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Theorem nrmhmph 22403
Description: Normality is a topological property. (Contributed by Mario Carneiro, 25-Aug-2015.)
Assertion
Ref Expression
nrmhmph (𝐽𝐾 → (𝐽 ∈ Nrm → 𝐾 ∈ Nrm))

Proof of Theorem nrmhmph
Dummy variables 𝑤 𝑓 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 hmph 22385 . 2 (𝐽𝐾 ↔ (𝐽Homeo𝐾) ≠ ∅)
2 n0 4263 . . 3 ((𝐽Homeo𝐾) ≠ ∅ ↔ ∃𝑓 𝑓 ∈ (𝐽Homeo𝐾))
3 hmeocn 22369 . . . . . . . 8 (𝑓 ∈ (𝐽Homeo𝐾) → 𝑓 ∈ (𝐽 Cn 𝐾))
43adantl 485 . . . . . . 7 ((𝐽 ∈ Nrm ∧ 𝑓 ∈ (𝐽Homeo𝐾)) → 𝑓 ∈ (𝐽 Cn 𝐾))
5 cntop2 21850 . . . . . . 7 (𝑓 ∈ (𝐽 Cn 𝐾) → 𝐾 ∈ Top)
64, 5syl 17 . . . . . 6 ((𝐽 ∈ Nrm ∧ 𝑓 ∈ (𝐽Homeo𝐾)) → 𝐾 ∈ Top)
7 simpll 766 . . . . . . . . 9 (((𝐽 ∈ Nrm ∧ 𝑓 ∈ (𝐽Homeo𝐾)) ∧ (𝑥𝐾𝑦 ∈ ((Clsd‘𝐾) ∩ 𝒫 𝑥))) → 𝐽 ∈ Nrm)
84adantr 484 . . . . . . . . . 10 (((𝐽 ∈ Nrm ∧ 𝑓 ∈ (𝐽Homeo𝐾)) ∧ (𝑥𝐾𝑦 ∈ ((Clsd‘𝐾) ∩ 𝒫 𝑥))) → 𝑓 ∈ (𝐽 Cn 𝐾))
9 simprl 770 . . . . . . . . . 10 (((𝐽 ∈ Nrm ∧ 𝑓 ∈ (𝐽Homeo𝐾)) ∧ (𝑥𝐾𝑦 ∈ ((Clsd‘𝐾) ∩ 𝒫 𝑥))) → 𝑥𝐾)
10 cnima 21874 . . . . . . . . . 10 ((𝑓 ∈ (𝐽 Cn 𝐾) ∧ 𝑥𝐾) → (𝑓𝑥) ∈ 𝐽)
118, 9, 10syl2anc 587 . . . . . . . . 9 (((𝐽 ∈ Nrm ∧ 𝑓 ∈ (𝐽Homeo𝐾)) ∧ (𝑥𝐾𝑦 ∈ ((Clsd‘𝐾) ∩ 𝒫 𝑥))) → (𝑓𝑥) ∈ 𝐽)
12 simprr 772 . . . . . . . . . . 11 (((𝐽 ∈ Nrm ∧ 𝑓 ∈ (𝐽Homeo𝐾)) ∧ (𝑥𝐾𝑦 ∈ ((Clsd‘𝐾) ∩ 𝒫 𝑥))) → 𝑦 ∈ ((Clsd‘𝐾) ∩ 𝒫 𝑥))
1312elin1d 4128 . . . . . . . . . 10 (((𝐽 ∈ Nrm ∧ 𝑓 ∈ (𝐽Homeo𝐾)) ∧ (𝑥𝐾𝑦 ∈ ((Clsd‘𝐾) ∩ 𝒫 𝑥))) → 𝑦 ∈ (Clsd‘𝐾))
14 cnclima 21877 . . . . . . . . . 10 ((𝑓 ∈ (𝐽 Cn 𝐾) ∧ 𝑦 ∈ (Clsd‘𝐾)) → (𝑓𝑦) ∈ (Clsd‘𝐽))
158, 13, 14syl2anc 587 . . . . . . . . 9 (((𝐽 ∈ Nrm ∧ 𝑓 ∈ (𝐽Homeo𝐾)) ∧ (𝑥𝐾𝑦 ∈ ((Clsd‘𝐾) ∩ 𝒫 𝑥))) → (𝑓𝑦) ∈ (Clsd‘𝐽))
1612elin2d 4129 . . . . . . . . . . 11 (((𝐽 ∈ Nrm ∧ 𝑓 ∈ (𝐽Homeo𝐾)) ∧ (𝑥𝐾𝑦 ∈ ((Clsd‘𝐾) ∩ 𝒫 𝑥))) → 𝑦 ∈ 𝒫 𝑥)
1716elpwid 4511 . . . . . . . . . 10 (((𝐽 ∈ Nrm ∧ 𝑓 ∈ (𝐽Homeo𝐾)) ∧ (𝑥𝐾𝑦 ∈ ((Clsd‘𝐾) ∩ 𝒫 𝑥))) → 𝑦𝑥)
18 imass2 5936 . . . . . . . . . 10 (𝑦𝑥 → (𝑓𝑦) ⊆ (𝑓𝑥))
1917, 18syl 17 . . . . . . . . 9 (((𝐽 ∈ Nrm ∧ 𝑓 ∈ (𝐽Homeo𝐾)) ∧ (𝑥𝐾𝑦 ∈ ((Clsd‘𝐾) ∩ 𝒫 𝑥))) → (𝑓𝑦) ⊆ (𝑓𝑥))
20 nrmsep3 21964 . . . . . . . . 9 ((𝐽 ∈ Nrm ∧ ((𝑓𝑥) ∈ 𝐽 ∧ (𝑓𝑦) ∈ (Clsd‘𝐽) ∧ (𝑓𝑦) ⊆ (𝑓𝑥))) → ∃𝑤𝐽 ((𝑓𝑦) ⊆ 𝑤 ∧ ((cls‘𝐽)‘𝑤) ⊆ (𝑓𝑥)))
217, 11, 15, 19, 20syl13anc 1369 . . . . . . . 8 (((𝐽 ∈ Nrm ∧ 𝑓 ∈ (𝐽Homeo𝐾)) ∧ (𝑥𝐾𝑦 ∈ ((Clsd‘𝐾) ∩ 𝒫 𝑥))) → ∃𝑤𝐽 ((𝑓𝑦) ⊆ 𝑤 ∧ ((cls‘𝐽)‘𝑤) ⊆ (𝑓𝑥)))
22 simpllr 775 . . . . . . . . . 10 ((((𝐽 ∈ Nrm ∧ 𝑓 ∈ (𝐽Homeo𝐾)) ∧ (𝑥𝐾𝑦 ∈ ((Clsd‘𝐾) ∩ 𝒫 𝑥))) ∧ (𝑤𝐽 ∧ ((𝑓𝑦) ⊆ 𝑤 ∧ ((cls‘𝐽)‘𝑤) ⊆ (𝑓𝑥)))) → 𝑓 ∈ (𝐽Homeo𝐾))
23 simprl 770 . . . . . . . . . 10 ((((𝐽 ∈ Nrm ∧ 𝑓 ∈ (𝐽Homeo𝐾)) ∧ (𝑥𝐾𝑦 ∈ ((Clsd‘𝐾) ∩ 𝒫 𝑥))) ∧ (𝑤𝐽 ∧ ((𝑓𝑦) ⊆ 𝑤 ∧ ((cls‘𝐽)‘𝑤) ⊆ (𝑓𝑥)))) → 𝑤𝐽)
24 hmeoima 22374 . . . . . . . . . 10 ((𝑓 ∈ (𝐽Homeo𝐾) ∧ 𝑤𝐽) → (𝑓𝑤) ∈ 𝐾)
2522, 23, 24syl2anc 587 . . . . . . . . 9 ((((𝐽 ∈ Nrm ∧ 𝑓 ∈ (𝐽Homeo𝐾)) ∧ (𝑥𝐾𝑦 ∈ ((Clsd‘𝐾) ∩ 𝒫 𝑥))) ∧ (𝑤𝐽 ∧ ((𝑓𝑦) ⊆ 𝑤 ∧ ((cls‘𝐽)‘𝑤) ⊆ (𝑓𝑥)))) → (𝑓𝑤) ∈ 𝐾)
26 simprrl 780 . . . . . . . . . 10 ((((𝐽 ∈ Nrm ∧ 𝑓 ∈ (𝐽Homeo𝐾)) ∧ (𝑥𝐾𝑦 ∈ ((Clsd‘𝐾) ∩ 𝒫 𝑥))) ∧ (𝑤𝐽 ∧ ((𝑓𝑦) ⊆ 𝑤 ∧ ((cls‘𝐽)‘𝑤) ⊆ (𝑓𝑥)))) → (𝑓𝑦) ⊆ 𝑤)
27 eqid 2801 . . . . . . . . . . . . . 14 𝐽 = 𝐽
28 eqid 2801 . . . . . . . . . . . . . 14 𝐾 = 𝐾
2927, 28hmeof1o 22373 . . . . . . . . . . . . 13 (𝑓 ∈ (𝐽Homeo𝐾) → 𝑓: 𝐽1-1-onto 𝐾)
3022, 29syl 17 . . . . . . . . . . . 12 ((((𝐽 ∈ Nrm ∧ 𝑓 ∈ (𝐽Homeo𝐾)) ∧ (𝑥𝐾𝑦 ∈ ((Clsd‘𝐾) ∩ 𝒫 𝑥))) ∧ (𝑤𝐽 ∧ ((𝑓𝑦) ⊆ 𝑤 ∧ ((cls‘𝐽)‘𝑤) ⊆ (𝑓𝑥)))) → 𝑓: 𝐽1-1-onto 𝐾)
31 f1ofun 6596 . . . . . . . . . . . 12 (𝑓: 𝐽1-1-onto 𝐾 → Fun 𝑓)
3230, 31syl 17 . . . . . . . . . . 11 ((((𝐽 ∈ Nrm ∧ 𝑓 ∈ (𝐽Homeo𝐾)) ∧ (𝑥𝐾𝑦 ∈ ((Clsd‘𝐾) ∩ 𝒫 𝑥))) ∧ (𝑤𝐽 ∧ ((𝑓𝑦) ⊆ 𝑤 ∧ ((cls‘𝐽)‘𝑤) ⊆ (𝑓𝑥)))) → Fun 𝑓)
3313adantr 484 . . . . . . . . . . . . 13 ((((𝐽 ∈ Nrm ∧ 𝑓 ∈ (𝐽Homeo𝐾)) ∧ (𝑥𝐾𝑦 ∈ ((Clsd‘𝐾) ∩ 𝒫 𝑥))) ∧ (𝑤𝐽 ∧ ((𝑓𝑦) ⊆ 𝑤 ∧ ((cls‘𝐽)‘𝑤) ⊆ (𝑓𝑥)))) → 𝑦 ∈ (Clsd‘𝐾))
3428cldss 21638 . . . . . . . . . . . . 13 (𝑦 ∈ (Clsd‘𝐾) → 𝑦 𝐾)
3533, 34syl 17 . . . . . . . . . . . 12 ((((𝐽 ∈ Nrm ∧ 𝑓 ∈ (𝐽Homeo𝐾)) ∧ (𝑥𝐾𝑦 ∈ ((Clsd‘𝐾) ∩ 𝒫 𝑥))) ∧ (𝑤𝐽 ∧ ((𝑓𝑦) ⊆ 𝑤 ∧ ((cls‘𝐽)‘𝑤) ⊆ (𝑓𝑥)))) → 𝑦 𝐾)
36 f1ofo 6601 . . . . . . . . . . . . 13 (𝑓: 𝐽1-1-onto 𝐾𝑓: 𝐽onto 𝐾)
37 forn 6572 . . . . . . . . . . . . 13 (𝑓: 𝐽onto 𝐾 → ran 𝑓 = 𝐾)
3830, 36, 373syl 18 . . . . . . . . . . . 12 ((((𝐽 ∈ Nrm ∧ 𝑓 ∈ (𝐽Homeo𝐾)) ∧ (𝑥𝐾𝑦 ∈ ((Clsd‘𝐾) ∩ 𝒫 𝑥))) ∧ (𝑤𝐽 ∧ ((𝑓𝑦) ⊆ 𝑤 ∧ ((cls‘𝐽)‘𝑤) ⊆ (𝑓𝑥)))) → ran 𝑓 = 𝐾)
3935, 38sseqtrrd 3959 . . . . . . . . . . 11 ((((𝐽 ∈ Nrm ∧ 𝑓 ∈ (𝐽Homeo𝐾)) ∧ (𝑥𝐾𝑦 ∈ ((Clsd‘𝐾) ∩ 𝒫 𝑥))) ∧ (𝑤𝐽 ∧ ((𝑓𝑦) ⊆ 𝑤 ∧ ((cls‘𝐽)‘𝑤) ⊆ (𝑓𝑥)))) → 𝑦 ⊆ ran 𝑓)
40 funimass1 6410 . . . . . . . . . . 11 ((Fun 𝑓𝑦 ⊆ ran 𝑓) → ((𝑓𝑦) ⊆ 𝑤𝑦 ⊆ (𝑓𝑤)))
4132, 39, 40syl2anc 587 . . . . . . . . . 10 ((((𝐽 ∈ Nrm ∧ 𝑓 ∈ (𝐽Homeo𝐾)) ∧ (𝑥𝐾𝑦 ∈ ((Clsd‘𝐾) ∩ 𝒫 𝑥))) ∧ (𝑤𝐽 ∧ ((𝑓𝑦) ⊆ 𝑤 ∧ ((cls‘𝐽)‘𝑤) ⊆ (𝑓𝑥)))) → ((𝑓𝑦) ⊆ 𝑤𝑦 ⊆ (𝑓𝑤)))
4226, 41mpd 15 . . . . . . . . 9 ((((𝐽 ∈ Nrm ∧ 𝑓 ∈ (𝐽Homeo𝐾)) ∧ (𝑥𝐾𝑦 ∈ ((Clsd‘𝐾) ∩ 𝒫 𝑥))) ∧ (𝑤𝐽 ∧ ((𝑓𝑦) ⊆ 𝑤 ∧ ((cls‘𝐽)‘𝑤) ⊆ (𝑓𝑥)))) → 𝑦 ⊆ (𝑓𝑤))
43 elssuni 4833 . . . . . . . . . . . 12 (𝑤𝐽𝑤 𝐽)
4443ad2antrl 727 . . . . . . . . . . 11 ((((𝐽 ∈ Nrm ∧ 𝑓 ∈ (𝐽Homeo𝐾)) ∧ (𝑥𝐾𝑦 ∈ ((Clsd‘𝐾) ∩ 𝒫 𝑥))) ∧ (𝑤𝐽 ∧ ((𝑓𝑦) ⊆ 𝑤 ∧ ((cls‘𝐽)‘𝑤) ⊆ (𝑓𝑥)))) → 𝑤 𝐽)
4527hmeocls 22377 . . . . . . . . . . 11 ((𝑓 ∈ (𝐽Homeo𝐾) ∧ 𝑤 𝐽) → ((cls‘𝐾)‘(𝑓𝑤)) = (𝑓 “ ((cls‘𝐽)‘𝑤)))
4622, 44, 45syl2anc 587 . . . . . . . . . 10 ((((𝐽 ∈ Nrm ∧ 𝑓 ∈ (𝐽Homeo𝐾)) ∧ (𝑥𝐾𝑦 ∈ ((Clsd‘𝐾) ∩ 𝒫 𝑥))) ∧ (𝑤𝐽 ∧ ((𝑓𝑦) ⊆ 𝑤 ∧ ((cls‘𝐽)‘𝑤) ⊆ (𝑓𝑥)))) → ((cls‘𝐾)‘(𝑓𝑤)) = (𝑓 “ ((cls‘𝐽)‘𝑤)))
47 simprrr 781 . . . . . . . . . . 11 ((((𝐽 ∈ Nrm ∧ 𝑓 ∈ (𝐽Homeo𝐾)) ∧ (𝑥𝐾𝑦 ∈ ((Clsd‘𝐾) ∩ 𝒫 𝑥))) ∧ (𝑤𝐽 ∧ ((𝑓𝑦) ⊆ 𝑤 ∧ ((cls‘𝐽)‘𝑤) ⊆ (𝑓𝑥)))) → ((cls‘𝐽)‘𝑤) ⊆ (𝑓𝑥))
48 nrmtop 21945 . . . . . . . . . . . . . . 15 (𝐽 ∈ Nrm → 𝐽 ∈ Top)
4948ad3antrrr 729 . . . . . . . . . . . . . 14 ((((𝐽 ∈ Nrm ∧ 𝑓 ∈ (𝐽Homeo𝐾)) ∧ (𝑥𝐾𝑦 ∈ ((Clsd‘𝐾) ∩ 𝒫 𝑥))) ∧ (𝑤𝐽 ∧ ((𝑓𝑦) ⊆ 𝑤 ∧ ((cls‘𝐽)‘𝑤) ⊆ (𝑓𝑥)))) → 𝐽 ∈ Top)
5027clsss3 21668 . . . . . . . . . . . . . 14 ((𝐽 ∈ Top ∧ 𝑤 𝐽) → ((cls‘𝐽)‘𝑤) ⊆ 𝐽)
5149, 44, 50syl2anc 587 . . . . . . . . . . . . 13 ((((𝐽 ∈ Nrm ∧ 𝑓 ∈ (𝐽Homeo𝐾)) ∧ (𝑥𝐾𝑦 ∈ ((Clsd‘𝐾) ∩ 𝒫 𝑥))) ∧ (𝑤𝐽 ∧ ((𝑓𝑦) ⊆ 𝑤 ∧ ((cls‘𝐽)‘𝑤) ⊆ (𝑓𝑥)))) → ((cls‘𝐽)‘𝑤) ⊆ 𝐽)
52 f1odm 6598 . . . . . . . . . . . . . 14 (𝑓: 𝐽1-1-onto 𝐾 → dom 𝑓 = 𝐽)
5330, 52syl 17 . . . . . . . . . . . . 13 ((((𝐽 ∈ Nrm ∧ 𝑓 ∈ (𝐽Homeo𝐾)) ∧ (𝑥𝐾𝑦 ∈ ((Clsd‘𝐾) ∩ 𝒫 𝑥))) ∧ (𝑤𝐽 ∧ ((𝑓𝑦) ⊆ 𝑤 ∧ ((cls‘𝐽)‘𝑤) ⊆ (𝑓𝑥)))) → dom 𝑓 = 𝐽)
5451, 53sseqtrrd 3959 . . . . . . . . . . . 12 ((((𝐽 ∈ Nrm ∧ 𝑓 ∈ (𝐽Homeo𝐾)) ∧ (𝑥𝐾𝑦 ∈ ((Clsd‘𝐾) ∩ 𝒫 𝑥))) ∧ (𝑤𝐽 ∧ ((𝑓𝑦) ⊆ 𝑤 ∧ ((cls‘𝐽)‘𝑤) ⊆ (𝑓𝑥)))) → ((cls‘𝐽)‘𝑤) ⊆ dom 𝑓)
55 funimass3 6805 . . . . . . . . . . . 12 ((Fun 𝑓 ∧ ((cls‘𝐽)‘𝑤) ⊆ dom 𝑓) → ((𝑓 “ ((cls‘𝐽)‘𝑤)) ⊆ 𝑥 ↔ ((cls‘𝐽)‘𝑤) ⊆ (𝑓𝑥)))
5632, 54, 55syl2anc 587 . . . . . . . . . . 11 ((((𝐽 ∈ Nrm ∧ 𝑓 ∈ (𝐽Homeo𝐾)) ∧ (𝑥𝐾𝑦 ∈ ((Clsd‘𝐾) ∩ 𝒫 𝑥))) ∧ (𝑤𝐽 ∧ ((𝑓𝑦) ⊆ 𝑤 ∧ ((cls‘𝐽)‘𝑤) ⊆ (𝑓𝑥)))) → ((𝑓 “ ((cls‘𝐽)‘𝑤)) ⊆ 𝑥 ↔ ((cls‘𝐽)‘𝑤) ⊆ (𝑓𝑥)))
5747, 56mpbird 260 . . . . . . . . . 10 ((((𝐽 ∈ Nrm ∧ 𝑓 ∈ (𝐽Homeo𝐾)) ∧ (𝑥𝐾𝑦 ∈ ((Clsd‘𝐾) ∩ 𝒫 𝑥))) ∧ (𝑤𝐽 ∧ ((𝑓𝑦) ⊆ 𝑤 ∧ ((cls‘𝐽)‘𝑤) ⊆ (𝑓𝑥)))) → (𝑓 “ ((cls‘𝐽)‘𝑤)) ⊆ 𝑥)
5846, 57eqsstrd 3956 . . . . . . . . 9 ((((𝐽 ∈ Nrm ∧ 𝑓 ∈ (𝐽Homeo𝐾)) ∧ (𝑥𝐾𝑦 ∈ ((Clsd‘𝐾) ∩ 𝒫 𝑥))) ∧ (𝑤𝐽 ∧ ((𝑓𝑦) ⊆ 𝑤 ∧ ((cls‘𝐽)‘𝑤) ⊆ (𝑓𝑥)))) → ((cls‘𝐾)‘(𝑓𝑤)) ⊆ 𝑥)
59 sseq2 3944 . . . . . . . . . . 11 (𝑧 = (𝑓𝑤) → (𝑦𝑧𝑦 ⊆ (𝑓𝑤)))
60 fveq2 6649 . . . . . . . . . . . 12 (𝑧 = (𝑓𝑤) → ((cls‘𝐾)‘𝑧) = ((cls‘𝐾)‘(𝑓𝑤)))
6160sseq1d 3949 . . . . . . . . . . 11 (𝑧 = (𝑓𝑤) → (((cls‘𝐾)‘𝑧) ⊆ 𝑥 ↔ ((cls‘𝐾)‘(𝑓𝑤)) ⊆ 𝑥))
6259, 61anbi12d 633 . . . . . . . . . 10 (𝑧 = (𝑓𝑤) → ((𝑦𝑧 ∧ ((cls‘𝐾)‘𝑧) ⊆ 𝑥) ↔ (𝑦 ⊆ (𝑓𝑤) ∧ ((cls‘𝐾)‘(𝑓𝑤)) ⊆ 𝑥)))
6362rspcev 3574 . . . . . . . . 9 (((𝑓𝑤) ∈ 𝐾 ∧ (𝑦 ⊆ (𝑓𝑤) ∧ ((cls‘𝐾)‘(𝑓𝑤)) ⊆ 𝑥)) → ∃𝑧𝐾 (𝑦𝑧 ∧ ((cls‘𝐾)‘𝑧) ⊆ 𝑥))
6425, 42, 58, 63syl12anc 835 . . . . . . . 8 ((((𝐽 ∈ Nrm ∧ 𝑓 ∈ (𝐽Homeo𝐾)) ∧ (𝑥𝐾𝑦 ∈ ((Clsd‘𝐾) ∩ 𝒫 𝑥))) ∧ (𝑤𝐽 ∧ ((𝑓𝑦) ⊆ 𝑤 ∧ ((cls‘𝐽)‘𝑤) ⊆ (𝑓𝑥)))) → ∃𝑧𝐾 (𝑦𝑧 ∧ ((cls‘𝐾)‘𝑧) ⊆ 𝑥))
6521, 64rexlimddv 3253 . . . . . . 7 (((𝐽 ∈ Nrm ∧ 𝑓 ∈ (𝐽Homeo𝐾)) ∧ (𝑥𝐾𝑦 ∈ ((Clsd‘𝐾) ∩ 𝒫 𝑥))) → ∃𝑧𝐾 (𝑦𝑧 ∧ ((cls‘𝐾)‘𝑧) ⊆ 𝑥))
6665ralrimivva 3159 . . . . . 6 ((𝐽 ∈ Nrm ∧ 𝑓 ∈ (𝐽Homeo𝐾)) → ∀𝑥𝐾𝑦 ∈ ((Clsd‘𝐾) ∩ 𝒫 𝑥)∃𝑧𝐾 (𝑦𝑧 ∧ ((cls‘𝐾)‘𝑧) ⊆ 𝑥))
67 isnrm 21944 . . . . . 6 (𝐾 ∈ Nrm ↔ (𝐾 ∈ Top ∧ ∀𝑥𝐾𝑦 ∈ ((Clsd‘𝐾) ∩ 𝒫 𝑥)∃𝑧𝐾 (𝑦𝑧 ∧ ((cls‘𝐾)‘𝑧) ⊆ 𝑥)))
686, 66, 67sylanbrc 586 . . . . 5 ((𝐽 ∈ Nrm ∧ 𝑓 ∈ (𝐽Homeo𝐾)) → 𝐾 ∈ Nrm)
6968expcom 417 . . . 4 (𝑓 ∈ (𝐽Homeo𝐾) → (𝐽 ∈ Nrm → 𝐾 ∈ Nrm))
7069exlimiv 1931 . . 3 (∃𝑓 𝑓 ∈ (𝐽Homeo𝐾) → (𝐽 ∈ Nrm → 𝐾 ∈ Nrm))
712, 70sylbi 220 . 2 ((𝐽Homeo𝐾) ≠ ∅ → (𝐽 ∈ Nrm → 𝐾 ∈ Nrm))
721, 71sylbi 220 1 (𝐽𝐾 → (𝐽 ∈ Nrm → 𝐾 ∈ Nrm))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399   = wceq 1538  wex 1781  wcel 2112  wne 2990  wral 3109  wrex 3110  cin 3883  wss 3884  c0 4246  𝒫 cpw 4500   cuni 4803   class class class wbr 5033  ccnv 5522  dom cdm 5523  ran crn 5524  cima 5526  Fun wfun 6322  ontowfo 6326  1-1-ontowf1o 6327  cfv 6328  (class class class)co 7139  Topctop 21502  Clsdccld 21625  clsccl 21627   Cn ccn 21833  Nrmcnrm 21919  Homeochmeo 22362  chmph 22363
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-rep 5157  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298  ax-un 7445
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-ral 3114  df-rex 3115  df-reu 3116  df-rab 3118  df-v 3446  df-sbc 3724  df-csb 3832  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4804  df-int 4842  df-iun 4886  df-iin 4887  df-br 5034  df-opab 5096  df-mpt 5114  df-id 5428  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-suc 6169  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336  df-ov 7142  df-oprab 7143  df-mpo 7144  df-1st 7675  df-2nd 7676  df-1o 8089  df-map 8395  df-top 21503  df-topon 21520  df-cld 21628  df-cls 21630  df-cn 21836  df-nrm 21926  df-hmeo 22364  df-hmph 22365
This theorem is referenced by: (None)
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