Theorem grporndm 31142

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 2763	. . 3 ⊢ ran 𝐺 = ran 𝐺
2	1	grpofo 31131	. 2 ⊢ (𝐺 ∈ GrpOp → 𝐺:(ran 𝐺 × ran 𝐺)⟶ontoran 𝐺)
3		fof 6818	. . . . 5 ⊢ (𝐺:(ran 𝐺 × ran 𝐺)⟶ontoran 𝐺 → 𝐺:(ran 𝐺 × ran 𝐺)⟶ran 𝐺)
4	3	fdmd 6742	. . . 4 ⊢ (𝐺:(ran 𝐺 × ran 𝐺)⟶ontoran 𝐺 → dom 𝐺 = (ran 𝐺 × ran 𝐺))
5	4	dmeqd 5894	. . 3 ⊢ (𝐺:(ran 𝐺 × ran 𝐺)⟶ontoran 𝐺 → dom dom 𝐺 = dom (ran 𝐺 × ran 𝐺))
6		dmxpid 5919	. . 3 ⊢ dom (ran 𝐺 × ran 𝐺) = ran 𝐺
7	5, 6	eqtr2di 2815	. 2 ⊢ (𝐺:(ran 𝐺 × ran 𝐺)⟶ontoran 𝐺 → ran 𝐺 = dom dom 𝐺)
8	2, 7	syl 18	1 ⊢ (𝐺 ∈ GrpOp → ran 𝐺 = dom dom 𝐺)

This proof depends on syntax axioms:   → wi 4   = wceq 1570   ∈ wcel 2147   × cxp 5653  dom cdm 5655  ran crn 5656  ⟶ontowfo 6556  GrpOpcgr 31121

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-sep 5252  ax-nul 5263  ax-pr 5395  ax-un 7775

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-iun 4956  df-br 5107  df-opab 5171  df-mpt 5190  df-id 5550  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-iota 6512  df-fun 6560  df-fn 6561  df-f 6562  df-fo 6564  df-fv 6566  df-ov 7446  df-grpo 31125

This theorem is used by:  hhshsslem1  31899  rngorn1  39059  divrngcl  39083  isdrngo2  39084