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Theorem grporn 27715

Description: The range of a group operation. Useful for satisfying group base set hypotheses of the form 𝑋 = ran 𝐺. (Contributed by NM, 5-Nov-2006.) (New usage is discouraged.)

**Hypotheses**
Ref	Expression
grprn.1	⊢ 𝐺 ∈ GrpOp
grprn.2	⊢ dom 𝐺 = (𝑋 × 𝑋)

**Assertion**
Ref	Expression
grporn	⊢ 𝑋 = ran 𝐺

**Proof of Theorem grporn**
Step	Hyp	Ref	Expression
1		grprn.1	. . . 4 ⊢ 𝐺 ∈ GrpOp
2		eqid 2771	. . . . 5 ⊢ ran 𝐺 = ran 𝐺
3	2	grpofo 27693	. . . 4 ⊢ (𝐺 ∈ GrpOp → 𝐺:(ran 𝐺 × ran 𝐺)–onto→ran 𝐺)
4		fofun 6258	. . . 4 ⊢ (𝐺:(ran 𝐺 × ran 𝐺)–onto→ran 𝐺 → Fun 𝐺)
5	1, 3, 4	mp2b 10	. . 3 ⊢ Fun 𝐺
6		grprn.2	. . 3 ⊢ dom 𝐺 = (𝑋 × 𝑋)
7		df-fn 6033	. . 3 ⊢ (𝐺 Fn (𝑋 × 𝑋) ↔ (Fun 𝐺 ∧ dom 𝐺 = (𝑋 × 𝑋)))
8	5, 6, 7	mpbir2an 690	. 2 ⊢ 𝐺 Fn (𝑋 × 𝑋)
9		fofn 6259	. . 3 ⊢ (𝐺:(ran 𝐺 × ran 𝐺)–onto→ran 𝐺 → 𝐺 Fn (ran 𝐺 × ran 𝐺))
10	1, 3, 9	mp2b 10	. 2 ⊢ 𝐺 Fn (ran 𝐺 × ran 𝐺)
11		fndmu 6131	. . 3 ⊢ ((𝐺 Fn (𝑋 × 𝑋) ∧ 𝐺 Fn (ran 𝐺 × ran 𝐺)) → (𝑋 × 𝑋) = (ran 𝐺 × ran 𝐺))
12		xpid11 5484	. . 3 ⊢ ((𝑋 × 𝑋) = (ran 𝐺 × ran 𝐺) ↔ 𝑋 = ran 𝐺)
13	11, 12	sylib 208	. 2 ⊢ ((𝐺 Fn (𝑋 × 𝑋) ∧ 𝐺 Fn (ran 𝐺 × ran 𝐺)) → 𝑋 = ran 𝐺)
14	8, 10, 13	mp2an 672	1 ⊢ 𝑋 = ran 𝐺

Colors of variables: wff setvar class

Syntax hints: ∧ wa 382 = wceq 1631 ∈ wcel 2145 × cxp 5248 dom cdm 5250 ran crn 5251 Fun wfun 6024 Fn wfn 6025 –onto→wfo 6028 GrpOpcgr 27683

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1870 ax-4 1885 ax-5 1991 ax-6 2057 ax-7 2093 ax-8 2147 ax-9 2154 ax-10 2174 ax-11 2190 ax-12 2203 ax-13 2408 ax-ext 2751 ax-sep 4916 ax-nul 4924 ax-pr 5035 ax-un 7100

This theorem depends on definitions: df-bi 197 df-an 383 df-or 837 df-3an 1073 df-tru 1634 df-ex 1853 df-nf 1858 df-sb 2050 df-eu 2622 df-mo 2623 df-clab 2758 df-cleq 2764 df-clel 2767 df-nfc 2902 df-ne 2944 df-ral 3066 df-rex 3067 df-rab 3070 df-v 3353 df-sbc 3588 df-csb 3683 df-dif 3726 df-un 3728 df-in 3730 df-ss 3737 df-nul 4064 df-if 4227 df-sn 4318 df-pr 4320 df-op 4324 df-uni 4576 df-iun 4657 df-br 4788 df-opab 4848 df-mpt 4865 df-id 5158 df-xp 5256 df-rel 5257 df-cnv 5258 df-co 5259 df-dm 5260 df-rn 5261 df-iota 5993 df-fun 6032 df-fn 6033 df-f 6034 df-fo 6036 df-fv 6038 df-ov 6799 df-grpo 27687

This theorem is referenced by: isabloi 27745 isvciOLD 27775 cnidOLD 27777 cnnv 27872 cnnvba 27874 cncph 28014 hilid 28358 hhnv 28362 hhba 28364 hhph 28375 hhssnv 28461

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