Theorem grporndm 28279

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 2819	. . 3 ⊢ ran 𝐺 = ran 𝐺
2	1	grpofo 28268	. 2 ⊢ (𝐺 ∈ GrpOp → 𝐺:(ran 𝐺 × ran 𝐺)–onto→ran 𝐺)
3		fof 6583	. . . . 5 ⊢ (𝐺:(ran 𝐺 × ran 𝐺)–onto→ran 𝐺 → 𝐺:(ran 𝐺 × ran 𝐺)⟶ran 𝐺)
4	3	fdmd 6516	. . . 4 ⊢ (𝐺:(ran 𝐺 × ran 𝐺)–onto→ran 𝐺 → dom 𝐺 = (ran 𝐺 × ran 𝐺))
5	4	dmeqd 5767	. . 3 ⊢ (𝐺:(ran 𝐺 × ran 𝐺)–onto→ran 𝐺 → dom dom 𝐺 = dom (ran 𝐺 × ran 𝐺))
6		dmxpid 5793	. . 3 ⊢ dom (ran 𝐺 × ran 𝐺) = ran 𝐺
7	5, 6	syl6req 2871	. 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 1531   ∈ wcel 2108   × cxp 5546  dom cdm 5548  ran crn 5549  –onto→wfo 6346  GrpOpcgr 28258

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1905  ax-6 1964  ax-7 2009  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2154  ax-12 2170  ax-ext 2791  ax-sep 5194  ax-nul 5201  ax-pr 5320  ax-un 7453

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1084  df-tru 1534  df-ex 1775  df-nf 1779  df-sb 2064  df-mo 2616  df-eu 2648  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-ne 3015  df-ral 3141  df-rex 3142  df-rab 3145  df-v 3495  df-sbc 3771  df-csb 3882  df-dif 3937  df-un 3939  df-in 3941  df-ss 3950  df-nul 4290  df-if 4466  df-sn 4560  df-pr 4562  df-op 4566  df-uni 4831  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-fo 6354  df-fv 6356  df-ov 7151  df-grpo 28262

This theorem is referenced by:  hhshsslem1  29036  rngorn1  35203  divrngcl  35227  isdrngo2  35228