grpinvf1o - Intuitionistic Logic Explorer

	Intuitionistic Logic Explorer		< Previous Next > Nearby theorems
Mirrors > Home > ILE Home > Th. List > grpinvf1o		GIF version

Theorem grpinvf1o 13772

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 13749	. . . 4 ⊢ (𝐺 ∈ Grp → 𝑁:𝐵⟶𝐵)
5	1, 4	syl 14	. . 3 ⊢ (𝜑 → 𝑁:𝐵⟶𝐵)
6	5	ffnd 5508	. 2 ⊢ (𝜑 → 𝑁 Fn 𝐵)
7	2, 3	grpinvcnv 13770	. . . . 5 ⊢ (𝐺 ∈ Grp → ^◡𝑁 = 𝑁)
8	1, 7	syl 14	. . . 4 ⊢ (𝜑 → ^◡𝑁 = 𝑁)
9	8	fneq1d 5445	. . 3 ⊢ (𝜑 → (^◡𝑁 Fn 𝐵 ↔ 𝑁 Fn 𝐵))
10	6, 9	mpbird 167	. 2 ⊢ (𝜑 → ^◡𝑁 Fn 𝐵)
11		dff1o4 5621	. 2 ⊢ (𝑁:𝐵–1-1-onto→𝐵 ↔ (𝑁 Fn 𝐵 ∧ ^◡𝑁 Fn 𝐵))
12	6, 10, 11	sylanbrc 417	1 ⊢ (𝜑 → 𝑁:𝐵–1-1-onto→𝐵)

Colors of variables: wff set class

Syntax hints: → wi 4 = wceq 1398 ∈ wcel 2203 ^◡ccnv 4747 Fn wfn 5346 ⟶wf 5347 –1-1-onto→wf1o 5350 ‘cfv 5351 Basecbs 13201 Grpcgrp 13702 inv_gcminusg 13703

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 717 ax-5 1496 ax-7 1497 ax-gen 1498 ax-ie1 1542 ax-ie2 1543 ax-8 1553 ax-10 1554 ax-11 1555 ax-i12 1556 ax-bndl 1558 ax-4 1559 ax-17 1575 ax-i9 1579 ax-ial 1583 ax-i5r 1584 ax-13 2205 ax-14 2206 ax-ext 2214 ax-coll 4224 ax-sep 4227 ax-pow 4286 ax-pr 4321 ax-un 4553 ax-cnex 8214 ax-resscn 8215 ax-1re 8217 ax-addrcl 8220

This theorem depends on definitions: df-bi 117 df-3an 1007 df-tru 1401 df-nf 1510 df-sb 1812 df-eu 2083 df-mo 2084 df-clab 2219 df-cleq 2225 df-clel 2228 df-nfc 2373 df-ral 2525 df-rex 2526 df-reu 2527 df-rmo 2528 df-rab 2529 df-v 2814 df-sbc 3042 df-csb 3138 df-un 3214 df-in 3216 df-ss 3223 df-pw 3670 df-sn 3694 df-pr 3695 df-op 3697 df-uni 3914 df-int 3949 df-iun 3992 df-br 4109 df-opab 4171 df-mpt 4172 df-id 4413 df-xp 4754 df-rel 4755 df-cnv 4756 df-co 4757 df-dm 4758 df-rn 4759 df-res 4760 df-ima 4761 df-iota 5311 df-fun 5353 df-fn 5354 df-f 5355 df-f1 5356 df-fo 5357 df-f1o 5358 df-fv 5359 df-riota 6002 df-ov 6052 df-inn 9234 df-2 9292 df-ndx 13204 df-slot 13205 df-base 13207 df-plusg 13292 df-0g 13460 df-mgm 13558 df-sgrp 13604 df-mnd 13619 df-grp 13705 df-minusg 13706

This theorem is referenced by: psrnegcl 14825 psrlinv 14826

W3C validator