![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > map0b | Structured version Visualization version GIF version |
Description: Set exponentiation with an empty base is the empty set, provided the exponent is nonempty. Theorem 96 of [Suppes] p. 89. (Contributed by NM, 10-Dec-2003.) (Revised by Mario Carneiro, 26-Apr-2015.) |
Ref | Expression |
---|---|
map0b | β’ (π΄ β β β (β βm π΄) = β ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elmapi 8867 | . . . 4 β’ (π β (β βm π΄) β π:π΄βΆβ ) | |
2 | fdm 6731 | . . . . 5 β’ (π:π΄βΆβ β dom π = π΄) | |
3 | frn 6729 | . . . . . . 7 β’ (π:π΄βΆβ β ran π β β ) | |
4 | ss0 4399 | . . . . . . 7 β’ (ran π β β β ran π = β ) | |
5 | 3, 4 | syl 17 | . . . . . 6 β’ (π:π΄βΆβ β ran π = β ) |
6 | dm0rn0 5927 | . . . . . 6 β’ (dom π = β β ran π = β ) | |
7 | 5, 6 | sylibr 233 | . . . . 5 β’ (π:π΄βΆβ β dom π = β ) |
8 | 2, 7 | eqtr3d 2770 | . . . 4 β’ (π:π΄βΆβ β π΄ = β ) |
9 | 1, 8 | syl 17 | . . 3 β’ (π β (β βm π΄) β π΄ = β ) |
10 | 9 | necon3ai 2962 | . 2 β’ (π΄ β β β Β¬ π β (β βm π΄)) |
11 | 10 | eq0rdv 4405 | 1 β’ (π΄ β β β (β βm π΄) = β ) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 = wceq 1534 β wcel 2099 β wne 2937 β wss 3947 β c0 4323 dom cdm 5678 ran crn 5679 βΆwf 6544 (class class class)co 7420 βm cmap 8844 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-sep 5299 ax-nul 5306 ax-pow 5365 ax-pr 5429 ax-un 7740 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-ral 3059 df-rex 3068 df-rab 3430 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4909 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5576 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-iota 6500 df-fun 6550 df-fn 6551 df-f 6552 df-fv 6556 df-ov 7423 df-oprab 7424 df-mpo 7425 df-1st 7993 df-2nd 7994 df-map 8846 |
This theorem is referenced by: map0g 8902 mapdom2 9172 ply1plusgfvi 22159 satf0 34982 prv0 35040 |
Copyright terms: Public domain | W3C validator |