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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 23123 | . . 3 ⊢ (𝐶 ∈ (∞Met‘𝑋) → 𝐽 ∈ 1stω) |
4 | 1, 3 | syl 17 | . 2 ⊢ (𝜑 → 𝐽 ∈ 1stω) |
5 | 2 | mopntopon 23041 | . . 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 23041 | . . 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 22062 | 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 208 ∧ wa 398 ∀wal 1529 = wceq 1531 ∈ wcel 2108 class class class wbr 5057 ∘ ccom 5552 ⟶wf 6344 ‘cfv 6348 (class class class)co 7148 ℕcn 11630 ∞Metcxmet 20522 MetOpencmopn 20527 TopOnctopon 21510 CnP ccnp 21825 ⇝𝑡clm 21826 1stωc1stc 22037 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1905 ax-6 1964 ax-7 2009 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2154 ax-12 2170 ax-ext 2791 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7453 ax-inf2 9096 ax-cc 9849 ax-cnex 10585 ax-resscn 10586 ax-1cn 10587 ax-icn 10588 ax-addcl 10589 ax-addrcl 10590 ax-mulcl 10591 ax-mulrcl 10592 ax-mulcom 10593 ax-addass 10594 ax-mulass 10595 ax-distr 10596 ax-i2m1 10597 ax-1ne0 10598 ax-1rid 10599 ax-rnegex 10600 ax-rrecex 10601 ax-cnre 10602 ax-pre-lttri 10603 ax-pre-lttrn 10604 ax-pre-ltadd 10605 ax-pre-mulgt0 10606 ax-pre-sup 10607 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1083 df-3an 1084 df-tru 1534 df-fal 1544 df-ex 1775 df-nf 1779 df-sb 2064 df-mo 2616 df-eu 2648 df-clab 2798 df-cleq 2812 df-clel 2891 df-nfc 2961 df-ne 3015 df-nel 3122 df-ral 3141 df-rex 3142 df-reu 3143 df-rmo 3144 df-rab 3145 df-v 3495 df-sbc 3771 df-csb 3882 df-dif 3937 df-un 3939 df-in 3941 df-ss 3950 df-pss 3952 df-nul 4290 df-if 4466 df-pw 4539 df-sn 4560 df-pr 4562 df-tp 4564 df-op 4566 df-uni 4831 df-int 4868 df-iun 4912 df-iin 4913 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-se 5508 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-isom 6357 df-riota 7106 df-ov 7151 df-oprab 7152 df-mpo 7153 df-om 7573 df-1st 7681 df-2nd 7682 df-wrecs 7939 df-recs 8000 df-rdg 8038 df-1o 8094 df-oadd 8098 df-er 8281 df-map 8400 df-pm 8401 df-en 8502 df-dom 8503 df-sdom 8504 df-fin 8505 df-sup 8898 df-inf 8899 df-card 9360 df-acn 9363 df-pnf 10669 df-mnf 10670 df-xr 10671 df-ltxr 10672 df-le 10673 df-sub 10864 df-neg 10865 df-div 11290 df-nn 11631 df-2 11692 df-n0 11890 df-z 11974 df-uz 12236 df-q 12341 df-rp 12382 df-xneg 12499 df-xadd 12500 df-xmul 12501 df-fz 12885 df-topgen 16709 df-psmet 20529 df-xmet 20530 df-bl 20532 df-mopn 20533 df-top 21494 df-topon 21511 df-bases 21546 df-cld 21619 df-ntr 21620 df-cls 21621 df-cnp 21828 df-lm 21829 df-1stc 22039 |
This theorem is referenced by: (None) |
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