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Mirrors > Home > MPE Home > Th. List > reldmmpt2 | 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 |
---|---|
reldmmpt2 | ⊢ Rel dom 𝐹 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | reldmoprab 6643 | . 2 ⊢ Rel dom {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = 𝐶)} | |
2 | rngop.1 | . . . . 5 ⊢ 𝐹 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) | |
3 | df-mpt2 6554 | . . . . 5 ⊢ (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) = {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = 𝐶)} | |
4 | 2, 3 | eqtri 2632 | . . . 4 ⊢ 𝐹 = {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = 𝐶)} |
5 | 4 | dmeqi 5247 | . . 3 ⊢ dom 𝐹 = dom {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = 𝐶)} |
6 | 5 | releqi 5125 | . 2 ⊢ (Rel dom 𝐹 ↔ Rel dom {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = 𝐶)}) |
7 | 1, 6 | mpbir 220 | 1 ⊢ Rel dom 𝐹 |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 383 = wceq 1475 ∈ wcel 1977 dom cdm 5038 Rel wrel 5043 {coprab 6550 ↦ cmpt2 6551 |
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 |
This theorem is referenced by: reldmmap 7753 reldmsets 15718 reldmress 15753 reldmprds 15932 gsum0 17101 reldmghm 17482 oppglsm 17880 reldmdprd 18219 reldmlmhm 18846 reldmpsr 19182 reldmmpl 19248 reldmopsr 19294 reldmevls 19338 vr1val 19383 reldmevls1 19503 evl1fval 19513 zrhval 19675 reldmdsmm 19896 frlmrcl 19920 matbas0pc 20034 mdetfval 20211 madufval 20262 qtopres 21311 fgabs 21493 reldmtng 22252 reldmnghm 22326 reldmnmhm 22327 dvbsss 23472 reldmmdeg 23621 reldmresv 29157 bj-restsnid 32221 mzpmfp 36328 brovmptimex 37345 nbgrprc0 40555 wwlksn 41040 wwlks2onv 41158 clwwlksn 41189 |
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