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Theorem hmeoimaf1o 12472
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 12468 . 2 ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝑥𝐽) → (𝐹𝑥) ∈ 𝐾)
3 hmeocn 12463 . . 3 (𝐹 ∈ (𝐽Homeo𝐾) → 𝐹 ∈ (𝐽 Cn 𝐾))
4 cnima 12378 . . 3 ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝑦𝐾) → (𝐹𝑦) ∈ 𝐽)
53, 4sylan 281 . 2 ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝑦𝐾) → (𝐹𝑦) ∈ 𝐽)
6 eqid 2137 . . . . . . 7 𝐽 = 𝐽
7 eqid 2137 . . . . . . 7 𝐾 = 𝐾
86, 7hmeof1o 12467 . . . . . 6 (𝐹 ∈ (𝐽Homeo𝐾) → 𝐹: 𝐽1-1-onto 𝐾)
98adantr 274 . . . . 5 ((𝐹 ∈ (𝐽Homeo𝐾) ∧ (𝑥𝐽𝑦𝐾)) → 𝐹: 𝐽1-1-onto 𝐾)
10 f1of1 5359 . . . . 5 (𝐹: 𝐽1-1-onto 𝐾𝐹: 𝐽1-1 𝐾)
119, 10syl 14 . . . 4 ((𝐹 ∈ (𝐽Homeo𝐾) ∧ (𝑥𝐽𝑦𝐾)) → 𝐹: 𝐽1-1 𝐾)
12 elssuni 3759 . . . . 5 (𝑥𝐽𝑥 𝐽)
1312ad2antrl 481 . . . 4 ((𝐹 ∈ (𝐽Homeo𝐾) ∧ (𝑥𝐽𝑦𝐾)) → 𝑥 𝐽)
14 cnvimass 4897 . . . . 5 (𝐹𝑦) ⊆ dom 𝐹
15 f1dm 5328 . . . . . 6 (𝐹: 𝐽1-1 𝐾 → dom 𝐹 = 𝐽)
1611, 15syl 14 . . . . 5 ((𝐹 ∈ (𝐽Homeo𝐾) ∧ (𝑥𝐽𝑦𝐾)) → dom 𝐹 = 𝐽)
1714, 16sseqtrid 3142 . . . 4 ((𝐹 ∈ (𝐽Homeo𝐾) ∧ (𝑥𝐽𝑦𝐾)) → (𝐹𝑦) ⊆ 𝐽)
18 f1imaeq 5669 . . . 4 ((𝐹: 𝐽1-1 𝐾 ∧ (𝑥 𝐽 ∧ (𝐹𝑦) ⊆ 𝐽)) → ((𝐹𝑥) = (𝐹 “ (𝐹𝑦)) ↔ 𝑥 = (𝐹𝑦)))
1911, 13, 17, 18syl12anc 1214 . . 3 ((𝐹 ∈ (𝐽Homeo𝐾) ∧ (𝑥𝐽𝑦𝐾)) → ((𝐹𝑥) = (𝐹 “ (𝐹𝑦)) ↔ 𝑥 = (𝐹𝑦)))
20 f1ofo 5367 . . . . . . 7 (𝐹: 𝐽1-1-onto 𝐾𝐹: 𝐽onto 𝐾)
219, 20syl 14 . . . . . 6 ((𝐹 ∈ (𝐽Homeo𝐾) ∧ (𝑥𝐽𝑦𝐾)) → 𝐹: 𝐽onto 𝐾)
22 elssuni 3759 . . . . . . 7 (𝑦𝐾𝑦 𝐾)
2322ad2antll 482 . . . . . 6 ((𝐹 ∈ (𝐽Homeo𝐾) ∧ (𝑥𝐽𝑦𝐾)) → 𝑦 𝐾)
24 foimacnv 5378 . . . . . 6 ((𝐹: 𝐽onto 𝐾𝑦 𝐾) → (𝐹 “ (𝐹𝑦)) = 𝑦)
2521, 23, 24syl2anc 408 . . . . 5 ((𝐹 ∈ (𝐽Homeo𝐾) ∧ (𝑥𝐽𝑦𝐾)) → (𝐹 “ (𝐹𝑦)) = 𝑦)
2625eqeq2d 2149 . . . 4 ((𝐹 ∈ (𝐽Homeo𝐾) ∧ (𝑥𝐽𝑦𝐾)) → ((𝐹𝑥) = (𝐹 “ (𝐹𝑦)) ↔ (𝐹𝑥) = 𝑦))
27 eqcom 2139 . . . 4 ((𝐹𝑥) = 𝑦𝑦 = (𝐹𝑥))
2826, 27syl6bb 195 . . 3 ((𝐹 ∈ (𝐽Homeo𝐾) ∧ (𝑥𝐽𝑦𝐾)) → ((𝐹𝑥) = (𝐹 “ (𝐹𝑦)) ↔ 𝑦 = (𝐹𝑥)))
2919, 28bitr3d 189 . 2 ((𝐹 ∈ (𝐽Homeo𝐾) ∧ (𝑥𝐽𝑦𝐾)) → (𝑥 = (𝐹𝑦) ↔ 𝑦 = (𝐹𝑥)))
301, 2, 5, 29f1o2d 5968 1 (𝐹 ∈ (𝐽Homeo𝐾) → 𝐺:𝐽1-1-onto𝐾)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wb 104   = wceq 1331  wcel 1480  wss 3066   cuni 3731  cmpt 3984  ccnv 4533  dom cdm 4534  cima 4537  1-1wf1 5115  ontowfo 5116  1-1-ontowf1o 5117  (class class class)co 5767   Cn ccn 12343  Homeochmeo 12458
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-sep 4041  ax-pow 4093  ax-pr 4126  ax-un 4350  ax-setind 4447
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2000  df-mo 2001  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ne 2307  df-ral 2419  df-rex 2420  df-rab 2423  df-v 2683  df-sbc 2905  df-csb 2999  df-dif 3068  df-un 3070  df-in 3072  df-ss 3079  df-pw 3507  df-sn 3528  df-pr 3529  df-op 3531  df-uni 3732  df-iun 3810  df-br 3925  df-opab 3985  df-mpt 3986  df-id 4210  df-xp 4540  df-rel 4541  df-cnv 4542  df-co 4543  df-dm 4544  df-rn 4545  df-res 4546  df-ima 4547  df-iota 5083  df-fun 5120  df-fn 5121  df-f 5122  df-f1 5123  df-fo 5124  df-f1o 5125  df-fv 5126  df-ov 5770  df-oprab 5771  df-mpo 5772  df-1st 6031  df-2nd 6032  df-map 6537  df-top 12154  df-topon 12167  df-cn 12346  df-hmeo 12459
This theorem is referenced by: (None)
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