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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 7539 | . 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 7436 | . . . . 5 ⊢ (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) = {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = 𝐶)} | |
4 | 2, 3 | eqtri 2763 | . . . 4 ⊢ 𝐹 = {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = 𝐶)} |
5 | 4 | dmeqi 5918 | . . 3 ⊢ dom 𝐹 = dom {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = 𝐶)} |
6 | 5 | releqi 5790 | . 2 ⊢ (Rel dom 𝐹 ↔ Rel dom {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = 𝐶)}) |
7 | 1, 6 | mpbir 231 | 1 ⊢ Rel dom 𝐹 |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 395 = wceq 1537 ∈ wcel 2106 dom cdm 5689 Rel wrel 5694 {coprab 7432 ∈ cmpo 7433 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-br 5149 df-opab 5211 df-xp 5695 df-rel 5696 df-dm 5699 df-oprab 7435 df-mpo 7436 |
This theorem is referenced by: reldmmap 8874 reldmrelexp 15057 reldmsets 17199 reldmress 17276 reldmprds 17495 gsum0 18710 reldmghm 19245 oppglsm 19675 reldmdprd 20032 reldmlmhm 21042 zrhval 21536 reldmdsmm 21771 frlmrcl 21795 reldmpsr 21952 reldmmpl 22026 reldmopsr 22081 reldmevls 22126 reldmmhp 22159 vr1val 22209 reldmevls1 22337 evl1fval 22348 matbas0pc 22429 mdetfval 22608 madufval 22659 qtopres 23722 fgabs 23903 reldmtng 24667 reldmnghm 24749 reldmnmhm 24750 dvbsss 25952 reldmmdeg 26111 nbgrprc0 29366 wwlksn 29867 of0r 32695 reldmrloc 33244 erlval 33245 reldmresv 33332 bj-restsnid 37070 mzpmfp 42735 brovmptimex 44017 clnbgrprc0 47745 grimdmrel 47804 grlimdmrel 47883 1aryenef 48495 2aryenef 48506 |
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