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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 6689 | . 2 β’ (π β dom πΉ = π΄) |
4 | limcmptdm.c | . . . 4 β’ (π β πΆ β (πΉ limβ π·)) | |
5 | limcrcl 25758 | . . . 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 4016 | 1 β’ (π β π΄ β β) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 395 β§ w3a 1084 = wceq 1533 β wcel 2098 β wss 3943 β¦ cmpt 5224 dom cdm 5669 βΆwf 6533 (class class class)co 7405 βcc 11110 limβ climc 25746 |
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 2163 ax-ext 2697 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7722 ax-cnex 11168 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 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 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-ral 3056 df-rex 3065 df-rab 3427 df-v 3470 df-sbc 3773 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-br 5142 df-opab 5204 df-mpt 5225 df-id 5567 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-iota 6489 df-fun 6539 df-fn 6540 df-f 6541 df-fv 6545 df-ov 7408 df-oprab 7409 df-mpo 7410 df-pm 8825 df-limc 25750 |
This theorem is referenced by: neglimc 44935 addlimc 44936 0ellimcdiv 44937 reclimc 44941 |
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