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Theorem gimfn 18858
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 18856 . 2 GrpIso = (𝑠 ∈ Grp, 𝑡 ∈ Grp ↦ {𝑔 ∈ (𝑠 GrpHom 𝑡) ∣ 𝑔:(Base‘𝑠)–1-1-onto→(Base‘𝑡)})
2 ovex 7301 . . 3 (𝑠 GrpHom 𝑡) ∈ V
32rabex 5259 . 2 {𝑔 ∈ (𝑠 GrpHom 𝑡) ∣ 𝑔:(Base‘𝑠)–1-1-onto→(Base‘𝑡)} ∈ V
41, 3fnmpoi 7896 1 GrpIso Fn (Grp × Grp)
Colors of variables: wff setvar class
Syntax hints:  {crab 3069   × cxp 5586   Fn wfn 6425  1-1-ontowf1o 6429  cfv 6430  (class class class)co 7268  Basecbs 16893  Grpcgrp 18558   GrpHom cghm 18812   GrpIso cgim 18854
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1801  ax-4 1815  ax-5 1916  ax-6 1974  ax-7 2014  ax-8 2111  ax-9 2119  ax-10 2140  ax-11 2157  ax-12 2174  ax-ext 2710  ax-sep 5226  ax-nul 5233  ax-pr 5355  ax-un 7579
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1544  df-fal 1554  df-ex 1786  df-nf 1790  df-sb 2071  df-mo 2541  df-eu 2570  df-clab 2717  df-cleq 2731  df-clel 2817  df-nfc 2890  df-ne 2945  df-ral 3070  df-rex 3071  df-rab 3074  df-v 3432  df-sbc 3720  df-csb 3837  df-dif 3894  df-un 3896  df-in 3898  df-ss 3908  df-nul 4262  df-if 4465  df-sn 4567  df-pr 4569  df-op 4573  df-uni 4845  df-iun 4931  df-br 5079  df-opab 5141  df-mpt 5162  df-id 5488  df-xp 5594  df-rel 5595  df-cnv 5596  df-co 5597  df-dm 5598  df-rn 5599  df-res 5600  df-ima 5601  df-iota 6388  df-fun 6432  df-fn 6433  df-f 6434  df-fv 6438  df-ov 7271  df-oprab 7272  df-mpo 7273  df-1st 7817  df-2nd 7818  df-gim 18856
This theorem is referenced by:  brgic  18866  gicer  18873
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