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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 6672 | . . . . 5 β’ (πΉ:π΄βΆβ β Fun πΉ) | |
2 | frn 6676 | . . . . . . 7 β’ (πΉ:π΄βΆβ β ran πΉ β β ) | |
3 | ss0 4359 | . . . . . . 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 5881 | . . . . . 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 6500 | . . . . 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 6633 | . . . 4 β’ (πΉ Fn β β πΉ = β ) | |
10 | 8, 9 | sylib 217 | . . 3 β’ (πΉ:π΄βΆβ β πΉ = β ) |
11 | fdm 6678 | . . . 4 β’ (πΉ:π΄βΆβ β dom πΉ = π΄) | |
12 | 11, 6 | eqtr3d 2779 | . . 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 6724 | . . 3 β’ β :β βΆβ | |
15 | feq1 6650 | . . . 4 β’ (πΉ = β β (πΉ:π΄βΆβ β β :π΄βΆβ )) | |
16 | feq2 6651 | . . . 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 3911 β c0 4283 dom cdm 5634 ran crn 5635 Fun wfun 6491 Fn wfn 6492 βΆwf 6493 |
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 2708 ax-sep 5257 ax-nul 5264 ax-pr 5385 |
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 2539 df-clab 2715 df-cleq 2729 df-clel 2815 df-ral 3066 df-rex 3075 df-rab 3409 df-v 3448 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-nul 4284 df-if 4488 df-sn 4588 df-pr 4590 df-op 4594 df-br 5107 df-opab 5169 df-id 5532 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-fun 6499 df-fn 6500 df-f 6501 |
This theorem is referenced by: dom0 9047 cantnff 9611 0wrd0 14429 supcvg 15742 ram0 16895 itgsubstlem 25415 uhgr0vb 28026 lfuhgr1v0e 28205 wlkv0 28602 sate0fv0 34014 prv0 34027 ismgmOLD 36312 mof0 46911 mof0ALT 46913 mofeu 46921 fdomne0 46923 f002 46927 fullthinc 47073 |
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