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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 6717 | . . . . 5 β’ (πΉ:π΄βΆβ β Fun πΉ) | |
2 | frn 6721 | . . . . . . 7 β’ (πΉ:π΄βΆβ β ran πΉ β β ) | |
3 | ss0 4397 | . . . . . . 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 5922 | . . . . . 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 6543 | . . . . 5 β’ (πΉ Fn β β (Fun πΉ β§ dom πΉ = β )) | |
8 | 1, 6, 7 | sylanbrc 583 | . . . 4 β’ (πΉ:π΄βΆβ β πΉ Fn β ) |
9 | fn0 6678 | . . . 4 β’ (πΉ Fn β β πΉ = β ) | |
10 | 8, 9 | sylib 217 | . . 3 β’ (πΉ:π΄βΆβ β πΉ = β ) |
11 | fdm 6723 | . . . 4 β’ (πΉ:π΄βΆβ β dom πΉ = π΄) | |
12 | 11, 6 | eqtr3d 2774 | . . 3 β’ (πΉ:π΄βΆβ β π΄ = β ) |
13 | 10, 12 | jca 512 | . 2 β’ (πΉ:π΄βΆβ β (πΉ = β β§ π΄ = β )) |
14 | f0 6769 | . . 3 β’ β :β βΆβ | |
15 | feq1 6695 | . . . 4 β’ (πΉ = β β (πΉ:π΄βΆβ β β :π΄βΆβ )) | |
16 | feq2 6696 | . . . 4 β’ (π΄ = β β (β :π΄βΆβ β β :β βΆβ )) | |
17 | 15, 16 | sylan9bb 510 | . . 3 β’ ((πΉ = β β§ π΄ = β ) β (πΉ:π΄βΆβ β β :β βΆβ )) |
18 | 14, 17 | mpbiri 257 | . 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 396 = wceq 1541 β wss 3947 β c0 4321 dom cdm 5675 ran crn 5676 Fun wfun 6534 Fn wfn 6535 βΆwf 6536 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-sep 5298 ax-nul 5305 ax-pr 5426 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-clab 2710 df-cleq 2724 df-clel 2810 df-ral 3062 df-rex 3071 df-rab 3433 df-v 3476 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4322 df-if 4528 df-sn 4628 df-pr 4630 df-op 4634 df-br 5148 df-opab 5210 df-id 5573 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-fun 6542 df-fn 6543 df-f 6544 |
This theorem is referenced by: dom0 9098 cantnff 9665 0wrd0 14486 supcvg 15798 ram0 16951 itgsubstlem 25556 uhgr0vb 28321 lfuhgr1v0e 28500 wlkv0 28897 sate0fv0 34396 prv0 34409 ismgmOLD 36706 mof0 47457 mof0ALT 47459 mofeu 47467 fdomne0 47469 f002 47473 fullthinc 47619 |
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