![]() |
Mathbox for Zhi Wang |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > Mathboxes > mof0 | Structured version Visualization version GIF version |
Description: There is at most one function into the empty set. (Contributed by Zhi Wang, 19-Sep-2024.) |
Ref | Expression |
---|---|
mof0 | β’ β*π π:π΄βΆβ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0ex 5301 | . . . 4 β’ β β V | |
2 | eqeq2 2740 | . . . . . 6 β’ (π = β β (π = π β π = β )) | |
3 | 2 | imbi2d 340 | . . . . 5 β’ (π = β β ((π:π΄βΆβ β π = π) β (π:π΄βΆβ β π = β ))) |
4 | 3 | albidv 1916 | . . . 4 β’ (π = β β (βπ(π:π΄βΆβ β π = π) β βπ(π:π΄βΆβ β π = β ))) |
5 | 1, 4 | spcev 3592 | . . 3 β’ (βπ(π:π΄βΆβ β π = β ) β βπβπ(π:π΄βΆβ β π = π)) |
6 | f00 6773 | . . . 4 β’ (π:π΄βΆβ β (π = β β§ π΄ = β )) | |
7 | 6 | simplbi 497 | . . 3 β’ (π:π΄βΆβ β π = β ) |
8 | 5, 7 | mpg 1792 | . 2 β’ βπβπ(π:π΄βΆβ β π = π) |
9 | df-mo 2530 | . 2 β’ (β*π π:π΄βΆβ β βπβπ(π:π΄βΆβ β π = π)) | |
10 | 8, 9 | mpbir 230 | 1 β’ β*π π:π΄βΆβ |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 βwal 1532 = wceq 1534 βwex 1774 β*wmo 2528 β c0 4318 βΆwf 6538 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-sep 5293 ax-nul 5300 ax-pr 5423 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-clab 2706 df-cleq 2720 df-clel 2806 df-ral 3058 df-rex 3067 df-rab 3429 df-v 3472 df-dif 3948 df-un 3950 df-in 3952 df-ss 3962 df-nul 4319 df-if 4525 df-sn 4625 df-pr 4627 df-op 4631 df-br 5143 df-opab 5205 df-id 5570 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-fun 6544 df-fn 6545 df-f 6546 |
This theorem is referenced by: mof02 47885 |
Copyright terms: Public domain | W3C validator |