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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 6647 | . 2 β’ (π β dom πΉ = π΄) |
4 | limcmptdm.c | . . . 4 β’ (π β πΆ β (πΉ limβ π·)) | |
5 | limcrcl 25241 | . . . 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 1144 | . 2 β’ (π β dom πΉ β β) |
8 | 3, 7 | eqsstrrd 3984 | 1 β’ (π β π΄ β β) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 397 β§ w3a 1088 = wceq 1542 β wcel 2107 β wss 3911 β¦ cmpt 5189 dom cdm 5634 βΆwf 6493 (class class class)co 7358 βcc 11050 limβ climc 25229 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2708 ax-sep 5257 ax-nul 5264 ax-pow 5321 ax-pr 5385 ax-un 7673 ax-cnex 11108 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ne 2945 df-ral 3066 df-rex 3075 df-rab 3409 df-v 3448 df-sbc 3741 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-br 5107 df-opab 5169 df-mpt 5190 df-id 5532 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-fv 6505 df-ov 7361 df-oprab 7362 df-mpo 7363 df-pm 8769 df-limc 25233 |
This theorem is referenced by: neglimc 43895 addlimc 43896 0ellimcdiv 43897 reclimc 43901 |
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