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Theorem grporn 30540

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 2737	. . . . 5 ⊢ ran 𝐺 = ran 𝐺
3	2	grpofo 30518	. . . 4 ⊢ (𝐺 ∈ GrpOp → 𝐺:(ran 𝐺 × ran 𝐺)–onto→ran 𝐺)
4		fofun 6821	. . . 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 6564	. . 3 ⊢ (𝐺 Fn (𝑋 × 𝑋) ↔ (Fun 𝐺 ∧ dom 𝐺 = (𝑋 × 𝑋)))
8	5, 6, 7	mpbir2an 711	. 2 ⊢ 𝐺 Fn (𝑋 × 𝑋)
9		fofn 6822	. . 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 6675	. . 3 ⊢ ((𝐺 Fn (𝑋 × 𝑋) ∧ 𝐺 Fn (ran 𝐺 × ran 𝐺)) → (𝑋 × 𝑋) = (ran 𝐺 × ran 𝐺))
12		xpid11 5943	. . 3 ⊢ ((𝑋 × 𝑋) = (ran 𝐺 × ran 𝐺) ↔ 𝑋 = ran 𝐺)
13	11, 12	sylib 218	. 2 ⊢ ((𝐺 Fn (𝑋 × 𝑋) ∧ 𝐺 Fn (ran 𝐺 × ran 𝐺)) → 𝑋 = ran 𝐺)
14	8, 10, 13	mp2an 692	1 ⊢ 𝑋 = ran 𝐺

Colors of variables: wff setvar class

Syntax hints: ∧ wa 395 = wceq 1540 ∈ wcel 2108 × cxp 5683 dom cdm 5685 ran crn 5686 Fun wfun 6555 Fn wfn 6556 –onto→wfo 6559 GrpOpcgr 30508

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 ax-un 7755

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-fo 6567 df-fv 6569 df-ov 7434 df-grpo 30512

This theorem is referenced by: isabloi 30570 isvciOLD 30599 cnidOLD 30601 cnnv 30696 cnnvba 30698 cncph 30838 hilid 31180 hhnv 31184 hhba 31186 hhph 31197 hhssnv 31283

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