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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 7078 | . . 3 β’ (π΄ β πΆ β (πΉβπ΄) β π« π΅) |
3 | 2 | elpwid 4606 | . 2 β’ (π΄ β πΆ β (πΉβπ΄) β π΅) |
4 | 1 | fdmi 6722 | . . . . 5 β’ dom πΉ = πΆ |
5 | 4 | eleq2i 2819 | . . . 4 β’ (π΄ β dom πΉ β π΄ β πΆ) |
6 | ndmfv 6919 | . . . 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 4391 | . . 3 β’ β β π΅ | |
9 | 7, 8 | eqsstrdi 4031 | . 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 1533 β wcel 2098 β wss 3943 β c0 4317 π« cpw 4597 dom cdm 5669 βΆwf 6532 βcfv 6536 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 ax-sep 5292 ax-nul 5299 ax-pr 5420 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-ne 2935 df-ral 3056 df-rex 3065 df-rab 3427 df-v 3470 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-br 5142 df-opab 5204 df-id 5567 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-iota 6488 df-fun 6538 df-fn 6539 df-f 6540 df-fv 6544 |
This theorem is referenced by: (None) |
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