grpinvf1o - Metamath Proof Explorer

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Theorem grpinvf1o 19040

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 19017	. . . 4 ⊢ (𝐺 ∈ Grp → 𝑁:𝐵⟶𝐵)
5	1, 4	syl 17	. . 3 ⊢ (𝜑 → 𝑁:𝐵⟶𝐵)
6	5	ffnd 6738	. 2 ⊢ (𝜑 → 𝑁 Fn 𝐵)
7	2, 3	grpinvcnv 19037	. . . . 5 ⊢ (𝐺 ∈ Grp → ^◡𝑁 = 𝑁)
8	1, 7	syl 17	. . . 4 ⊢ (𝜑 → ^◡𝑁 = 𝑁)
9	8	fneq1d 6662	. . 3 ⊢ (𝜑 → (^◡𝑁 Fn 𝐵 ↔ 𝑁 Fn 𝐵))
10	6, 9	mpbird 257	. 2 ⊢ (𝜑 → ^◡𝑁 Fn 𝐵)
11		dff1o4 6857	. 2 ⊢ (𝑁:𝐵–1-1-onto→𝐵 ↔ (𝑁 Fn 𝐵 ∧ ^◡𝑁 Fn 𝐵))
12	6, 10, 11	sylanbrc 583	1 ⊢ (𝜑 → 𝑁:𝐵–1-1-onto→𝐵)

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2106 ^◡ccnv 5688 Fn wfn 6558 ⟶wf 6559 –1-1-onto→wf1o 6562 ‘cfv 6563 Basecbs 17245 Grpcgrp 18964 inv_gcminusg 18965

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-riota 7388 df-ov 7434 df-0g 17488 df-mgm 18666 df-sgrp 18745 df-mnd 18761 df-grp 18967 df-minusg 18968

This theorem is referenced by: invoppggim 19394 gsumsub 19981 dprdfsub 20056 psrnegcl 21992 psrlinv 21993 mdetleib2 22610 ply1divalg3 35627 lflnegl 39058

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