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Mirrors > Home > MPE Home > Th. List > dmfi | Structured version Visualization version GIF version |
Description: The domain of a finite set is finite. (Contributed by Mario Carneiro, 24-Sep-2013.) |
Ref | Expression |
---|---|
dmfi | ⊢ (𝐴 ∈ Fin → dom 𝐴 ∈ Fin) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fidomdm 9372 | . 2 ⊢ (𝐴 ∈ Fin → dom 𝐴 ≼ 𝐴) | |
2 | domfi 9227 | . 2 ⊢ ((𝐴 ∈ Fin ∧ dom 𝐴 ≼ 𝐴) → dom 𝐴 ∈ Fin) | |
3 | 1, 2 | mpdan 687 | 1 ⊢ (𝐴 ∈ Fin → dom 𝐴 ∈ Fin) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2106 class class class wbr 5148 dom cdm 5689 ≼ cdom 8982 Fincfn 8984 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-om 7888 df-1st 8013 df-2nd 8014 df-1o 8505 df-en 8985 df-dom 8986 df-fin 8988 |
This theorem is referenced by: fundmfibi 9374 residfi 9376 rnfi 9378 hashfun 14473 hashreshashfun 14475 psgnprfval 19554 gsum2dlem2 20004 gsum2d 20005 tsmsxp 24179 numedglnl 29176 vtxdginducedm1fi 29577 finsumvtxdg2ssteplem2 29579 finsumvtxdg2ssteplem4 29581 finsumvtxdg2sstep 29582 vtxdgoddnumeven 29586 relfi 32622 gsumfs2d 33041 fedgmullem2 33658 esum2d 34074 imadomfi 41984 etransclem27 46217 |
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