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Mirrors > Home > MPE Home > Th. List > reldmmpo | Structured version Visualization version GIF version |
Description: The domain of an operation defined by maps-to notation is a relation. (Contributed by Stefan O'Rear, 27-Nov-2014.) |
Ref | Expression |
---|---|
rngop.1 | ⊢ 𝐹 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) |
Ref | Expression |
---|---|
reldmmpo | ⊢ Rel dom 𝐹 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | reldmoprab 7516 | . 2 ⊢ Rel dom {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = 𝐶)} | |
2 | rngop.1 | . . . . 5 ⊢ 𝐹 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) | |
3 | df-mpo 7416 | . . . . 5 ⊢ (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = 𝐶)} | |
4 | 2, 3 | eqtri 2758 | . . . 4 ⊢ 𝐹 = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = 𝐶)} |
5 | 4 | dmeqi 5903 | . . 3 ⊢ dom 𝐹 = dom {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = 𝐶)} |
6 | 5 | releqi 5776 | . 2 ⊢ (Rel dom 𝐹 ↔ Rel dom {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = 𝐶)}) |
7 | 1, 6 | mpbir 230 | 1 ⊢ Rel dom 𝐹 |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 394 = wceq 1539 ∈ wcel 2104 dom cdm 5675 Rel wrel 5680 {coprab 7412 ∈ cmpo 7413 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-10 2135 ax-11 2152 ax-12 2169 ax-ext 2701 ax-sep 5298 ax-nul 5305 ax-pr 5426 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 844 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2532 df-eu 2561 df-clab 2708 df-cleq 2722 df-clel 2808 df-nfc 2883 df-rab 3431 df-v 3474 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-br 5148 df-opab 5210 df-xp 5681 df-rel 5682 df-dm 5685 df-oprab 7415 df-mpo 7416 |
This theorem is referenced by: reldmmap 8831 reldmrelexp 14972 reldmsets 17102 reldmress 17179 reldmprds 17398 gsum0 18609 reldmghm 19129 oppglsm 19551 reldmdprd 19908 reldmlmhm 20780 zrhval 21276 reldmdsmm 21507 frlmrcl 21531 reldmpsr 21686 reldmmpl 21766 reldmopsr 21819 reldmevls 21866 vr1val 21935 reldmevls1 22056 evl1fval 22067 matbas0pc 22129 mdetfval 22308 madufval 22359 qtopres 23422 fgabs 23603 reldmtng 24367 reldmnghm 24449 reldmnmhm 24450 dvbsss 25651 reldmmdeg 25807 nbgrprc0 28858 wwlksn 29358 reldmresv 32710 bj-restsnid 36271 mzpmfp 41787 brovmptimex 43080 1aryenef 47418 2aryenef 47429 |
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