Theorem grporndm 30569

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 2735	. . 3 ⊢ ran 𝐺 = ran 𝐺
2	1	grpofo 30558	. 2 ⊢ (𝐺 ∈ GrpOp → 𝐺:(ran 𝐺 × ran 𝐺)–onto→ran 𝐺)
3		fof 6741	. . . . 5 ⊢ (𝐺:(ran 𝐺 × ran 𝐺)–onto→ran 𝐺 → 𝐺:(ran 𝐺 × ran 𝐺)⟶ran 𝐺)
4	3	fdmd 6667	. . . 4 ⊢ (𝐺:(ran 𝐺 × ran 𝐺)–onto→ran 𝐺 → dom 𝐺 = (ran 𝐺 × ran 𝐺))
5	4	dmeqd 5849	. . 3 ⊢ (𝐺:(ran 𝐺 × ran 𝐺)–onto→ran 𝐺 → dom dom 𝐺 = dom (ran 𝐺 × ran 𝐺))
6		dmxpid 5874	. . 3 ⊢ dom (ran 𝐺 × ran 𝐺) = ran 𝐺
7	5, 6	eqtr2di 2787	. 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 1542   ∈ wcel 2114   × cxp 5618  dom cdm 5620  ran crn 5621  –onto→wfo 6485  GrpOpcgr 30548

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2184  ax-ext 2707  ax-sep 5220  ax-nul 5230  ax-pr 5364  ax-un 7678

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2538  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2810  df-nfc 2884  df-ne 2931  df-ral 3050  df-rex 3060  df-rab 3388  df-v 3429  df-sbc 3726  df-csb 3834  df-dif 3888  df-un 3890  df-in 3892  df-ss 3902  df-nul 4264  df-if 4457  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4841  df-iun 4925  df-br 5075  df-opab 5137  df-mpt 5156  df-id 5515  df-xp 5626  df-rel 5627  df-cnv 5628  df-co 5629  df-dm 5630  df-rn 5631  df-iota 6443  df-fun 6489  df-fn 6490  df-f 6491  df-fo 6493  df-fv 6495  df-ov 7359  df-grpo 30552

This theorem is referenced by:  hhshsslem1  31326  rngorn1  38242  divrngcl  38266  isdrngo2  38267