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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 8840 | . . . 4 β’ (π β (β βm π΄) β π:π΄βΆβ ) | |
2 | fdm 6717 | . . . . 5 β’ (π:π΄βΆβ β dom π = π΄) | |
3 | frn 6715 | . . . . . . 7 β’ (π:π΄βΆβ β ran π β β ) | |
4 | ss0 4391 | . . . . . . 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 5915 | . . . . . 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 2766 | . . . 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 2957 | . 2 β’ (π΄ β β β Β¬ π β (β βm π΄)) |
11 | 10 | eq0rdv 4397 | 1 β’ (π΄ β β β (β βm π΄) = β ) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 = wceq 1533 β wcel 2098 β wne 2932 β wss 3941 β c0 4315 dom cdm 5667 ran crn 5668 βΆwf 6530 (class class class)co 7402 βm cmap 8817 |
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 2695 ax-sep 5290 ax-nul 5297 ax-pow 5354 ax-pr 5418 ax-un 7719 |
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 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-ral 3054 df-rex 3063 df-rab 3425 df-v 3468 df-sbc 3771 df-csb 3887 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-nul 4316 df-if 4522 df-pw 4597 df-sn 4622 df-pr 4624 df-op 4628 df-uni 4901 df-iun 4990 df-br 5140 df-opab 5202 df-mpt 5223 df-id 5565 df-xp 5673 df-rel 5674 df-cnv 5675 df-co 5676 df-dm 5677 df-rn 5678 df-res 5679 df-ima 5680 df-iota 6486 df-fun 6536 df-fn 6537 df-f 6538 df-fv 6542 df-ov 7405 df-oprab 7406 df-mpo 7407 df-1st 7969 df-2nd 7970 df-map 8819 |
This theorem is referenced by: map0g 8875 mapdom2 9145 ply1plusgfvi 22104 satf0 34880 prv0 34938 |
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