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Mirrors > Home > MPE Home > Th. List > dprddomcld | Structured version Visualization version GIF version |
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.) |
Ref | Expression |
---|---|
dprddomcld.1 | ⊢ (𝜑 → 𝐺dom DProd 𝑆) |
dprddomcld.2 | ⊢ (𝜑 → dom 𝑆 = 𝐼) |
Ref | Expression |
---|---|
dprddomcld | ⊢ (𝜑 → 𝐼 ∈ V) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dprddomcld.2 | . 2 ⊢ (𝜑 → dom 𝑆 = 𝐼) | |
2 | dprddomcld.1 | . 2 ⊢ (𝜑 → 𝐺dom DProd 𝑆) | |
3 | df-nel 3047 | . . . . 5 ⊢ (dom 𝑆 ∉ V ↔ ¬ dom 𝑆 ∈ V) | |
4 | dprddomprc 19864 | . . . . 5 ⊢ (dom 𝑆 ∉ V → ¬ 𝐺dom DProd 𝑆) | |
5 | 3, 4 | sylbir 234 | . . . 4 ⊢ (¬ dom 𝑆 ∈ V → ¬ 𝐺dom DProd 𝑆) |
6 | 5 | con4i 114 | . . 3 ⊢ (𝐺dom DProd 𝑆 → dom 𝑆 ∈ V) |
7 | eleq1 2821 | . . 3 ⊢ (dom 𝑆 = 𝐼 → (dom 𝑆 ∈ V ↔ 𝐼 ∈ V)) | |
8 | 6, 7 | imbitrid 243 | . 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 1541 ∈ wcel 2106 ∉ wnel 3046 Vcvv 3474 class class class wbr 5147 dom cdm 5675 DProd cdprd 19857 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-sep 5298 ax-nul 5305 ax-pr 5426 ax-un 7721 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-nel 3047 df-ral 3062 df-rex 3071 df-rab 3433 df-v 3476 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4322 df-if 4528 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-br 5148 df-opab 5210 df-xp 5681 df-rel 5682 df-cnv 5683 df-dm 5685 df-rn 5686 df-oprab 7409 df-mpo 7410 df-dprd 19859 |
This theorem is referenced by: dprdcntz 19872 dprddisj 19873 dprdw 19874 dprdwd 19875 dprdfid 19881 dprdfinv 19883 dprdfadd 19884 dprdfsub 19885 dprdfeq0 19886 dprdf11 19887 dprdlub 19890 dprdres 19892 dprdss 19893 dprdf1o 19896 dmdprdsplitlem 19901 dprddisj2 19903 dmdprdsplit2 19910 dpjfval 19919 dpjidcl 19922 |
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