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Theorem grporn 30501

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 2729	. . . . 5 ⊢ ran 𝐺 = ran 𝐺
3	2	grpofo 30479	. . . 4 ⊢ (𝐺 ∈ GrpOp → 𝐺:(ran 𝐺 × ran 𝐺)–onto→ran 𝐺)
4		fofun 6755	. . . 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 6502	. . 3 ⊢ (𝐺 Fn (𝑋 × 𝑋) ↔ (Fun 𝐺 ∧ dom 𝐺 = (𝑋 × 𝑋)))
8	5, 6, 7	mpbir2an 711	. 2 ⊢ 𝐺 Fn (𝑋 × 𝑋)
9		fofn 6756	. . 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 6607	. . 3 ⊢ ((𝐺 Fn (𝑋 × 𝑋) ∧ 𝐺 Fn (ran 𝐺 × ran 𝐺)) → (𝑋 × 𝑋) = (ran 𝐺 × ran 𝐺))
12		xpid11 5885	. . 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 2109 × cxp 5629 dom cdm 5631 ran crn 5632 Fun wfun 6493 Fn wfn 6494 –onto→wfo 6497 GrpOpcgr 30469

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 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-sep 5246 ax-nul 5256 ax-pr 5382 ax-un 7691

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-ral 3045 df-rex 3054 df-rab 3403 df-v 3446 df-sbc 3751 df-csb 3860 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-nul 4293 df-if 4485 df-sn 4586 df-pr 4588 df-op 4592 df-uni 4868 df-iun 4953 df-br 5103 df-opab 5165 df-mpt 5184 df-id 5526 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-iota 6452 df-fun 6501 df-fn 6502 df-f 6503 df-fo 6505 df-fv 6507 df-ov 7372 df-grpo 30473

This theorem is referenced by: isabloi 30531 isvciOLD 30560 cnidOLD 30562 cnnv 30657 cnnvba 30659 cncph 30799 hilid 31141 hhnv 31145 hhba 31147 hhph 31158 hhssnv 31244

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