grpinvf1o - Metamath Proof Explorer

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Theorem grpinvf1o 19063

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 19041	. . . 4 ⊢ (𝐺 ∈ Grp → 𝑁:𝐵⟶𝐵)
5	1, 4	syl 18	. . 3 ⊢ (𝜑 → 𝑁:𝐵⟶𝐵)
6	5	ffnd 6696	. 2 ⊢ (𝜑 → 𝑁 Fn 𝐵)
7	2, 3	grpinvcnv 19061	. . . . 5 ⊢ (𝐺 ∈ Grp → ^◡𝑁 = 𝑁)
8	1, 7	syl 18	. . . 4 ⊢ (𝜑 → ^◡𝑁 = 𝑁)
9	8	fneq1d 6618	. . 3 ⊢ (𝜑 → (^◡𝑁 Fn 𝐵 ↔ 𝑁 Fn 𝐵))
10	6, 9	mpbird 260	. 2 ⊢ (𝜑 → ^◡𝑁 Fn 𝐵)
11		dff1o4 6819	. 2 ⊢ (𝑁:𝐵–1-1-onto→𝐵 ↔ (𝑁 Fn 𝐵 ∧ ^◡𝑁 Fn 𝐵))
12	6, 10, 11	sylanbrc 594	1 ⊢ (𝜑 → 𝑁:𝐵–1-1-onto→𝐵)

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1563 ∈ wcel 2145 ^◡ccnv 5650 Fn wfn 6520 ⟶wf 6521 –1-1-onto→wf1o 6524 ‘cfv 6525 Basecbs 17257 Grpcgrp 18988 inv_gcminusg 18989

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1818 ax-4 1832 ax-5 1933 ax-6 1990 ax-7 2031 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2215 ax-ext 2737 ax-sep 5250 ax-nul 5260 ax-pow 5326 ax-pr 5394 ax-un 7722

This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3an 1103 df-tru 1566 df-fal 1576 df-ex 1803 df-nf 1807 df-sb 2094 df-mo 2569 df-eu 2599 df-clab 2744 df-cleq 2757 df-clel 2840 df-nfc 2914 df-ne 2961 df-ral 3080 df-rex 3090 df-rmo 3370 df-reu 3371 df-rab 3418 df-v 3459 df-sbc 3748 df-dif 3910 df-un 3912 df-in 3914 df-ss 3924 df-nul 4289 df-if 4484 df-pw 4560 df-sn 4586 df-pr 4588 df-op 4592 df-uni 4868 df-br 5105 df-opab 5167 df-mpt 5186 df-id 5546 df-xp 5657 df-rel 5658 df-cnv 5659 df-co 5660 df-dm 5661 df-rn 5662 df-res 5663 df-ima 5664 df-iota 6481 df-fun 6527 df-fn 6528 df-f 6529 df-f1 6530 df-fo 6531 df-f1o 6532 df-fv 6533 df-riota 7357 df-ov 7403 df-0g 17482 df-mgm 18686 df-sgrp 18765 df-mnd 18781 df-grp 18991 df-minusg 18992

This theorem is referenced by: invoppggim 19418 gsumsub 20006 dprdfsub 20081 psrnegcl 22061 psrlinv 22062 mdetleib2 22702 ply1divalg3 36000 lflnegl 39707

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