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Mirrors > Home > MPE Home > Th. List > unirnffid | Structured version Visualization version GIF version |
Description: The union of the range of a function from a finite set into the class of finite sets is finite. Deduction form. (Contributed by David Moews, 1-May-2017.) |
Ref | Expression |
---|---|
unirnffid.1 | β’ (π β πΉ:πβΆFin) |
unirnffid.2 | β’ (π β π β Fin) |
Ref | Expression |
---|---|
unirnffid | β’ (π β βͺ ran πΉ β Fin) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | unirnffid.1 | . . . . 5 β’ (π β πΉ:πβΆFin) | |
2 | 1 | ffnd 6718 | . . . 4 β’ (π β πΉ Fn π) |
3 | unirnffid.2 | . . . 4 β’ (π β π β Fin) | |
4 | fnfi 9204 | . . . 4 β’ ((πΉ Fn π β§ π β Fin) β πΉ β Fin) | |
5 | 2, 3, 4 | syl2anc 582 | . . 3 β’ (π β πΉ β Fin) |
6 | rnfi 9359 | . . 3 β’ (πΉ β Fin β ran πΉ β Fin) | |
7 | 5, 6 | syl 17 | . 2 β’ (π β ran πΉ β Fin) |
8 | 1 | frnd 6725 | . 2 β’ (π β ran πΉ β Fin) |
9 | unifi 9365 | . 2 β’ ((ran πΉ β Fin β§ ran πΉ β Fin) β βͺ ran πΉ β Fin) | |
10 | 7, 8, 9 | syl2anc 582 | 1 β’ (π β βͺ ran πΉ β Fin) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β wcel 2098 β wss 3939 βͺ cuni 4903 ran crn 5673 Fn wfn 6538 βΆwf 6539 Fincfn 8962 |
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 2166 ax-ext 2696 ax-sep 5294 ax-nul 5301 ax-pow 5359 ax-pr 5423 ax-un 7738 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 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 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2931 df-ral 3052 df-rex 3061 df-reu 3365 df-rab 3420 df-v 3465 df-sbc 3769 df-csb 3885 df-dif 3942 df-un 3944 df-in 3946 df-ss 3956 df-pss 3959 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5227 df-tr 5261 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-om 7869 df-1st 7991 df-2nd 7992 df-1o 8485 df-er 8723 df-en 8963 df-dom 8964 df-fin 8966 |
This theorem is referenced by: marypha2 9462 acsinfd 18547 |
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