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Theorem grpinvf1o 19027
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 = (invg𝐺)
grpinv11.g (𝜑𝐺 ∈ Grp)
Assertion
Ref Expression
grpinvf1o (𝜑𝑁:𝐵1-1-onto𝐵)

Proof of Theorem grpinvf1o
StepHypRef Expression
1 grpinv11.g . . . 4 (𝜑𝐺 ∈ Grp)
2 grpinvinv.b . . . . 5 𝐵 = (Base‘𝐺)
3 grpinvinv.n . . . . 5 𝑁 = (invg𝐺)
42, 3grpinvf 19004 . . . 4 (𝐺 ∈ Grp → 𝑁:𝐵𝐵)
51, 4syl 17 . . 3 (𝜑𝑁:𝐵𝐵)
65ffnd 6737 . 2 (𝜑𝑁 Fn 𝐵)
72, 3grpinvcnv 19024 . . . . 5 (𝐺 ∈ Grp → 𝑁 = 𝑁)
81, 7syl 17 . . . 4 (𝜑𝑁 = 𝑁)
98fneq1d 6661 . . 3 (𝜑 → (𝑁 Fn 𝐵𝑁 Fn 𝐵))
106, 9mpbird 257 . 2 (𝜑𝑁 Fn 𝐵)
11 dff1o4 6856 . 2 (𝑁:𝐵1-1-onto𝐵 ↔ (𝑁 Fn 𝐵𝑁 Fn 𝐵))
126, 10, 11sylanbrc 583 1 (𝜑𝑁:𝐵1-1-onto𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1540  wcel 2108  ccnv 5684   Fn wfn 6556  wf 6557  1-1-ontowf1o 6560  cfv 6561  Basecbs 17247  Grpcgrp 18951  invgcminusg 18952
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-riota 7388  df-ov 7434  df-0g 17486  df-mgm 18653  df-sgrp 18732  df-mnd 18748  df-grp 18954  df-minusg 18955
This theorem is referenced by:  invoppggim  19379  gsumsub  19966  dprdfsub  20041  psrnegcl  21974  psrlinv  21975  mdetleib2  22594  ply1divalg3  35647  lflnegl  39077
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