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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 3048 | . . . . 5 ⊢ (dom 𝑆 ∉ V ↔ ¬ dom 𝑆 ∈ V) | |
4 | dprddomprc 19870 | . . . . 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 2822 | . . 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 1542 ∈ wcel 2107 ∉ wnel 3047 Vcvv 3475 class class class wbr 5149 dom cdm 5677 DProd cdprd 19863 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5300 ax-nul 5307 ax-pr 5428 ax-un 7725 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-nel 3048 df-ral 3063 df-rex 3072 df-rab 3434 df-v 3477 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4324 df-if 4530 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-br 5150 df-opab 5212 df-xp 5683 df-rel 5684 df-cnv 5685 df-dm 5687 df-rn 5688 df-oprab 7413 df-mpo 7414 df-dprd 19865 |
This theorem is referenced by: dprdcntz 19878 dprddisj 19879 dprdw 19880 dprdwd 19881 dprdfid 19887 dprdfinv 19889 dprdfadd 19890 dprdfsub 19891 dprdfeq0 19892 dprdf11 19893 dprdlub 19896 dprdres 19898 dprdss 19899 dprdf1o 19902 dmdprdsplitlem 19907 dprddisj2 19909 dmdprdsplit2 19916 dpjfval 19925 dpjidcl 19928 |
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