grpinvf1o - Metamath Proof Explorer

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Theorem grpinvf1o 18560

Description: The group inverse is a one-to-one onto function. (Contributed by NM, 22-Oct-2014.) (Proof shortened by Mario Carneiro, 14-Aug-2015.)

**Hypotheses**
Ref	Expression
grpinvinv.b	⊢ 𝐵 = (Base‘𝐺)
grpinvinv.n	⊢ 𝑁 = (inv_g‘𝐺)
grpinv11.g	⊢ (𝜑 → 𝐺 ∈ Grp)

**Assertion**
Ref	Expression
grpinvf1o	⊢ (𝜑 → 𝑁:𝐵–1-1-onto→𝐵)

**Proof of Theorem grpinvf1o**
Step	Hyp	Ref	Expression
1		grpinv11.g	. . . 4 ⊢ (𝜑 → 𝐺 ∈ Grp)
2		grpinvinv.b	. . . . 5 ⊢ 𝐵 = (Base‘𝐺)
3		grpinvinv.n	. . . . 5 ⊢ 𝑁 = (inv_g‘𝐺)
4	2, 3	grpinvf 18541	. . . 4 ⊢ (𝐺 ∈ Grp → 𝑁:𝐵⟶𝐵)
5	1, 4	syl 17	. . 3 ⊢ (𝜑 → 𝑁:𝐵⟶𝐵)
6	5	ffnd 6585	. 2 ⊢ (𝜑 → 𝑁 Fn 𝐵)
7	2, 3	grpinvcnv 18558	. . . . 5 ⊢ (𝐺 ∈ Grp → ^◡𝑁 = 𝑁)
8	1, 7	syl 17	. . . 4 ⊢ (𝜑 → ^◡𝑁 = 𝑁)
9	8	fneq1d 6510	. . 3 ⊢ (𝜑 → (^◡𝑁 Fn 𝐵 ↔ 𝑁 Fn 𝐵))
10	6, 9	mpbird 256	. 2 ⊢ (𝜑 → ^◡𝑁 Fn 𝐵)
11		dff1o4 6708	. 2 ⊢ (𝑁:𝐵–1-1-onto→𝐵 ↔ (𝑁 Fn 𝐵 ∧ ^◡𝑁 Fn 𝐵))
12	6, 10, 11	sylanbrc 582	1 ⊢ (𝜑 → 𝑁:𝐵–1-1-onto→𝐵)

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1539 ∈ wcel 2108 ^◡ccnv 5579 Fn wfn 6413 ⟶wf 6414 –1-1-onto→wf1o 6417 ‘cfv 6418 Basecbs 16840 Grpcgrp 18492 inv_gcminusg 18493

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1799 ax-4 1813 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2156 ax-12 2173 ax-ext 2709 ax-sep 5218 ax-nul 5225 ax-pow 5283 ax-pr 5347 ax-un 7566

This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1784 df-nf 1788 df-sb 2069 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2817 df-nfc 2888 df-ne 2943 df-ral 3068 df-rex 3069 df-reu 3070 df-rmo 3071 df-rab 3072 df-v 3424 df-sbc 3712 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-nul 4254 df-if 4457 df-pw 4532 df-sn 4559 df-pr 4561 df-op 4565 df-uni 4837 df-br 5071 df-opab 5133 df-mpt 5154 df-id 5480 df-xp 5586 df-rel 5587 df-cnv 5588 df-co 5589 df-dm 5590 df-rn 5591 df-res 5592 df-ima 5593 df-iota 6376 df-fun 6420 df-fn 6421 df-f 6422 df-f1 6423 df-fo 6424 df-f1o 6425 df-fv 6426 df-riota 7212 df-ov 7258 df-0g 17069 df-mgm 18241 df-sgrp 18290 df-mnd 18301 df-grp 18495 df-minusg 18496

This theorem is referenced by: invoppggim 18882 gsumsub 19464 dprdfsub 19539 psrnegcl 21075 psrlinv 21076 mdetleib2 21645 lflnegl 37017

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