Theorem subgex 13882

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 13876	. . 3 ⊢ SubGrp = (𝑤 ∈ Grp ↦ {𝑠 ∈ 𝒫 (Base‘𝑤) ∣ (𝑤 ↾s 𝑠) ∈ Grp})
2		fveq2 5669	. . . . 5 ⊢ (𝑤 = 𝐺 → (Base‘𝑤) = (Base‘𝐺))
3	2	pweqd 3673	. . . 4 ⊢ (𝑤 = 𝐺 → 𝒫 (Base‘𝑤) = 𝒫 (Base‘𝐺))
4		oveq1 6056	. . . . 5 ⊢ (𝑤 = 𝐺 → (𝑤 ↾s 𝑠) = (𝐺 ↾s 𝑠))
5	4	eleq1d 2301	. . . 4 ⊢ (𝑤 = 𝐺 → ((𝑤 ↾s 𝑠) ∈ Grp ↔ (𝐺 ↾s 𝑠) ∈ Grp))
6	3, 5	rabeqbidv 2807	. . 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 13260	. . . . . 6 ⊢ Base Fn V
9		elex 2824	. . . . . 6 ⊢ (𝐺 ∈ Grp → 𝐺 ∈ V)
10		funfvex 5686	. . . . . . 7 ⊢ ((Fun Base ∧ 𝐺 ∈ dom Base) → (Base‘𝐺) ∈ V)
11	10	funfni 5457	. . . . . 6 ⊢ ((Base Fn V ∧ 𝐺 ∈ V) → (Base‘𝐺) ∈ V)
12	8, 9, 11	sylancr 414	. . . . 5 ⊢ (𝐺 ∈ Grp → (Base‘𝐺) ∈ V)
13	12	pwexd 4293	. . . 4 ⊢ (𝐺 ∈ Grp → 𝒫 (Base‘𝐺) ∈ V)
14		rabexg 4254	. . . 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 5770	. 2 ⊢ (𝐺 ∈ Grp → (SubGrp‘𝐺) = {𝑠 ∈ 𝒫 (Base‘𝐺) ∣ (𝐺 ↾s 𝑠) ∈ Grp})
17	16, 15	eqeltrd 2309	1 ⊢ (𝐺 ∈ Grp → (SubGrp‘𝐺) ∈ V)

Syntax hints:   → wi 4   = wceq 1398   ∈ wcel 2203  {crab 2524  Vcvv 2812  𝒫 cpw 3668   Fn wfn 5346  ‘cfv 5351  (class class class)co 6049  Basecbs 13201   ↾_s cress 13202  Grpcgrp 13702  SubGrpcsubg 13873

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2205  ax-14 2206  ax-ext 2214  ax-sep 4227  ax-pow 4286  ax-pr 4321  ax-un 4553  ax-cnex 8214  ax-resscn 8215  ax-1re 8217  ax-addrcl 8220

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ral 2525  df-rex 2526  df-rab 2529  df-v 2814  df-sbc 3042  df-csb 3138  df-un 3214  df-in 3216  df-ss 3223  df-pw 3670  df-sn 3694  df-pr 3695  df-op 3697  df-uni 3914  df-int 3949  df-br 4109  df-opab 4171  df-mpt 4172  df-id 4413  df-xp 4754  df-rel 4755  df-cnv 4756  df-co 4757  df-dm 4758  df-rn 4759  df-res 4760  df-iota 5311  df-fun 5353  df-fn 5354  df-fv 5359  df-ov 6052  df-inn 9234  df-ndx 13204  df-slot 13205  df-base 13207  df-subg 13876

This theorem is referenced by:  isnsg  13908