Theorem subgex 13382

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 13376	. . 3 ⊢ SubGrp = (𝑤 ∈ Grp ↦ {𝑠 ∈ 𝒫 (Base‘𝑤) ∣ (𝑤 ↾s 𝑠) ∈ Grp})
2		fveq2 5561	. . . . 5 ⊢ (𝑤 = 𝐺 → (Base‘𝑤) = (Base‘𝐺))
3	2	pweqd 3611	. . . 4 ⊢ (𝑤 = 𝐺 → 𝒫 (Base‘𝑤) = 𝒫 (Base‘𝐺))
4		oveq1 5932	. . . . 5 ⊢ (𝑤 = 𝐺 → (𝑤 ↾s 𝑠) = (𝐺 ↾s 𝑠))
5	4	eleq1d 2265	. . . 4 ⊢ (𝑤 = 𝐺 → ((𝑤 ↾s 𝑠) ∈ Grp ↔ (𝐺 ↾s 𝑠) ∈ Grp))
6	3, 5	rabeqbidv 2758	. . 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 12761	. . . . . 6 ⊢ Base Fn V
9		elex 2774	. . . . . 6 ⊢ (𝐺 ∈ Grp → 𝐺 ∈ V)
10		funfvex 5578	. . . . . . 7 ⊢ ((Fun Base ∧ 𝐺 ∈ dom Base) → (Base‘𝐺) ∈ V)
11	10	funfni 5361	. . . . . 6 ⊢ ((Base Fn V ∧ 𝐺 ∈ V) → (Base‘𝐺) ∈ V)
12	8, 9, 11	sylancr 414	. . . . 5 ⊢ (𝐺 ∈ Grp → (Base‘𝐺) ∈ V)
13	12	pwexd 4215	. . . 4 ⊢ (𝐺 ∈ Grp → 𝒫 (Base‘𝐺) ∈ V)
14		rabexg 4177	. . . 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 5658	. 2 ⊢ (𝐺 ∈ Grp → (SubGrp‘𝐺) = {𝑠 ∈ 𝒫 (Base‘𝐺) ∣ (𝐺 ↾s 𝑠) ∈ Grp})
17	16, 15	eqeltrd 2273	1 ⊢ (𝐺 ∈ Grp → (SubGrp‘𝐺) ∈ V)

Syntax hints:   → wi 4   = wceq 1364   ∈ wcel 2167  {crab 2479  Vcvv 2763  𝒫 cpw 3606   Fn wfn 5254  ‘cfv 5259  (class class class)co 5925  Basecbs 12703   ↾_s cress 12704  Grpcgrp 13202  SubGrpcsubg 13373

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 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-sep 4152  ax-pow 4208  ax-pr 4243  ax-un 4469  ax-cnex 7987  ax-resscn 7988  ax-1re 7990  ax-addrcl 7993

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480  df-rex 2481  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-un 3161  df-in 3163  df-ss 3170  df-pw 3608  df-sn 3629  df-pr 3630  df-op 3632  df-uni 3841  df-int 3876  df-br 4035  df-opab 4096  df-mpt 4097  df-id 4329  df-xp 4670  df-rel 4671  df-cnv 4672  df-co 4673  df-dm 4674  df-rn 4675  df-res 4676  df-iota 5220  df-fun 5261  df-fn 5262  df-fv 5267  df-ov 5928  df-inn 9008  df-ndx 12706  df-slot 12707  df-base 12709  df-subg 13376

This theorem is referenced by:  isnsg  13408