cmpt 4922  ccnv 5311  dom cdm 5312  cima 5315  1-1wf1 6098  ontowfo 6099  1-1-ontowf1o 6100  (class class class)co 6878   Cn ccn 21357  Homeochmeo 21885
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-8 2159  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777  ax-sep 4975  ax-nul 4983  ax-pow 5035  ax-pr 5097  ax-un 7183
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2591  df-eu 2609  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-ral 3094  df-rex 3095  df-rab 3098  df-v 3387  df-sbc 3634  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-nul 4116  df-if 4278  df-pw 4351  df-sn 4369  df-pr 4371  df-op 4375  df-uni 4629  df-br 4844  df-opab 4906  df-mpt 4923  df-id 5220  df-xp 5318  df-rel 5319  df-cnv 5320  df-co 5321  df-dm 5322  df-rn 5323  df-res 5324  df-ima 5325  df-iota 6064  df-fun 6103  df-fn 6104  df-f 6105  df-f1 6106  df-fo 6107  df-f1o 6108  df-fv 6109  df-ov 6881  df-oprab 6882  df-mpt2 6883  df-map 8097  df-top 21027  df-topon 21044  df-cn 21360  df-hmeo 21887
This theorem is referenced by:  hmphen  21917
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