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Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > limcmptdm | Structured version Visualization version GIF version |
Description: The domain of a maps-to function with a limit. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
Ref | Expression |
---|---|
limcmptdm.f | β’ πΉ = (π₯ β π΄ β¦ π΅) |
limcmptdm.b | β’ ((π β§ π₯ β π΄) β π΅ β β) |
limcmptdm.c | β’ (π β πΆ β (πΉ limβ π·)) |
Ref | Expression |
---|---|
limcmptdm | β’ (π β π΄ β β) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | limcmptdm.f | . . 3 β’ πΉ = (π₯ β π΄ β¦ π΅) | |
2 | limcmptdm.b | . . 3 β’ ((π β§ π₯ β π΄) β π΅ β β) | |
3 | 1, 2 | dmmptd 6695 | . 2 β’ (π β dom πΉ = π΄) |
4 | limcmptdm.c | . . . 4 β’ (π β πΆ β (πΉ limβ π·)) | |
5 | limcrcl 25821 | . . . 4 β’ (πΆ β (πΉ limβ π·) β (πΉ:dom πΉβΆβ β§ dom πΉ β β β§ π· β β)) | |
6 | 4, 5 | syl 17 | . . 3 β’ (π β (πΉ:dom πΉβΆβ β§ dom πΉ β β β§ π· β β)) |
7 | 6 | simp2d 1140 | . 2 β’ (π β dom πΉ β β) |
8 | 3, 7 | eqsstrrd 4012 | 1 β’ (π β π΄ β β) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 394 β§ w3a 1084 = wceq 1533 β wcel 2098 β wss 3939 β¦ cmpt 5226 dom cdm 5672 βΆwf 6539 (class class class)co 7416 βcc 11136 limβ climc 25809 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-sep 5294 ax-nul 5301 ax-pow 5359 ax-pr 5423 ax-un 7738 ax-cnex 11194 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2931 df-ral 3052 df-rex 3061 df-rab 3420 df-v 3465 df-sbc 3769 df-dif 3942 df-un 3944 df-in 3946 df-ss 3956 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-br 5144 df-opab 5206 df-mpt 5227 df-id 5570 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-fv 6551 df-ov 7419 df-oprab 7420 df-mpo 7421 df-pm 8846 df-limc 25813 |
This theorem is referenced by: neglimc 45098 addlimc 45099 0ellimcdiv 45100 reclimc 45104 |
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