Theorem subgex 13962

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 13956	. . 3 ⊢ SubGrp = (𝑤 ∈ Grp ↦ {𝑠 ∈ 𝒫 (Base‘𝑤) ∣ (𝑤 ↾s 𝑠) ∈ Grp})
2		fveq2 5693	. . . . 5 ⊢ (𝑤 = 𝐺 → (Base‘𝑤) = (Base‘𝐺))
3	2	pweqd 3693	. . . 4 ⊢ (𝑤 = 𝐺 → 𝒫 (Base‘𝑤) = 𝒫 (Base‘𝐺))
4		oveq1 6086	. . . . 5 ⊢ (𝑤 = 𝐺 → (𝑤 ↾s 𝑠) = (𝐺 ↾s 𝑠))
5	4	eleq1d 2307	. . . 4 ⊢ (𝑤 = 𝐺 → ((𝑤 ↾s 𝑠) ∈ Grp ↔ (𝐺 ↾s 𝑠) ∈ Grp))
6	3, 5	rabeqbidv 2816	. . 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 13394	. . . . . 6 ⊢ Base Fn V
9		elex 2833	. . . . . 6 ⊢ (𝐺 ∈ Grp → 𝐺 ∈ V)
10		funfvex 5710	. . . . . . 7 ⊢ ((Fun Base ∧ 𝐺 ∈ dom Base) → (Base‘𝐺) ∈ V)
11	10	funfni 5481	. . . . . 6 ⊢ ((Base Fn V ∧ 𝐺 ∈ V) → (Base‘𝐺) ∈ V)
12	8, 9, 11	sylancr 418	. . . . 5 ⊢ (𝐺 ∈ Grp → (Base‘𝐺) ∈ V)
13	12	pwexd 4316	. . . 4 ⊢ (𝐺 ∈ Grp → 𝒫 (Base‘𝐺) ∈ V)
14		rabexg 4277	. . . 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 5796	. 2 ⊢ (𝐺 ∈ Grp → (SubGrp‘𝐺) = {𝑠 ∈ 𝒫 (Base‘𝐺) ∣ (𝐺 ↾s 𝑠) ∈ Grp})
17	16, 15	eqeltrd 2315	1 ⊢ (𝐺 ∈ Grp → (SubGrp‘𝐺) ∈ V)

Syntax hints:   → wi 4   = wceq 1402   ∈ wcel 2209  {crab 2532  Vcvv 2821  𝒫 cpw 3688   Fn wfn 5370  ‘cfv 5375  (class class class)co 6079  Basecbs 13335   ↾_s cress 13336  Grpcgrp 13788  SubGrpcsubg 13953

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4247  ax-pow 4309  ax-pr 4344  ax-un 4576  ax-cnex 8264  ax-resscn 8265  ax-1re 8267  ax-addrcl 8270

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-un 3224  df-in 3226  df-ss 3233  df-pw 3690  df-sn 3714  df-pr 3715  df-op 3717  df-uni 3934  df-int 3969  df-br 4129  df-opab 4191  df-mpt 4192  df-id 4436  df-xp 4778  df-rel 4779  df-cnv 4780  df-co 4781  df-dm 4782  df-rn 4783  df-res 4784  df-iota 5335  df-fun 5377  df-fn 5378  df-fv 5383  df-ov 6082  df-inn 9288  df-ndx 13338  df-slot 13339  df-base 13341  df-subg 13956

This theorem is referenced by:  isnsg  13988