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Mirrors > Home > MPE Home > Th. List > grporndm | Structured version Visualization version GIF version |
Description: A group's range in terms of its domain. (Contributed by NM, 6-Apr-2008.) (New usage is discouraged.) |
Ref | Expression |
---|---|
grporndm | β’ (πΊ β GrpOp β ran πΊ = dom dom πΊ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2725 | . . 3 β’ ran πΊ = ran πΊ | |
2 | 1 | grpofo 30353 | . 2 β’ (πΊ β GrpOp β πΊ:(ran πΊ Γ ran πΊ)βontoβran πΊ) |
3 | fof 6806 | . . . . 5 β’ (πΊ:(ran πΊ Γ ran πΊ)βontoβran πΊ β πΊ:(ran πΊ Γ ran πΊ)βΆran πΊ) | |
4 | 3 | fdmd 6728 | . . . 4 β’ (πΊ:(ran πΊ Γ ran πΊ)βontoβran πΊ β dom πΊ = (ran πΊ Γ ran πΊ)) |
5 | 4 | dmeqd 5902 | . . 3 β’ (πΊ:(ran πΊ Γ ran πΊ)βontoβran πΊ β dom dom πΊ = dom (ran πΊ Γ ran πΊ)) |
6 | dmxpid 5926 | . . 3 β’ dom (ran πΊ Γ ran πΊ) = ran πΊ | |
7 | 5, 6 | eqtr2di 2782 | . 2 β’ (πΊ:(ran πΊ Γ ran πΊ)βontoβran πΊ β ran πΊ = dom dom πΊ) |
8 | 2, 7 | syl 17 | 1 β’ (πΊ β GrpOp β ran πΊ = dom dom πΊ) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 = wceq 1533 β wcel 2098 Γ cxp 5670 dom cdm 5672 ran crn 5673 βontoβwfo 6541 GrpOpcgr 30343 |
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-pr 5423 ax-un 7738 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 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-rab 3420 df-v 3465 df-sbc 3769 df-csb 3885 df-dif 3942 df-un 3944 df-in 3946 df-ss 3956 df-nul 4319 df-if 4525 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-id 5570 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-fo 6549 df-fv 6551 df-ov 7419 df-grpo 30347 |
This theorem is referenced by: hhshsslem1 31121 rngorn1 37463 divrngcl 37487 isdrngo2 37488 |
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