Theorem grporndm 28293

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 2798	. . 3 ⊢ ran 𝐺 = ran 𝐺
2	1	grpofo 28282	. 2 ⊢ (𝐺 ∈ GrpOp → 𝐺:(ran 𝐺 × ran 𝐺)–onto→ran 𝐺)
3		fof 6565	. . . . 5 ⊢ (𝐺:(ran 𝐺 × ran 𝐺)–onto→ran 𝐺 → 𝐺:(ran 𝐺 × ran 𝐺)⟶ran 𝐺)
4	3	fdmd 6497	. . . 4 ⊢ (𝐺:(ran 𝐺 × ran 𝐺)–onto→ran 𝐺 → dom 𝐺 = (ran 𝐺 × ran 𝐺))
5	4	dmeqd 5738	. . 3 ⊢ (𝐺:(ran 𝐺 × ran 𝐺)–onto→ran 𝐺 → dom dom 𝐺 = dom (ran 𝐺 × ran 𝐺))
6		dmxpid 5764	. . 3 ⊢ dom (ran 𝐺 × ran 𝐺) = ran 𝐺
7	5, 6	eqtr2di 2850	. 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 1538   ∈ wcel 2111   × cxp 5517  dom cdm 5519  ran crn 5520  –onto→wfo 6322  GrpOpcgr 28272

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pr 5295  ax-un 7441

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-id 5425  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-fo 6330  df-fv 6332  df-ov 7138  df-grpo 28276

This theorem is referenced by:  hhshsslem1  29050  rngorn1  35371  divrngcl  35395  isdrngo2  35396