grporn - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > grporn		Structured version Visualization version GIF version

Theorem grporn 29240

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 29218	. . . 4 ⊢ (𝐺 ∈ GrpOp → 𝐺:(ran 𝐺 × ran 𝐺)–onto→ran 𝐺)
4		fofun 6752	. . . 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 6494	. . 3 ⊢ (𝐺 Fn (𝑋 × 𝑋) ↔ (Fun 𝐺 ∧ dom 𝐺 = (𝑋 × 𝑋)))
8	5, 6, 7	mpbir2an 709	. 2 ⊢ 𝐺 Fn (𝑋 × 𝑋)
9		fofn 6753	. . 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 6604	. . 3 ⊢ ((𝐺 Fn (𝑋 × 𝑋) ∧ 𝐺 Fn (ran 𝐺 × ran 𝐺)) → (𝑋 × 𝑋) = (ran 𝐺 × ran 𝐺))
12		xpid11 5883	. . 3 ⊢ ((𝑋 × 𝑋) = (ran 𝐺 × ran 𝐺) ↔ 𝑋 = ran 𝐺)
13	11, 12	sylib 217	. 2 ⊢ ((𝐺 Fn (𝑋 × 𝑋) ∧ 𝐺 Fn (ran 𝐺 × ran 𝐺)) → 𝑋 = ran 𝐺)
14	8, 10, 13	mp2an 690	1 ⊢ 𝑋 = ran 𝐺

Colors of variables: wff setvar class

Syntax hints: ∧ wa 397 = wceq 1541 ∈ wcel 2106 × cxp 5628 dom cdm 5630 ran crn 5631 Fun wfun 6485 Fn wfn 6486 –onto→wfo 6489 GrpOpcgr 29208

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2708 ax-sep 5254 ax-nul 5261 ax-pr 5382 ax-un 7662

This theorem depends on definitions: df-bi 206 df-an 398 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2887 df-ne 2942 df-ral 3063 df-rex 3072 df-rab 3406 df-v 3445 df-sbc 3738 df-csb 3854 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-nul 4281 df-if 4485 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4864 df-iun 4954 df-br 5104 df-opab 5166 df-mpt 5187 df-id 5528 df-xp 5636 df-rel 5637 df-cnv 5638 df-co 5639 df-dm 5640 df-rn 5641 df-iota 6443 df-fun 6493 df-fn 6494 df-f 6495 df-fo 6497 df-fv 6499 df-ov 7352 df-grpo 29212

This theorem is referenced by: isabloi 29270 isvciOLD 29299 cnidOLD 29301 cnnv 29396 cnnvba 29398 cncph 29538 hilid 29880 hhnv 29884 hhba 29886 hhph 29897 hhssnv 29983

W3C validator