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Mirrors > Home > MPE Home > Th. List > 1stdm | Structured version Visualization version GIF version |
Description: The first ordered pair component of a member of a relation belongs to the domain of the relation. (Contributed by NM, 17-Sep-2006.) |
Ref | Expression |
---|---|
1stdm | ⊢ ((Rel 𝑅 ∧ 𝐴 ∈ 𝑅) → (1st ‘𝐴) ∈ dom 𝑅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-rel 5696 | . . . . 5 ⊢ (Rel 𝑅 ↔ 𝑅 ⊆ (V × V)) | |
2 | 1 | biimpi 216 | . . . 4 ⊢ (Rel 𝑅 → 𝑅 ⊆ (V × V)) |
3 | 2 | sselda 3995 | . . 3 ⊢ ((Rel 𝑅 ∧ 𝐴 ∈ 𝑅) → 𝐴 ∈ (V × V)) |
4 | 1stval2 8030 | . . 3 ⊢ (𝐴 ∈ (V × V) → (1st ‘𝐴) = ∩ ∩ 𝐴) | |
5 | 3, 4 | syl 17 | . 2 ⊢ ((Rel 𝑅 ∧ 𝐴 ∈ 𝑅) → (1st ‘𝐴) = ∩ ∩ 𝐴) |
6 | elreldm 5949 | . 2 ⊢ ((Rel 𝑅 ∧ 𝐴 ∈ 𝑅) → ∩ ∩ 𝐴 ∈ dom 𝑅) | |
7 | 5, 6 | eqeltrd 2839 | 1 ⊢ ((Rel 𝑅 ∧ 𝐴 ∈ 𝑅) → (1st ‘𝐴) ∈ dom 𝑅) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1537 ∈ wcel 2106 Vcvv 3478 ⊆ wss 3963 ∩ cint 4951 × cxp 5687 dom cdm 5689 Rel wrel 5694 ‘cfv 6563 1st c1st 8011 |
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 ax-un 7754 |
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-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-int 4952 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-iota 6516 df-fun 6565 df-fv 6571 df-1st 8013 |
This theorem is referenced by: releldmdifi 8069 funeldmdif 8072 frxp 8150 dprd2dlem2 20075 dprd2da 20077 gsummpt2d 33035 gsumhashmul 33047 gsumwrd2dccat 33053 satfdmlem 35353 satffunlem1lem2 35388 satffunlem2lem2 35391 |
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