Theorem subgex 13041

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 13035	. . 3 ⊢ SubGrp = (𝑤 ∈ Grp ↦ {𝑠 ∈ 𝒫 (Base‘𝑤) ∣ (𝑤 ↾s 𝑠) ∈ Grp})
2		fveq2 5517	. . . . 5 ⊢ (𝑤 = 𝐺 → (Base‘𝑤) = (Base‘𝐺))
3	2	pweqd 3582	. . . 4 ⊢ (𝑤 = 𝐺 → 𝒫 (Base‘𝑤) = 𝒫 (Base‘𝐺))
4		oveq1 5884	. . . . 5 ⊢ (𝑤 = 𝐺 → (𝑤 ↾s 𝑠) = (𝐺 ↾s 𝑠))
5	4	eleq1d 2246	. . . 4 ⊢ (𝑤 = 𝐺 → ((𝑤 ↾s 𝑠) ∈ Grp ↔ (𝐺 ↾s 𝑠) ∈ Grp))
6	3, 5	rabeqbidv 2734	. . 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 12522	. . . . . 6 ⊢ Base Fn V
9		elex 2750	. . . . . 6 ⊢ (𝐺 ∈ Grp → 𝐺 ∈ V)
10		funfvex 5534	. . . . . . 7 ⊢ ((Fun Base ∧ 𝐺 ∈ dom Base) → (Base‘𝐺) ∈ V)
11	10	funfni 5318	. . . . . 6 ⊢ ((Base Fn V ∧ 𝐺 ∈ V) → (Base‘𝐺) ∈ V)
12	8, 9, 11	sylancr 414	. . . . 5 ⊢ (𝐺 ∈ Grp → (Base‘𝐺) ∈ V)
13	12	pwexd 4183	. . . 4 ⊢ (𝐺 ∈ Grp → 𝒫 (Base‘𝐺) ∈ V)
14		rabexg 4148	. . . 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 5611	. 2 ⊢ (𝐺 ∈ Grp → (SubGrp‘𝐺) = {𝑠 ∈ 𝒫 (Base‘𝐺) ∣ (𝐺 ↾s 𝑠) ∈ Grp})
17	16, 15	eqeltrd 2254	1 ⊢ (𝐺 ∈ Grp → (SubGrp‘𝐺) ∈ V)

Syntax hints:   → wi 4   = wceq 1353   ∈ wcel 2148  {crab 2459  Vcvv 2739  𝒫 cpw 3577   Fn wfn 5213  ‘cfv 5218  (class class class)co 5877  Basecbs 12464   ↾_s cress 12465  Grpcgrp 12882  SubGrpcsubg 13032

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4123  ax-pow 4176  ax-pr 4211  ax-un 4435  ax-cnex 7904  ax-resscn 7905  ax-1re 7907  ax-addrcl 7910

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-rab 2464  df-v 2741  df-sbc 2965  df-csb 3060  df-un 3135  df-in 3137  df-ss 3144  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-int 3847  df-br 4006  df-opab 4067  df-mpt 4068  df-id 4295  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-rn 4639  df-res 4640  df-iota 5180  df-fun 5220  df-fn 5221  df-fv 5226  df-ov 5880  df-inn 8922  df-ndx 12467  df-slot 12468  df-base 12470  df-subg 13035

This theorem is referenced by:  isnsg  13067