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Theorem dprddomcld 19123

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 3124	. . . . 5 ⊢ (dom 𝑆 ∉ V ↔ ¬ dom 𝑆 ∈ V)
4		dprddomprc 19122	. . . . 5 ⊢ (dom 𝑆 ∉ V → ¬ 𝐺dom DProd 𝑆)
5	3, 4	sylbir 237	. . . 4 ⊢ (¬ dom 𝑆 ∈ V → ¬ 𝐺dom DProd 𝑆)
6	5	con4i 114	. . 3 ⊢ (𝐺dom DProd 𝑆 → dom 𝑆 ∈ V)
7		eleq1 2900	. . 3 ⊢ (dom 𝑆 = 𝐼 → (dom 𝑆 ∈ V ↔ 𝐼 ∈ V))
8	6, 7	syl5ib 246	. 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 1537 ∈ wcel 2114 ∉ wnel 3123 Vcvv 3494 class class class wbr 5066 dom cdm 5555 DProd cdprd 19115

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pr 5330 ax-un 7461

This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-nel 3124 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-br 5067 df-opab 5129 df-xp 5561 df-rel 5562 df-cnv 5563 df-dm 5565 df-rn 5566 df-oprab 7160 df-mpo 7161 df-dprd 19117

This theorem is referenced by: dprdcntz 19130 dprddisj 19131 dprdw 19132 dprdwd 19133 dprdfid 19139 dprdfinv 19141 dprdfadd 19142 dprdfsub 19143 dprdfeq0 19144 dprdf11 19145 dprdlub 19148 dprdres 19150 dprdss 19151 dprdf1o 19154 dmdprdsplitlem 19159 dprddisj2 19161 dmdprdsplit2 19168 dpjfval 19177 dpjidcl 19180

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