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Theorem gimfn 18397
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 18395 . 2 GrpIso = (𝑠 ∈ Grp, 𝑡 ∈ Grp ↦ {𝑔 ∈ (𝑠 GrpHom 𝑡) ∣ 𝑔:(Base‘𝑠)–1-1-onto→(Base‘𝑡)})
2 ovex 7178 . . 3 (𝑠 GrpHom 𝑡) ∈ V
32rabex 5221 . 2 {𝑔 ∈ (𝑠 GrpHom 𝑡) ∣ 𝑔:(Base‘𝑠)–1-1-onto→(Base‘𝑡)} ∈ V
41, 3fnmpoi 7758 1 GrpIso Fn (Grp × Grp)
Colors of variables: wff setvar class
Syntax hints:  {crab 3137   × cxp 5540   Fn wfn 6338  1-1-ontowf1o 6342  cfv 6343  (class class class)co 7145  Basecbs 16479  Grpcgrp 18099   GrpHom cghm 18351   GrpIso cgim 18393
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5189  ax-nul 5196  ax-pow 5253  ax-pr 5317  ax-un 7451
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-ral 3138  df-rex 3139  df-rab 3142  df-v 3482  df-sbc 3759  df-csb 3867  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4276  df-if 4450  df-sn 4550  df-pr 4552  df-op 4556  df-uni 4825  df-iun 4907  df-br 5053  df-opab 5115  df-mpt 5133  df-id 5447  df-xp 5548  df-rel 5549  df-cnv 5550  df-co 5551  df-dm 5552  df-rn 5553  df-res 5554  df-ima 5555  df-iota 6302  df-fun 6345  df-fn 6346  df-f 6347  df-fv 6351  df-ov 7148  df-oprab 7149  df-mpo 7150  df-1st 7679  df-2nd 7680  df-gim 18395
This theorem is referenced by:  brgic  18405  gicer  18412
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