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Theorem dprddomcld 19242

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 3039	. . . . 5 ⊢ (dom 𝑆 ∉ V ↔ ¬ dom 𝑆 ∈ V)
4		dprddomprc 19241	. . . . 5 ⊢ (dom 𝑆 ∉ V → ¬ 𝐺dom DProd 𝑆)
5	3, 4	sylbir 238	. . . 4 ⊢ (¬ dom 𝑆 ∈ V → ¬ 𝐺dom DProd 𝑆)
6	5	con4i 114	. . 3 ⊢ (𝐺dom DProd 𝑆 → dom 𝑆 ∈ V)
7		eleq1 2820	. . 3 ⊢ (dom 𝑆 = 𝐼 → (dom 𝑆 ∈ V ↔ 𝐼 ∈ V))
8	6, 7	syl5ib 247	. 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 1542 ∈ wcel 2114 ∉ wnel 3038 Vcvv 3398 class class class wbr 5030 dom cdm 5525 DProd cdprd 19234

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-ext 2710 ax-sep 5167 ax-nul 5174 ax-pr 5296 ax-un 7479

This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2075 df-mo 2540 df-eu 2570 df-clab 2717 df-cleq 2730 df-clel 2811 df-nfc 2881 df-nel 3039 df-ral 3058 df-rex 3059 df-rab 3062 df-v 3400 df-dif 3846 df-un 3848 df-in 3850 df-ss 3860 df-nul 4212 df-if 4415 df-sn 4517 df-pr 4519 df-op 4523 df-uni 4797 df-br 5031 df-opab 5093 df-xp 5531 df-rel 5532 df-cnv 5533 df-dm 5535 df-rn 5536 df-oprab 7174 df-mpo 7175 df-dprd 19236

This theorem is referenced by: dprdcntz 19249 dprddisj 19250 dprdw 19251 dprdwd 19252 dprdfid 19258 dprdfinv 19260 dprdfadd 19261 dprdfsub 19262 dprdfeq0 19263 dprdf11 19264 dprdlub 19267 dprdres 19269 dprdss 19270 dprdf1o 19273 dmdprdsplitlem 19278 dprddisj2 19280 dmdprdsplit2 19287 dpjfval 19296 dpjidcl 19299

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