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| Mirrors > Home > MPE Home > Th. List > cnf2 | Structured version Visualization version GIF version | ||
| Description: A continuous function is a mapping. (Contributed by Mario Carneiro, 21-Aug-2015.) | 
| Ref | Expression | 
|---|---|
| cnf2 | ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → 𝐹:𝑋⟶𝑌) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | iscn 23244 | . . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → (𝐹 ∈ (𝐽 Cn 𝐾) ↔ (𝐹:𝑋⟶𝑌 ∧ ∀𝑥 ∈ 𝐾 (◡𝐹 “ 𝑥) ∈ 𝐽))) | |
| 2 | 1 | simprbda 498 | . 2 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → 𝐹:𝑋⟶𝑌) | 
| 3 | 2 | 3impa 1109 | 1 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → 𝐹:𝑋⟶𝑌) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1086 ∈ wcel 2107 ∀wral 3060 ◡ccnv 5683 “ cima 5687 ⟶wf 6556 ‘cfv 6560 (class class class)co 7432 TopOnctopon 22917 Cn ccn 23233 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-sep 5295 ax-nul 5305 ax-pow 5364 ax-pr 5431 ax-un 7756 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-ral 3061 df-rex 3070 df-rab 3436 df-v 3481 df-sbc 3788 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-nul 4333 df-if 4525 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-br 5143 df-opab 5205 df-mpt 5225 df-id 5577 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-iota 6513 df-fun 6562 df-fn 6563 df-f 6564 df-fv 6568 df-ov 7435 df-oprab 7436 df-mpo 7437 df-map 8869 df-top 22901 df-topon 22918 df-cn 23236 | 
| This theorem is referenced by: iscncl 23278 cncls2 23282 cncls 23283 cnntr 23284 cnrest2 23295 cnrest2r 23296 ptcn 23636 txdis1cn 23644 lmcn2 23658 cnmpt11 23672 cnmpt1t 23674 cnmpt12 23676 cnmpt21 23680 cnmpt2t 23682 cnmpt22 23683 cnmpt22f 23684 cnmptcom 23687 cnmptkp 23689 cnmptk1 23690 cnmpt1k 23691 cnmptkk 23692 cnmptk1p 23694 cnmptk2 23695 cnmpt2k 23697 qtopss 23724 qtopeu 23725 qtopomap 23727 qtopcmap 23728 hmeof1o2 23772 xpstopnlem1 23818 xkocnv 23823 xkohmeo 23824 qtophmeo 23826 cnmpt1plusg 24096 cnmpt2plusg 24097 tsmsmhm 24155 cnmpt1vsca 24203 cnmpt2vsca 24204 cnmpt1ds 24865 cnmpt2ds 24866 fsumcn 24895 cnmpopc 24956 htpyco1 25011 htpyco2 25012 phtpyco2 25023 pi1xfrf 25087 pi1xfr 25089 pi1xfrcnvlem 25090 pi1xfrcnv 25091 pi1cof 25093 pi1coghm 25095 cnmpt1ip 25282 cnmpt2ip 25283 txsconnlem 35246 txsconn 35247 cvmlift3lem6 35330 fcnre 45035 refsumcn 45040 refsum2cnlem1 45047 fprodcnlem 45619 icccncfext 45907 itgsubsticclem 45995 | 
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