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Theorem hmeoimaf1o 23765
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 23760 . 2 ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝑥𝐽) → (𝐹𝑥) ∈ 𝐾)
3 hmeocn 23755 . . 3 (𝐹 ∈ (𝐽Homeo𝐾) → 𝐹 ∈ (𝐽 Cn 𝐾))
4 cnima 23260 . . 3 ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝑦𝐾) → (𝐹𝑦) ∈ 𝐽)
53, 4sylan 578 . 2 ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝑦𝐾) → (𝐹𝑦) ∈ 𝐽)
6 eqid 2726 . . . . . . 7 𝐽 = 𝐽
7 eqid 2726 . . . . . . 7 𝐾 = 𝐾
86, 7hmeof1o 23759 . . . . . 6 (𝐹 ∈ (𝐽Homeo𝐾) → 𝐹: 𝐽1-1-onto 𝐾)
98adantr 479 . . . . 5 ((𝐹 ∈ (𝐽Homeo𝐾) ∧ (𝑥𝐽𝑦𝐾)) → 𝐹: 𝐽1-1-onto 𝐾)
10 f1of1 6842 . . . . 5 (𝐹: 𝐽1-1-onto 𝐾𝐹: 𝐽1-1 𝐾)
119, 10syl 17 . . . 4 ((𝐹 ∈ (𝐽Homeo𝐾) ∧ (𝑥𝐽𝑦𝐾)) → 𝐹: 𝐽1-1 𝐾)
12 elssuni 4945 . . . . 5 (𝑥𝐽𝑥 𝐽)
1312ad2antrl 726 . . . 4 ((𝐹 ∈ (𝐽Homeo𝐾) ∧ (𝑥𝐽𝑦𝐾)) → 𝑥 𝐽)
14 cnvimass 6091 . . . . 5 (𝐹𝑦) ⊆ dom 𝐹
15 f1dm 6802 . . . . . 6 (𝐹: 𝐽1-1 𝐾 → dom 𝐹 = 𝐽)
1611, 15syl 17 . . . . 5 ((𝐹 ∈ (𝐽Homeo𝐾) ∧ (𝑥𝐽𝑦𝐾)) → dom 𝐹 = 𝐽)
1714, 16sseqtrid 4032 . . . 4 ((𝐹 ∈ (𝐽Homeo𝐾) ∧ (𝑥𝐽𝑦𝐾)) → (𝐹𝑦) ⊆ 𝐽)
18 f1imaeq 7280 . . . 4 ((𝐹: 𝐽1-1 𝐾 ∧ (𝑥 𝐽 ∧ (𝐹𝑦) ⊆ 𝐽)) → ((𝐹𝑥) = (𝐹 “ (𝐹𝑦)) ↔ 𝑥 = (𝐹𝑦)))
1911, 13, 17, 18syl12anc 835 . . 3 ((𝐹 ∈ (𝐽Homeo𝐾) ∧ (𝑥𝐽𝑦𝐾)) → ((𝐹𝑥) = (𝐹 “ (𝐹𝑦)) ↔ 𝑥 = (𝐹𝑦)))
20 f1ofo 6850 . . . . . . 7 (𝐹: 𝐽1-1-onto 𝐾𝐹: 𝐽onto 𝐾)
219, 20syl 17 . . . . . 6 ((𝐹 ∈ (𝐽Homeo𝐾) ∧ (𝑥𝐽𝑦𝐾)) → 𝐹: 𝐽onto 𝐾)
22 elssuni 4945 . . . . . . 7 (𝑦𝐾𝑦 𝐾)
2322ad2antll 727 . . . . . 6 ((𝐹 ∈ (𝐽Homeo𝐾) ∧ (𝑥𝐽𝑦𝐾)) → 𝑦 𝐾)
24 foimacnv 6860 . . . . . 6 ((𝐹: 𝐽onto 𝐾𝑦 𝐾) → (𝐹 “ (𝐹𝑦)) = 𝑦)
2521, 23, 24syl2anc 582 . . . . 5 ((𝐹 ∈ (𝐽Homeo𝐾) ∧ (𝑥𝐽𝑦𝐾)) → (𝐹 “ (𝐹𝑦)) = 𝑦)
2625eqeq2d 2737 . . . 4 ((𝐹 ∈ (𝐽Homeo𝐾) ∧ (𝑥𝐽𝑦𝐾)) → ((𝐹𝑥) = (𝐹 “ (𝐹𝑦)) ↔ (𝐹𝑥) = 𝑦))
27 eqcom 2733 . . . 4 ((𝐹𝑥) = 𝑦𝑦 = (𝐹𝑥))
2826, 27bitrdi 286 . . 3 ((𝐹 ∈ (𝐽Homeo𝐾) ∧ (𝑥𝐽𝑦𝐾)) → ((𝐹𝑥) = (𝐹 “ (𝐹𝑦)) ↔ 𝑦 = (𝐹𝑥)))
2919, 28bitr3d 280 . 2 ((𝐹 ∈ (𝐽Homeo𝐾) ∧ (𝑥𝐽𝑦𝐾)) → (𝑥 = (𝐹𝑦) ↔ 𝑦 = (𝐹𝑥)))
301, 2, 5, 29f1o2d 7680 1 (𝐹 ∈ (𝐽Homeo𝐾) → 𝐺:𝐽1-1-onto𝐾)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 394   = wceq 1534  wcel 2099  wss 3947   cuni 4913  cmpt 5236  ccnv 5681  dom cdm 5682  cima 5685  1-1wf1 6551  ontowfo 6552  1-1-ontowf1o 6553  (class class class)co 7424   Cn ccn 23219  Homeochmeo 23748
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2697  ax-sep 5304  ax-nul 5311  ax-pow 5369  ax-pr 5433  ax-un 7746
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2704  df-cleq 2718  df-clel 2803  df-nfc 2878  df-ne 2931  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3464  df-sbc 3777  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4326  df-if 4534  df-pw 4609  df-sn 4634  df-pr 4636  df-op 4640  df-uni 4914  df-br 5154  df-opab 5216  df-mpt 5237  df-id 5580  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-iota 6506  df-fun 6556  df-fn 6557  df-f 6558  df-f1 6559  df-fo 6560  df-f1o 6561  df-fv 6562  df-ov 7427  df-oprab 7428  df-mpo 7429  df-map 8857  df-top 22887  df-topon 22904  df-cn 23222  df-hmeo 23750
This theorem is referenced by:  hmphen  23780
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