Theorem subgex 13765

Description: The class of subgroups of a group is a set. (Contributed by Jim Kingdon, 8-Mar-2025.)

Assertion

Ref	Expression
subgex	⊢ (𝐺 ∈ Grp → (SubGrp‘𝐺) ∈ V)

Proof of Theorem subgex

Dummy variables 𝑠 𝑤 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-subg 13759	. . 3 ⊢ SubGrp = (𝑤 ∈ Grp ↦ {𝑠 ∈ 𝒫 (Base‘𝑤) ∣ (𝑤 ↾s 𝑠) ∈ Grp})
2		fveq2 5639	. . . . 5 ⊢ (𝑤 = 𝐺 → (Base‘𝑤) = (Base‘𝐺))
3	2	pweqd 3657	. . . 4 ⊢ (𝑤 = 𝐺 → 𝒫 (Base‘𝑤) = 𝒫 (Base‘𝐺))
4		oveq1 6025	. . . . 5 ⊢ (𝑤 = 𝐺 → (𝑤 ↾s 𝑠) = (𝐺 ↾s 𝑠))
5	4	eleq1d 2300	. . . 4 ⊢ (𝑤 = 𝐺 → ((𝑤 ↾s 𝑠) ∈ Grp ↔ (𝐺 ↾s 𝑠) ∈ Grp))
6	3, 5	rabeqbidv 2797	. . 3 ⊢ (𝑤 = 𝐺 → {𝑠 ∈ 𝒫 (Base‘𝑤) ∣ (𝑤 ↾s 𝑠) ∈ Grp} = {𝑠 ∈ 𝒫 (Base‘𝐺) ∣ (𝐺 ↾s 𝑠) ∈ Grp})
7		id 19	. . 3 ⊢ (𝐺 ∈ Grp → 𝐺 ∈ Grp)
8		basfn 13143	. . . . . 6 ⊢ Base Fn V
9		elex 2814	. . . . . 6 ⊢ (𝐺 ∈ Grp → 𝐺 ∈ V)
10		funfvex 5656	. . . . . . 7 ⊢ ((Fun Base ∧ 𝐺 ∈ dom Base) → (Base‘𝐺) ∈ V)
11	10	funfni 5432	. . . . . 6 ⊢ ((Base Fn V ∧ 𝐺 ∈ V) → (Base‘𝐺) ∈ V)
12	8, 9, 11	sylancr 414	. . . . 5 ⊢ (𝐺 ∈ Grp → (Base‘𝐺) ∈ V)
13	12	pwexd 4271	. . . 4 ⊢ (𝐺 ∈ Grp → 𝒫 (Base‘𝐺) ∈ V)
14		rabexg 4233	. . . 4 ⊢ (𝒫 (Base‘𝐺) ∈ V → {𝑠 ∈ 𝒫 (Base‘𝐺) ∣ (𝐺 ↾s 𝑠) ∈ Grp} ∈ V)
15	13, 14	syl 14	. . 3 ⊢ (𝐺 ∈ Grp → {𝑠 ∈ 𝒫 (Base‘𝐺) ∣ (𝐺 ↾s 𝑠) ∈ Grp} ∈ V)
16	1, 6, 7, 15	fvmptd3 5740	. 2 ⊢ (𝐺 ∈ Grp → (SubGrp‘𝐺) = {𝑠 ∈ 𝒫 (Base‘𝐺) ∣ (𝐺 ↾s 𝑠) ∈ Grp})
17	16, 15	eqeltrd 2308	1 ⊢ (𝐺 ∈ Grp → (SubGrp‘𝐺) ∈ V)

Syntax hints:   → wi 4   = wceq 1397   ∈ wcel 2202  {crab 2514  Vcvv 2802  𝒫 cpw 3652   Fn wfn 5321  ‘cfv 5326  (class class class)co 6018  Basecbs 13084   ↾_s cress 13085  Grpcgrp 13585  SubGrpcsubg 13756

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-cnex 8123  ax-resscn 8124  ax-1re 8126  ax-addrcl 8129

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ral 2515  df-rex 2516  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-int 3929  df-br 4089  df-opab 4151  df-mpt 4152  df-id 4390  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-iota 5286  df-fun 5328  df-fn 5329  df-fv 5334  df-ov 6021  df-inn 9144  df-ndx 13087  df-slot 13088  df-base 13090  df-subg 13759

This theorem is referenced by:  isnsg  13791