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Theorem hmeoimaf1o 22066
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 22061 . 2 ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝑥𝐽) → (𝐹𝑥) ∈ 𝐾)
3 hmeocn 22056 . . 3 (𝐹 ∈ (𝐽Homeo𝐾) → 𝐹 ∈ (𝐽 Cn 𝐾))
4 cnima 21561 . . 3 ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝑦𝐾) → (𝐹𝑦) ∈ 𝐽)
53, 4sylan 580 . 2 ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝑦𝐾) → (𝐹𝑦) ∈ 𝐽)
6 eqid 2797 . . . . . . 7 𝐽 = 𝐽
7 eqid 2797 . . . . . . 7 𝐾 = 𝐾
86, 7hmeof1o 22060 . . . . . 6 (𝐹 ∈ (𝐽Homeo𝐾) → 𝐹: 𝐽1-1-onto 𝐾)
98adantr 481 . . . . 5 ((𝐹 ∈ (𝐽Homeo𝐾) ∧ (𝑥𝐽𝑦𝐾)) → 𝐹: 𝐽1-1-onto 𝐾)
10 f1of1 6489 . . . . 5 (𝐹: 𝐽1-1-onto 𝐾𝐹: 𝐽1-1 𝐾)
119, 10syl 17 . . . 4 ((𝐹 ∈ (𝐽Homeo𝐾) ∧ (𝑥𝐽𝑦𝐾)) → 𝐹: 𝐽1-1 𝐾)
12 elssuni 4780 . . . . 5 (𝑥𝐽𝑥 𝐽)
1312ad2antrl 724 . . . 4 ((𝐹 ∈ (𝐽Homeo𝐾) ∧ (𝑥𝐽𝑦𝐾)) → 𝑥 𝐽)
14 cnvimass 5832 . . . . 5 (𝐹𝑦) ⊆ dom 𝐹
15 f1dm 6454 . . . . . 6 (𝐹: 𝐽1-1 𝐾 → dom 𝐹 = 𝐽)
1611, 15syl 17 . . . . 5 ((𝐹 ∈ (𝐽Homeo𝐾) ∧ (𝑥𝐽𝑦𝐾)) → dom 𝐹 = 𝐽)
1714, 16sseqtrid 3946 . . . 4 ((𝐹 ∈ (𝐽Homeo𝐾) ∧ (𝑥𝐽𝑦𝐾)) → (𝐹𝑦) ⊆ 𝐽)
18 f1imaeq 6895 . . . 4 ((𝐹: 𝐽1-1 𝐾 ∧ (𝑥 𝐽 ∧ (𝐹𝑦) ⊆ 𝐽)) → ((𝐹𝑥) = (𝐹 “ (𝐹𝑦)) ↔ 𝑥 = (𝐹𝑦)))
1911, 13, 17, 18syl12anc 833 . . 3 ((𝐹 ∈ (𝐽Homeo𝐾) ∧ (𝑥𝐽𝑦𝐾)) → ((𝐹𝑥) = (𝐹 “ (𝐹𝑦)) ↔ 𝑥 = (𝐹𝑦)))
20 f1ofo 6497 . . . . . . 7 (𝐹: 𝐽1-1-onto 𝐾𝐹: 𝐽onto 𝐾)
219, 20syl 17 . . . . . 6 ((𝐹 ∈ (𝐽Homeo𝐾) ∧ (𝑥𝐽𝑦𝐾)) → 𝐹: 𝐽onto 𝐾)
22 elssuni 4780 . . . . . . 7 (𝑦𝐾𝑦 𝐾)
2322ad2antll 725 . . . . . 6 ((𝐹 ∈ (𝐽Homeo𝐾) ∧ (𝑥𝐽𝑦𝐾)) → 𝑦 𝐾)
24 foimacnv 6507 . . . . . 6 ((𝐹: 𝐽onto 𝐾𝑦 𝐾) → (𝐹 “ (𝐹𝑦)) = 𝑦)
2521, 23, 24syl2anc 584 . . . . 5 ((𝐹 ∈ (𝐽Homeo𝐾) ∧ (𝑥𝐽𝑦𝐾)) → (𝐹 “ (𝐹𝑦)) = 𝑦)
2625eqeq2d 2807 . . . 4 ((𝐹 ∈ (𝐽Homeo𝐾) ∧ (𝑥𝐽𝑦𝐾)) → ((𝐹𝑥) = (𝐹 “ (𝐹𝑦)) ↔ (𝐹𝑥) = 𝑦))
27 eqcom 2804 . . . 4 ((𝐹𝑥) = 𝑦𝑦 = (𝐹𝑥))
2826, 27syl6bb 288 . . 3 ((𝐹 ∈ (𝐽Homeo𝐾) ∧ (𝑥𝐽𝑦𝐾)) → ((𝐹𝑥) = (𝐹 “ (𝐹𝑦)) ↔ 𝑦 = (𝐹𝑥)))
2919, 28bitr3d 282 . 2 ((𝐹 ∈ (𝐽Homeo𝐾) ∧ (𝑥𝐽𝑦𝐾)) → (𝑥 = (𝐹𝑦) ↔ 𝑦 = (𝐹𝑥)))
301, 2, 5, 29f1o2d 7264 1 (𝐹 ∈ (𝐽Homeo𝐾) → 𝐺:𝐽1-1-onto𝐾)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396   = wceq 1525  wcel 2083  wss 3865   cuni 4751  cmpt 5047  ccnv 5449  dom cdm 5450  cima 5453  1-1wf1 6229  ontowfo 6230  1-1-ontowf1o 6231  (class class class)co 7023   Cn ccn 21520  Homeochmeo 22049
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1781  ax-4 1795  ax-5 1892  ax-6 1951  ax-7 1996  ax-8 2085  ax-9 2093  ax-10 2114  ax-11 2128  ax-12 2143  ax-13 2346  ax-ext 2771  ax-sep 5101  ax-nul 5108  ax-pow 5164  ax-pr 5228  ax-un 7326
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3an 1082  df-tru 1528  df-ex 1766  df-nf 1770  df-sb 2045  df-mo 2578  df-eu 2614  df-clab 2778  df-cleq 2790  df-clel 2865  df-nfc 2937  df-ral 3112  df-rex 3113  df-rab 3116  df-v 3442  df-sbc 3712  df-dif 3868  df-un 3870  df-in 3872  df-ss 3880  df-nul 4218  df-if 4388  df-pw 4461  df-sn 4479  df-pr 4481  df-op 4485  df-uni 4752  df-br 4969  df-opab 5031  df-mpt 5048  df-id 5355  df-xp 5456  df-rel 5457  df-cnv 5458  df-co 5459  df-dm 5460  df-rn 5461  df-res 5462  df-ima 5463  df-iota 6196  df-fun 6234  df-fn 6235  df-f 6236  df-f1 6237  df-fo 6238  df-f1o 6239  df-fv 6240  df-ov 7026  df-oprab 7027  df-mpo 7028  df-map 8265  df-top 21190  df-topon 21207  df-cn 21523  df-hmeo 22051
This theorem is referenced by:  hmphen  22081
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