Theorem gimfn 19501

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.

Step	Hyp	Ref	Expression
1		df-gim 19499	. 2 ⊢ GrpIso = (𝑠 ∈ Grp, 𝑡 ∈ Grp ↦ {𝑔 ∈ (𝑠 GrpHom 𝑡) ∣ 𝑔:(Base‘𝑠)⟶1-1onto(Base‘𝑡)})
2		ovex 7473	. . 3 ⊢ (𝑠 GrpHom 𝑡) ∈ V
3	2	rabex 5304	. 2 ⊢ {𝑔 ∈ (𝑠 GrpHom 𝑡) ∣ 𝑔:(Base‘𝑠)⟶1-1onto(Base‘𝑡)} ∈ V
4	1, 3	fnmpoi 8103	1 ⊢ GrpIso Fn (Grp × Grp)

This proof depends on syntax axioms:  {crab 3416   × cxp 5653   Fn wfn 6550  ⟶1-1ontowf1o 6554  ‘cfv 6555  (class class class)co 7440  Basecbs 17413  Grpcgrp 19170   GrpHom cghm 19453   GrpIso cgim 19497

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-sep 5252  ax-nul 5263  ax-pr 5395  ax-un 7772

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-iun 4956  df-br 5107  df-opab 5171  df-mpt 5190  df-id 5550  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-iota 6509  df-fun 6557  df-fn 6558  df-f 6559  df-fv 6563  df-ov 7443  df-oprab 7444  df-mpo 7445  df-1st 8022  df-2nd 8023  df-gim 19499

This theorem is used by:  brgic  19511  gicer  19518