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Mirrors > Home > MPE Home > Th. List > reldmdprd | Structured version Visualization version GIF version |
Description: The domain of the internal direct product operation is a relation. (Contributed by Mario Carneiro, 25-Apr-2016.) (Proof shortened by AV, 11-Jul-2019.) |
Ref | Expression |
---|---|
reldmdprd | ⊢ Rel dom DProd |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-dprd 18217 | . 2 ⊢ DProd = (𝑔 ∈ Grp, 𝑠 ∈ {ℎ ∣ (ℎ:dom ℎ⟶(SubGrp‘𝑔) ∧ ∀𝑥 ∈ dom ℎ(∀𝑦 ∈ (dom ℎ ∖ {𝑥})(ℎ‘𝑥) ⊆ ((Cntz‘𝑔)‘(ℎ‘𝑦)) ∧ ((ℎ‘𝑥) ∩ ((mrCls‘(SubGrp‘𝑔))‘∪ (ℎ “ (dom ℎ ∖ {𝑥})))) = {(0g‘𝑔)}))} ↦ ran (𝑓 ∈ {ℎ ∈ X𝑥 ∈ dom 𝑠(𝑠‘𝑥) ∣ ℎ finSupp (0g‘𝑔)} ↦ (𝑔 Σg 𝑓))) | |
2 | 1 | reldmmpt2 6669 | 1 ⊢ Rel dom DProd |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 383 = wceq 1475 {cab 2596 ∀wral 2896 {crab 2900 ∖ cdif 3537 ∩ cin 3539 ⊆ wss 3540 {csn 4125 ∪ cuni 4372 class class class wbr 4583 ↦ cmpt 4643 dom cdm 5038 ran crn 5039 “ cima 5041 Rel wrel 5043 ⟶wf 5800 ‘cfv 5804 (class class class)co 6549 Xcixp 7794 finSupp cfsupp 8158 0gc0g 15923 Σg cgsu 15924 mrClscmrc 16066 Grpcgrp 17245 SubGrpcsubg 17411 Cntzccntz 17571 DProd cdprd 18215 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pr 4833 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-rab 2905 df-v 3175 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-br 4584 df-opab 4644 df-xp 5044 df-rel 5045 df-dm 5048 df-oprab 6553 df-mpt2 6554 df-dprd 18217 |
This theorem is referenced by: dprddomprc 18222 dprdval0prc 18224 dprdval 18225 dprdgrp 18227 dprdf 18228 dprdssv 18238 subgdmdprd 18256 dprd2da 18264 dpjfval 18277 |
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