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Theorem grporn 30988

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 2762	. . . . 5 ⊢ ran 𝐺 = ran 𝐺
3	2	grpofo 30966	. . . 4 ⊢ (𝐺 ∈ GrpOp → 𝐺:(ran 𝐺 × ran 𝐺)–onto→ran 𝐺)
4		fofun 6794	. . . 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 6540	. . 3 ⊢ (𝐺 Fn (𝑋 × 𝑋) ↔ (Fun 𝐺 ∧ dom 𝐺 = (𝑋 × 𝑋)))
8	5, 6, 7	mpbir2an 724	. 2 ⊢ 𝐺 Fn (𝑋 × 𝑋)
9		fofn 6795	. . 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 6643	. . 3 ⊢ ((𝐺 Fn (𝑋 × 𝑋) ∧ 𝐺 Fn (ran 𝐺 × ran 𝐺)) → (𝑋 × 𝑋) = (ran 𝐺 × ran 𝐺))
12		xpid11 5920	. . 3 ⊢ ((𝑋 × 𝑋) = (ran 𝐺 × ran 𝐺) ↔ 𝑋 = ran 𝐺)
13	11, 12	sylib 221	. 2 ⊢ ((𝐺 Fn (𝑋 × 𝑋) ∧ 𝐺 Fn (ran 𝐺 × ran 𝐺)) → 𝑋 = ran 𝐺)
14	8, 10, 13	mp2an 705	1 ⊢ 𝑋 = ran 𝐺

Colors of variables: wff setvar class

This proof depends on syntax axioms: ∧ wa 401 = wceq 1570 ∈ wcel 2145 × cxp 5657 dom cdm 5659 ran crn 5660 Fun wfun 6531 Fn wfn 6532 –onto→wfo 6535 GrpOpcgr 30956

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2215 ax-ext 2734 ax-sep 5255 ax-nul 5267 ax-pr 5402 ax-un 7739

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2566 df-eu 2596 df-clab 2741 df-cleq 2754 df-clel 2837 df-nfc 2911 df-ne 2958 df-ral 3079 df-rex 3089 df-rab 3415 df-v 3455 df-sbc 3743 df-csb 3851 df-dif 3905 df-un 3907 df-in 3909 df-ss 3919 df-nul 4283 df-if 4486 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4871 df-iun 4956 df-br 5108 df-opab 5172 df-mpt 5191 df-id 5554 df-xp 5665 df-rel 5666 df-cnv 5667 df-co 5668 df-dm 5669 df-rn 5670 df-iota 6493 df-fun 6539 df-fn 6540 df-f 6541 df-fo 6543 df-fv 6545 df-ov 7419 df-grpo 30960

This theorem is used by: isabloi 31018 isvciOLD 31047 cnidOLD 31049 cnnv 31144 cnnvba 31146 cncph 31286 hilid 31628 hhnv 31632 hhba 31634 hhph 31645 hhssnv 31731

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