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Mirrors > Home > MPE Home > Th. List > f00 | Structured version Visualization version GIF version |
Description: A class is a function with empty codomain iff it and its domain are empty. (Contributed by NM, 10-Dec-2003.) |
Ref | Expression |
---|---|
f00 | β’ (πΉ:π΄βΆβ β (πΉ = β β§ π΄ = β )) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ffun 6721 | . . . . 5 β’ (πΉ:π΄βΆβ β Fun πΉ) | |
2 | frn 6725 | . . . . . . 7 β’ (πΉ:π΄βΆβ β ran πΉ β β ) | |
3 | ss0 4399 | . . . . . . 7 β’ (ran πΉ β β β ran πΉ = β ) | |
4 | 2, 3 | syl 17 | . . . . . 6 β’ (πΉ:π΄βΆβ β ran πΉ = β ) |
5 | dm0rn0 5925 | . . . . . 6 β’ (dom πΉ = β β ran πΉ = β ) | |
6 | 4, 5 | sylibr 233 | . . . . 5 β’ (πΉ:π΄βΆβ β dom πΉ = β ) |
7 | df-fn 6547 | . . . . 5 β’ (πΉ Fn β β (Fun πΉ β§ dom πΉ = β )) | |
8 | 1, 6, 7 | sylanbrc 584 | . . . 4 β’ (πΉ:π΄βΆβ β πΉ Fn β ) |
9 | fn0 6682 | . . . 4 β’ (πΉ Fn β β πΉ = β ) | |
10 | 8, 9 | sylib 217 | . . 3 β’ (πΉ:π΄βΆβ β πΉ = β ) |
11 | fdm 6727 | . . . 4 β’ (πΉ:π΄βΆβ β dom πΉ = π΄) | |
12 | 11, 6 | eqtr3d 2775 | . . 3 β’ (πΉ:π΄βΆβ β π΄ = β ) |
13 | 10, 12 | jca 513 | . 2 β’ (πΉ:π΄βΆβ β (πΉ = β β§ π΄ = β )) |
14 | f0 6773 | . . 3 β’ β :β βΆβ | |
15 | feq1 6699 | . . . 4 β’ (πΉ = β β (πΉ:π΄βΆβ β β :π΄βΆβ )) | |
16 | feq2 6700 | . . . 4 β’ (π΄ = β β (β :π΄βΆβ β β :β βΆβ )) | |
17 | 15, 16 | sylan9bb 511 | . . 3 β’ ((πΉ = β β§ π΄ = β ) β (πΉ:π΄βΆβ β β :β βΆβ )) |
18 | 14, 17 | mpbiri 258 | . 2 β’ ((πΉ = β β§ π΄ = β ) β πΉ:π΄βΆβ ) |
19 | 13, 18 | impbii 208 | 1 β’ (πΉ:π΄βΆβ β (πΉ = β β§ π΄ = β )) |
Colors of variables: wff setvar class |
Syntax hints: β wb 205 β§ wa 397 = wceq 1542 β wss 3949 β c0 4323 dom cdm 5677 ran crn 5678 Fun wfun 6538 Fn wfn 6539 βΆwf 6540 |
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 5300 ax-nul 5307 ax-pr 5428 |
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-clab 2711 df-cleq 2725 df-clel 2811 df-ral 3063 df-rex 3072 df-rab 3434 df-v 3477 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4324 df-if 4530 df-sn 4630 df-pr 4632 df-op 4636 df-br 5150 df-opab 5212 df-id 5575 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-fun 6546 df-fn 6547 df-f 6548 |
This theorem is referenced by: dom0 9102 cantnff 9669 0wrd0 14490 supcvg 15802 ram0 16955 itgsubstlem 25565 uhgr0vb 28332 lfuhgr1v0e 28511 wlkv0 28908 sate0fv0 34408 prv0 34421 ismgmOLD 36718 mof0 47504 mof0ALT 47506 mofeu 47514 fdomne0 47516 f002 47520 fullthinc 47666 |
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