MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  gimfn Structured version   Visualization version   GIF version

Theorem gimfn 17619
Description: The group isomorphism function is a well-defined function. (Contributed by Mario Carneiro, 23-Aug-2015.)
Assertion
Ref Expression
gimfn GrpIso Fn (Grp × Grp)

Proof of Theorem gimfn
Dummy variables 𝑔 𝑠 𝑡 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-gim 17617 . 2 GrpIso = (𝑠 ∈ Grp, 𝑡 ∈ Grp ↦ {𝑔 ∈ (𝑠 GrpHom 𝑡) ∣ 𝑔:(Base‘𝑠)–1-1-onto→(Base‘𝑡)})
2 ovex 6633 . . 3 (𝑠 GrpHom 𝑡) ∈ V
32rabex 4778 . 2 {𝑔 ∈ (𝑠 GrpHom 𝑡) ∣ 𝑔:(Base‘𝑠)–1-1-onto→(Base‘𝑡)} ∈ V
41, 3fnmpt2i 7185 1 GrpIso Fn (Grp × Grp)
Colors of variables: wff setvar class
Syntax hints:  {crab 2916   × cxp 5077   Fn wfn 5845  1-1-ontowf1o 5849  cfv 5850  (class class class)co 6605  Basecbs 15776  Grpcgrp 17338   GrpHom cghm 17573   GrpIso cgim 17615
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1841  ax-6 1890  ax-7 1937  ax-8 1994  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6903
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1883  df-eu 2478  df-mo 2479  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-ne 2797  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3193  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3897  df-if 4064  df-sn 4154  df-pr 4156  df-op 4160  df-uni 4408  df-iun 4492  df-br 4619  df-opab 4679  df-mpt 4680  df-id 4994  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-iota 5813  df-fun 5852  df-fn 5853  df-f 5854  df-fv 5858  df-ov 6608  df-oprab 6609  df-mpt2 6610  df-1st 7116  df-2nd 7117  df-gim 17617
This theorem is referenced by:  brgic  17627  gicer  17634  gicerOLD  17635
  Copyright terms: Public domain W3C validator