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Theorem dprddomcld 18400

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 2898	. . . . 5 ⊢ (dom 𝑆 ∉ V ↔ ¬ dom 𝑆 ∈ V)
4		dprddomprc 18399	. . . . 5 ⊢ (dom 𝑆 ∉ V → ¬ 𝐺dom DProd 𝑆)
5	3, 4	sylbir 225	. . . 4 ⊢ (¬ dom 𝑆 ∈ V → ¬ 𝐺dom DProd 𝑆)
6	5	con4i 113	. . 3 ⊢ (𝐺dom DProd 𝑆 → dom 𝑆 ∈ V)
7		eleq1 2689	. . 3 ⊢ (dom 𝑆 = 𝐼 → (dom 𝑆 ∈ V ↔ 𝐼 ∈ V))
8	6, 7	syl5ib 234	. 2 ⊢ (dom 𝑆 = 𝐼 → (𝐺dom DProd 𝑆 → 𝐼 ∈ V))
9	1, 2, 8	sylc 65	1 ⊢ (𝜑 → 𝐼 ∈ V)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 = wceq 1483 ∈ wcel 1990 ∉ wnel 2897 Vcvv 3200 class class class wbr 4653 dom cdm 5114 DProd cdprd 18392

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pr 4906 ax-un 6949

This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-nel 2898 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-br 4654 df-opab 4713 df-xp 5120 df-rel 5121 df-cnv 5122 df-dm 5124 df-rn 5125 df-oprab 6654 df-mpt2 6655 df-dprd 18394

This theorem is referenced by: dprdcntz 18407 dprddisj 18408 dprdw 18409 dprdwd 18410 dprdfid 18416 dprdfinv 18418 dprdfadd 18419 dprdfsub 18420 dprdfeq0 18421 dprdf11 18422 dprdlub 18425 dprdres 18427 dprdss 18428 dprdf1o 18431 dmdprdsplitlem 18436 dprddisj2 18438 dmdprdsplit2 18445 dpjfval 18454 dpjidcl 18457

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