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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 | ⊢ (𝐺 ∈ GrpOp → ran 𝐺 = dom dom 𝐺) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2725 | . . 3 ⊢ ran 𝐺 = ran 𝐺 | |
2 | 1 | grpofo 30401 | . 2 ⊢ (𝐺 ∈ GrpOp → 𝐺:(ran 𝐺 × ran 𝐺)–onto→ran 𝐺) |
3 | fof 6810 | . . . . 5 ⊢ (𝐺:(ran 𝐺 × ran 𝐺)–onto→ran 𝐺 → 𝐺:(ran 𝐺 × ran 𝐺)⟶ran 𝐺) | |
4 | 3 | fdmd 6733 | . . . 4 ⊢ (𝐺:(ran 𝐺 × ran 𝐺)–onto→ran 𝐺 → dom 𝐺 = (ran 𝐺 × ran 𝐺)) |
5 | 4 | dmeqd 5908 | . . 3 ⊢ (𝐺:(ran 𝐺 × ran 𝐺)–onto→ran 𝐺 → dom dom 𝐺 = dom (ran 𝐺 × ran 𝐺)) |
6 | dmxpid 5932 | . . 3 ⊢ dom (ran 𝐺 × ran 𝐺) = ran 𝐺 | |
7 | 5, 6 | eqtr2di 2782 | . 2 ⊢ (𝐺:(ran 𝐺 × ran 𝐺)–onto→ran 𝐺 → ran 𝐺 = dom dom 𝐺) |
8 | 2, 7 | syl 17 | 1 ⊢ (𝐺 ∈ GrpOp → ran 𝐺 = dom dom 𝐺) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2098 × cxp 5676 dom cdm 5678 ran crn 5679 –onto→wfo 6547 GrpOpcgr 30391 |
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 5300 ax-nul 5307 ax-pr 5429 ax-un 7741 |
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 2930 df-ral 3051 df-rex 3060 df-rab 3419 df-v 3463 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4323 df-if 4531 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4910 df-iun 4999 df-br 5150 df-opab 5212 df-mpt 5233 df-id 5576 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-iota 6501 df-fun 6551 df-fn 6552 df-f 6553 df-fo 6555 df-fv 6557 df-ov 7422 df-grpo 30395 |
This theorem is referenced by: hhshsslem1 31169 rngorn1 37557 divrngcl 37581 isdrngo2 37582 |
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