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Mirrors > Home > MPE Home > Th. List > metcnp4 | Structured version Visualization version GIF version |
Description: Two ways to say a mapping from metric 𝐶 to metric 𝐷 is continuous at point 𝑃. Theorem 14-4.3 of [Gleason] p. 240. (Contributed by NM, 17-May-2007.) (Revised by Mario Carneiro, 4-May-2014.) |
Ref | Expression |
---|---|
metcnp4.3 | ⊢ 𝐽 = (MetOpen‘𝐶) |
metcnp4.4 | ⊢ 𝐾 = (MetOpen‘𝐷) |
metcnp4.5 | ⊢ (𝜑 → 𝐶 ∈ (∞Met‘𝑋)) |
metcnp4.6 | ⊢ (𝜑 → 𝐷 ∈ (∞Met‘𝑌)) |
metcnp4.7 | ⊢ (𝜑 → 𝑃 ∈ 𝑋) |
Ref | Expression |
---|---|
metcnp4 | ⊢ (𝜑 → (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) ↔ (𝐹:𝑋⟶𝑌 ∧ ∀𝑓((𝑓:ℕ⟶𝑋 ∧ 𝑓(⇝𝑡‘𝐽)𝑃) → (𝐹 ∘ 𝑓)(⇝𝑡‘𝐾)(𝐹‘𝑃))))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | metcnp4.5 | . . 3 ⊢ (𝜑 → 𝐶 ∈ (∞Met‘𝑋)) | |
2 | metcnp4.3 | . . . 4 ⊢ 𝐽 = (MetOpen‘𝐶) | |
3 | 2 | met1stc 22546 | . . 3 ⊢ (𝐶 ∈ (∞Met‘𝑋) → 𝐽 ∈ 1st𝜔) |
4 | 1, 3 | syl 17 | . 2 ⊢ (𝜑 → 𝐽 ∈ 1st𝜔) |
5 | 2 | mopntopon 22464 | . . 3 ⊢ (𝐶 ∈ (∞Met‘𝑋) → 𝐽 ∈ (TopOn‘𝑋)) |
6 | 1, 5 | syl 17 | . 2 ⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋)) |
7 | metcnp4.6 | . . 3 ⊢ (𝜑 → 𝐷 ∈ (∞Met‘𝑌)) | |
8 | metcnp4.4 | . . . 4 ⊢ 𝐾 = (MetOpen‘𝐷) | |
9 | 8 | mopntopon 22464 | . . 3 ⊢ (𝐷 ∈ (∞Met‘𝑌) → 𝐾 ∈ (TopOn‘𝑌)) |
10 | 7, 9 | syl 17 | . 2 ⊢ (𝜑 → 𝐾 ∈ (TopOn‘𝑌)) |
11 | metcnp4.7 | . 2 ⊢ (𝜑 → 𝑃 ∈ 𝑋) | |
12 | 4, 6, 10, 11 | 1stccnp 21486 | 1 ⊢ (𝜑 → (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) ↔ (𝐹:𝑋⟶𝑌 ∧ ∀𝑓((𝑓:ℕ⟶𝑋 ∧ 𝑓(⇝𝑡‘𝐽)𝑃) → (𝐹 ∘ 𝑓)(⇝𝑡‘𝐾)(𝐹‘𝑃))))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 ∧ wa 382 ∀wal 1629 = wceq 1631 ∈ wcel 2145 class class class wbr 4786 ∘ ccom 5253 ⟶wf 6027 ‘cfv 6031 (class class class)co 6793 ℕcn 11222 ∞Metcxmt 19946 MetOpencmopn 19951 TopOnctopon 20935 CnP ccnp 21250 ⇝𝑡clm 21251 1st𝜔c1stc 21461 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1870 ax-4 1885 ax-5 1991 ax-6 2057 ax-7 2093 ax-8 2147 ax-9 2154 ax-10 2174 ax-11 2190 ax-12 2203 ax-13 2408 ax-ext 2751 ax-rep 4904 ax-sep 4915 ax-nul 4923 ax-pow 4974 ax-pr 5034 ax-un 7096 ax-inf2 8702 ax-cc 9459 ax-cnex 10194 ax-resscn 10195 ax-1cn 10196 ax-icn 10197 ax-addcl 10198 ax-addrcl 10199 ax-mulcl 10200 ax-mulrcl 10201 ax-mulcom 10202 ax-addass 10203 ax-mulass 10204 ax-distr 10205 ax-i2m1 10206 ax-1ne0 10207 ax-1rid 10208 ax-rnegex 10209 ax-rrecex 10210 ax-cnre 10211 ax-pre-lttri 10212 ax-pre-lttrn 10213 ax-pre-ltadd 10214 ax-pre-mulgt0 10215 ax-pre-sup 10216 |
This theorem depends on definitions: df-bi 197 df-an 383 df-or 835 df-3or 1072 df-3an 1073 df-tru 1634 df-fal 1637 df-ex 1853 df-nf 1858 df-sb 2050 df-eu 2622 df-mo 2623 df-clab 2758 df-cleq 2764 df-clel 2767 df-nfc 2902 df-ne 2944 df-nel 3047 df-ral 3066 df-rex 3067 df-reu 3068 df-rmo 3069 df-rab 3070 df-v 3353 df-sbc 3588 df-csb 3683 df-dif 3726 df-un 3728 df-in 3730 df-ss 3737 df-pss 3739 df-nul 4064 df-if 4226 df-pw 4299 df-sn 4317 df-pr 4319 df-tp 4321 df-op 4323 df-uni 4575 df-int 4612 df-iun 4656 df-iin 4657 df-br 4787 df-opab 4847 df-mpt 4864 df-tr 4887 df-id 5157 df-eprel 5162 df-po 5170 df-so 5171 df-fr 5208 df-se 5209 df-we 5210 df-xp 5255 df-rel 5256 df-cnv 5257 df-co 5258 df-dm 5259 df-rn 5260 df-res 5261 df-ima 5262 df-pred 5823 df-ord 5869 df-on 5870 df-lim 5871 df-suc 5872 df-iota 5994 df-fun 6033 df-fn 6034 df-f 6035 df-f1 6036 df-fo 6037 df-f1o 6038 df-fv 6039 df-isom 6040 df-riota 6754 df-ov 6796 df-oprab 6797 df-mpt2 6798 df-om 7213 df-1st 7315 df-2nd 7316 df-wrecs 7559 df-recs 7621 df-rdg 7659 df-1o 7713 df-oadd 7717 df-er 7896 df-map 8011 df-pm 8012 df-en 8110 df-dom 8111 df-sdom 8112 df-fin 8113 df-sup 8504 df-inf 8505 df-card 8965 df-acn 8968 df-pnf 10278 df-mnf 10279 df-xr 10280 df-ltxr 10281 df-le 10282 df-sub 10470 df-neg 10471 df-div 10887 df-nn 11223 df-2 11281 df-n0 11495 df-z 11580 df-uz 11889 df-q 11992 df-rp 12036 df-xneg 12151 df-xadd 12152 df-xmul 12153 df-fz 12534 df-topgen 16312 df-psmet 19953 df-xmet 19954 df-bl 19956 df-mopn 19957 df-top 20919 df-topon 20936 df-bases 20971 df-cld 21044 df-ntr 21045 df-cls 21046 df-cnp 21253 df-lm 21254 df-1stc 21463 |
This theorem is referenced by: (None) |
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