Theorem grporndm 28872

Description: A group's range in terms of its domain. (Contributed by NM, 6-Apr-2008.) (New usage is discouraged.)

Assertion

Ref	Expression
grporndm	⊢ (𝐺 ∈ GrpOp → ran 𝐺 = dom dom 𝐺)

Proof of Theorem grporndm

Step	Hyp	Ref	Expression
1		eqid 2738	. . 3 ⊢ ran 𝐺 = ran 𝐺
2	1	grpofo 28861	. 2 ⊢ (𝐺 ∈ GrpOp → 𝐺:(ran 𝐺 × ran 𝐺)–onto→ran 𝐺)
3		fof 6688	. . . . 5 ⊢ (𝐺:(ran 𝐺 × ran 𝐺)–onto→ran 𝐺 → 𝐺:(ran 𝐺 × ran 𝐺)⟶ran 𝐺)
4	3	fdmd 6611	. . . 4 ⊢ (𝐺:(ran 𝐺 × ran 𝐺)–onto→ran 𝐺 → dom 𝐺 = (ran 𝐺 × ran 𝐺))
5	4	dmeqd 5814	. . 3 ⊢ (𝐺:(ran 𝐺 × ran 𝐺)–onto→ran 𝐺 → dom dom 𝐺 = dom (ran 𝐺 × ran 𝐺))
6		dmxpid 5839	. . 3 ⊢ dom (ran 𝐺 × ran 𝐺) = ran 𝐺
7	5, 6	eqtr2di 2795	. 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)

Syntax hints:   → wi 4   = wceq 1539   ∈ wcel 2106   × cxp 5587  dom cdm 5589  ran crn 5590  –onto→wfo 6431  GrpOpcgr 28851

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352  ax-un 7588

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-fo 6439  df-fv 6441  df-ov 7278  df-grpo 28855

This theorem is referenced by:  hhshsslem1  29629  rngorn1  36091  divrngcl  36115  isdrngo2  36116