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Mirrors > Home > MPE Home > Th. List > Mathboxes > fpwfvss | Structured version Visualization version GIF version |
Description: Functions into a powerset always have values which are subsets. This is dependant on our convention when the argument is not part of the domain. (Contributed by RP, 13-Sep-2024.) |
Ref | Expression |
---|---|
fpwfvss.f | β’ πΉ:πΆβΆπ« π΅ |
Ref | Expression |
---|---|
fpwfvss | β’ (πΉβπ΄) β π΅ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fpwfvss.f | . . . 4 β’ πΉ:πΆβΆπ« π΅ | |
2 | 1 | ffvelcdmi 7039 | . . 3 β’ (π΄ β πΆ β (πΉβπ΄) β π« π΅) |
3 | 2 | elpwid 4574 | . 2 β’ (π΄ β πΆ β (πΉβπ΄) β π΅) |
4 | 1 | fdmi 6685 | . . . . 5 β’ dom πΉ = πΆ |
5 | 4 | eleq2i 2830 | . . . 4 β’ (π΄ β dom πΉ β π΄ β πΆ) |
6 | ndmfv 6882 | . . . 4 β’ (Β¬ π΄ β dom πΉ β (πΉβπ΄) = β ) | |
7 | 5, 6 | sylnbir 331 | . . 3 β’ (Β¬ π΄ β πΆ β (πΉβπ΄) = β ) |
8 | 0ss 4361 | . . 3 β’ β β π΅ | |
9 | 7, 8 | eqsstrdi 4003 | . 2 β’ (Β¬ π΄ β πΆ β (πΉβπ΄) β π΅) |
10 | 3, 9 | pm2.61i 182 | 1 β’ (πΉβπ΄) β π΅ |
Colors of variables: wff setvar class |
Syntax hints: Β¬ wn 3 = wceq 1542 β wcel 2107 β wss 3915 β c0 4287 π« cpw 4565 dom cdm 5638 βΆwf 6497 βcfv 6501 |
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 5261 ax-nul 5268 ax-pr 5389 |
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-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-ne 2945 df-ral 3066 df-rex 3075 df-rab 3411 df-v 3450 df-dif 3918 df-un 3920 df-in 3922 df-ss 3932 df-nul 4288 df-if 4492 df-pw 4567 df-sn 4592 df-pr 4594 df-op 4598 df-uni 4871 df-br 5111 df-opab 5173 df-id 5536 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-iota 6453 df-fun 6503 df-fn 6504 df-f 6505 df-fv 6509 |
This theorem is referenced by: (None) |
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