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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 8790 | . . . 4 β’ (π β (β βm π΄) β π:π΄βΆβ ) | |
2 | fdm 6678 | . . . . 5 β’ (π:π΄βΆβ β dom π = π΄) | |
3 | frn 6676 | . . . . . . 7 β’ (π:π΄βΆβ β ran π β β ) | |
4 | ss0 4359 | . . . . . . 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 5881 | . . . . . 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 2775 | . . . 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 2965 | . 2 β’ (π΄ β β β Β¬ π β (β βm π΄)) |
11 | 10 | eq0rdv 4365 | 1 β’ (π΄ β β β (β βm π΄) = β ) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 = wceq 1542 β wcel 2107 β wne 2940 β wss 3911 β c0 4283 dom cdm 5634 ran crn 5635 βΆwf 6493 (class class class)co 7358 βm cmap 8768 |
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 2704 ax-sep 5257 ax-nul 5264 ax-pow 5321 ax-pr 5385 ax-un 7673 |
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 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3407 df-v 3446 df-sbc 3741 df-csb 3857 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-iun 4957 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-1st 7922 df-2nd 7923 df-map 8770 |
This theorem is referenced by: map0g 8825 mapdom2 9095 ply1plusgfvi 21629 satf0 34023 prv0 34081 |
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