Theorem gimfn 19140

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 19138	. 2 ⊢ GrpIso = (𝑠 ∈ Grp, 𝑡 ∈ Grp ↦ {𝑔 ∈ (𝑠 GrpHom 𝑡) ∣ 𝑔:(Base‘𝑠)–1-1-onto→(Base‘𝑡)})
2		ovex 7382	. . 3 ⊢ (𝑠 GrpHom 𝑡) ∈ V
3	2	rabex 5278	. 2 ⊢ {𝑔 ∈ (𝑠 GrpHom 𝑡) ∣ 𝑔:(Base‘𝑠)–1-1-onto→(Base‘𝑡)} ∈ V
4	1, 3	fnmpoi 8005	1 ⊢ GrpIso Fn (Grp × Grp)

Syntax hints:  {crab 3394   × cxp 5617   Fn wfn 6477  –1-1-onto→wf1o 6481  ‘cfv 6482  (class class class)co 7349  Basecbs 17120  Grpcgrp 18812   GrpHom cghm 19091   GrpIso cgim 19136

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 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5235  ax-nul 5245  ax-pr 5371  ax-un 7671

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rab 3395  df-v 3438  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4285  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4859  df-iun 4943  df-br 5093  df-opab 5155  df-mpt 5174  df-id 5514  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-iota 6438  df-fun 6484  df-fn 6485  df-f 6486  df-fv 6490  df-ov 7352  df-oprab 7353  df-mpo 7354  df-1st 7924  df-2nd 7925  df-gim 19138

This theorem is referenced by:  brgic  19149  gicer  19156