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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 5264 | . . . 4 β’ β β V | |
2 | eqeq2 2748 | . . . . . 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 1923 | . . . 4 β’ (π = β β (βπ(π:π΄βΆβ β π = π) β βπ(π:π΄βΆβ β π = β ))) |
5 | 1, 4 | spcev 3565 | . . 3 β’ (βπ(π:π΄βΆβ β π = β ) β βπβπ(π:π΄βΆβ β π = π)) |
6 | f00 6724 | . . . 4 β’ (π:π΄βΆβ β (π = β β§ π΄ = β )) | |
7 | 6 | simplbi 498 | . . 3 β’ (π:π΄βΆβ β π = β ) |
8 | 5, 7 | mpg 1799 | . 2 β’ βπβπ(π:π΄βΆβ β π = π) |
9 | df-mo 2538 | . 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 1539 = wceq 1541 βwex 1781 β*wmo 2536 β c0 4282 βΆwf 6492 |
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 2707 ax-sep 5256 ax-nul 5263 ax-pr 5384 |
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 2538 df-clab 2714 df-cleq 2728 df-clel 2814 df-ral 3065 df-rex 3074 df-rab 3408 df-v 3447 df-dif 3913 df-un 3915 df-in 3917 df-ss 3927 df-nul 4283 df-if 4487 df-sn 4587 df-pr 4589 df-op 4593 df-br 5106 df-opab 5168 df-id 5531 df-xp 5639 df-rel 5640 df-cnv 5641 df-co 5642 df-dm 5643 df-rn 5644 df-fun 6498 df-fn 6499 df-f 6500 |
This theorem is referenced by: mof02 46895 |
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