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Mirrors > Home > MPE Home > Th. List > haushmphlem | Structured version Visualization version GIF version |
Description: Lemma for haushmph 22400 and similar theorems. If the topological property 𝐴 is preserved under injective preimages, then property 𝐴 is preserved under homeomorphisms. (Contributed by Mario Carneiro, 25-Aug-2015.) |
Ref | Expression |
---|---|
haushmphlem.1 | ⊢ (𝐽 ∈ 𝐴 → 𝐽 ∈ Top) |
haushmphlem.2 | ⊢ ((𝐽 ∈ 𝐴 ∧ 𝑓:∪ 𝐾–1-1→∪ 𝐽 ∧ 𝑓 ∈ (𝐾 Cn 𝐽)) → 𝐾 ∈ 𝐴) |
Ref | Expression |
---|---|
haushmphlem | ⊢ (𝐽 ≃ 𝐾 → (𝐽 ∈ 𝐴 → 𝐾 ∈ 𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | hmphsym 22390 | . 2 ⊢ (𝐽 ≃ 𝐾 → 𝐾 ≃ 𝐽) | |
2 | hmph 22384 | . . 3 ⊢ (𝐾 ≃ 𝐽 ↔ (𝐾Homeo𝐽) ≠ ∅) | |
3 | n0 4310 | . . . 4 ⊢ ((𝐾Homeo𝐽) ≠ ∅ ↔ ∃𝑓 𝑓 ∈ (𝐾Homeo𝐽)) | |
4 | simpl 485 | . . . . . . 7 ⊢ ((𝐽 ∈ 𝐴 ∧ 𝑓 ∈ (𝐾Homeo𝐽)) → 𝐽 ∈ 𝐴) | |
5 | eqid 2821 | . . . . . . . . . 10 ⊢ ∪ 𝐾 = ∪ 𝐾 | |
6 | eqid 2821 | . . . . . . . . . 10 ⊢ ∪ 𝐽 = ∪ 𝐽 | |
7 | 5, 6 | hmeof1o 22372 | . . . . . . . . 9 ⊢ (𝑓 ∈ (𝐾Homeo𝐽) → 𝑓:∪ 𝐾–1-1-onto→∪ 𝐽) |
8 | 7 | adantl 484 | . . . . . . . 8 ⊢ ((𝐽 ∈ 𝐴 ∧ 𝑓 ∈ (𝐾Homeo𝐽)) → 𝑓:∪ 𝐾–1-1-onto→∪ 𝐽) |
9 | f1of1 6614 | . . . . . . . 8 ⊢ (𝑓:∪ 𝐾–1-1-onto→∪ 𝐽 → 𝑓:∪ 𝐾–1-1→∪ 𝐽) | |
10 | 8, 9 | syl 17 | . . . . . . 7 ⊢ ((𝐽 ∈ 𝐴 ∧ 𝑓 ∈ (𝐾Homeo𝐽)) → 𝑓:∪ 𝐾–1-1→∪ 𝐽) |
11 | hmeocn 22368 | . . . . . . . 8 ⊢ (𝑓 ∈ (𝐾Homeo𝐽) → 𝑓 ∈ (𝐾 Cn 𝐽)) | |
12 | 11 | adantl 484 | . . . . . . 7 ⊢ ((𝐽 ∈ 𝐴 ∧ 𝑓 ∈ (𝐾Homeo𝐽)) → 𝑓 ∈ (𝐾 Cn 𝐽)) |
13 | haushmphlem.2 | . . . . . . 7 ⊢ ((𝐽 ∈ 𝐴 ∧ 𝑓:∪ 𝐾–1-1→∪ 𝐽 ∧ 𝑓 ∈ (𝐾 Cn 𝐽)) → 𝐾 ∈ 𝐴) | |
14 | 4, 10, 12, 13 | syl3anc 1367 | . . . . . 6 ⊢ ((𝐽 ∈ 𝐴 ∧ 𝑓 ∈ (𝐾Homeo𝐽)) → 𝐾 ∈ 𝐴) |
15 | 14 | expcom 416 | . . . . 5 ⊢ (𝑓 ∈ (𝐾Homeo𝐽) → (𝐽 ∈ 𝐴 → 𝐾 ∈ 𝐴)) |
16 | 15 | exlimiv 1931 | . . . 4 ⊢ (∃𝑓 𝑓 ∈ (𝐾Homeo𝐽) → (𝐽 ∈ 𝐴 → 𝐾 ∈ 𝐴)) |
17 | 3, 16 | sylbi 219 | . . 3 ⊢ ((𝐾Homeo𝐽) ≠ ∅ → (𝐽 ∈ 𝐴 → 𝐾 ∈ 𝐴)) |
18 | 2, 17 | sylbi 219 | . 2 ⊢ (𝐾 ≃ 𝐽 → (𝐽 ∈ 𝐴 → 𝐾 ∈ 𝐴)) |
19 | 1, 18 | syl 17 | 1 ⊢ (𝐽 ≃ 𝐾 → (𝐽 ∈ 𝐴 → 𝐾 ∈ 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 ∧ w3a 1083 ∃wex 1780 ∈ wcel 2114 ≠ wne 3016 ∅c0 4291 ∪ cuni 4838 class class class wbr 5066 –1-1→wf1 6352 –1-1-onto→wf1o 6354 (class class class)co 7156 Topctop 21501 Cn ccn 21832 Homeochmeo 22361 ≃ chmph 22362 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-id 5460 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-ov 7159 df-oprab 7160 df-mpo 7161 df-1st 7689 df-2nd 7690 df-1o 8102 df-map 8408 df-top 21502 df-topon 21519 df-cn 21835 df-hmeo 22363 df-hmph 22364 |
This theorem is referenced by: t0hmph 22398 t1hmph 22399 haushmph 22400 |
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