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Mirrors > Home > MPE Home > Th. List > Mathboxes > elbigodm | Structured version Visualization version GIF version |
Description: The domain of a function of order G(x) is a subset of the reals. (Contributed by AV, 18-May-2020.) |
Ref | Expression |
---|---|
elbigodm | β’ (πΉ β (ΞβπΊ) β dom πΉ β β) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elbigo 46315 | . 2 β’ (πΉ β (ΞβπΊ) β (πΉ β (β βpm β) β§ πΊ β (β βpm β) β§ βπ₯ β β βπ β β βπ¦ β (dom πΉ β© (π₯[,)+β))(πΉβπ¦) β€ (π Β· (πΊβπ¦)))) | |
2 | reex 11064 | . . . . 5 β’ β β V | |
3 | 2, 2 | elpm2 8734 | . . . 4 β’ (πΉ β (β βpm β) β (πΉ:dom πΉβΆβ β§ dom πΉ β β)) |
4 | 3 | simprbi 497 | . . 3 β’ (πΉ β (β βpm β) β dom πΉ β β) |
5 | 4 | 3ad2ant1 1132 | . 2 β’ ((πΉ β (β βpm β) β§ πΊ β (β βpm β) β§ βπ₯ β β βπ β β βπ¦ β (dom πΉ β© (π₯[,)+β))(πΉβπ¦) β€ (π Β· (πΊβπ¦))) β dom πΉ β β) |
6 | 1, 5 | sylbi 216 | 1 β’ (πΉ β (ΞβπΊ) β dom πΉ β β) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ w3a 1086 β wcel 2105 βwral 3061 βwrex 3070 β© cin 3897 β wss 3898 class class class wbr 5093 dom cdm 5621 βΆwf 6476 βcfv 6480 (class class class)co 7338 βpm cpm 8688 βcr 10972 Β· cmul 10978 +βcpnf 11108 β€ cle 11112 [,)cico 13183 Ξcbigo 46311 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2707 ax-sep 5244 ax-nul 5251 ax-pow 5309 ax-pr 5373 ax-un 7651 ax-cnex 11029 ax-resscn 11030 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2886 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3404 df-v 3443 df-sbc 3728 df-dif 3901 df-un 3903 df-in 3905 df-ss 3915 df-nul 4271 df-if 4475 df-pw 4550 df-sn 4575 df-pr 4577 df-op 4581 df-uni 4854 df-br 5094 df-opab 5156 df-mpt 5177 df-id 5519 df-xp 5627 df-rel 5628 df-cnv 5629 df-co 5630 df-dm 5631 df-rn 5632 df-res 5633 df-ima 5634 df-iota 6432 df-fun 6482 df-fn 6483 df-f 6484 df-fv 6488 df-ov 7341 df-oprab 7342 df-mpo 7343 df-pm 8690 df-bigo 46312 |
This theorem is referenced by: (None) |
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