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Theorem dprddomcld 18223

Description: If a family of subgroups is a family of subgroups for an internal direct product, then it is indexed by a set. (Contributed by AV, 13-Jul-2019.)

**Hypotheses**
Ref	Expression
dprddomcld.1	⊢ (𝜑 → 𝐺dom DProd 𝑆)
dprddomcld.2	⊢ (𝜑 → dom 𝑆 = 𝐼)

**Assertion**
Ref	Expression
dprddomcld	⊢ (𝜑 → 𝐼 ∈ V)

**Proof of Theorem dprddomcld**
Step	Hyp	Ref	Expression
1		dprddomcld.2	. 2 ⊢ (𝜑 → dom 𝑆 = 𝐼)
2		dprddomcld.1	. 2 ⊢ (𝜑 → 𝐺dom DProd 𝑆)
3		df-nel 2783	. . . . 5 ⊢ (dom 𝑆 ∉ V ↔ ¬ dom 𝑆 ∈ V)
4		dprddomprc 18222	. . . . 5 ⊢ (dom 𝑆 ∉ V → ¬ 𝐺dom DProd 𝑆)
5	3, 4	sylbir 224	. . . 4 ⊢ (¬ dom 𝑆 ∈ V → ¬ 𝐺dom DProd 𝑆)
6	5	con4i 112	. . 3 ⊢ (𝐺dom DProd 𝑆 → dom 𝑆 ∈ V)
7		eleq1 2676	. . 3 ⊢ (dom 𝑆 = 𝐼 → (dom 𝑆 ∈ V ↔ 𝐼 ∈ V))
8	6, 7	syl5ib 233	. 2 ⊢ (dom 𝑆 = 𝐼 → (𝐺dom DProd 𝑆 → 𝐼 ∈ V))
9	1, 2, 8	sylc 63	1 ⊢ (𝜑 → 𝐼 ∈ V)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 = wceq 1475 ∈ wcel 1977 ∉ wnel 2781 Vcvv 3173 class class class wbr 4583 dom cdm 5038 DProd cdprd 18215

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pr 4833 ax-un 6847

This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-nel 2783 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-xp 5044 df-rel 5045 df-cnv 5046 df-dm 5048 df-rn 5049 df-oprab 6553 df-mpt2 6554 df-dprd 18217

This theorem is referenced by: dprdcntz 18230 dprddisj 18231 dprdw 18232 dprdwd 18233 dprdfid 18239 dprdfinv 18241 dprdfadd 18242 dprdfsub 18243 dprdfeq0 18244 dprdf11 18245 dprdlub 18248 dprdres 18250 dprdss 18251 dprdf1o 18254 dmdprdsplitlem 18259 dprddisj2 18261 dmdprdsplit2 18268 dpjfval 18277 dpjidcl 18280

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