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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 5692 | . . . . 5 ⊢ (Rel 𝑅 ↔ 𝑅 ⊆ (V × V)) | |
| 2 | 1 | biimpi 216 | . . . 4 ⊢ (Rel 𝑅 → 𝑅 ⊆ (V × V)) |
| 3 | 2 | sselda 3983 | . . 3 ⊢ ((Rel 𝑅 ∧ 𝐴 ∈ 𝑅) → 𝐴 ∈ (V × V)) |
| 4 | 1stval2 8031 | . . 3 ⊢ (𝐴 ∈ (V × V) → (1st ‘𝐴) = ∩ ∩ 𝐴) | |
| 5 | 3, 4 | syl 17 | . 2 ⊢ ((Rel 𝑅 ∧ 𝐴 ∈ 𝑅) → (1st ‘𝐴) = ∩ ∩ 𝐴) |
| 6 | elreldm 5946 | . 2 ⊢ ((Rel 𝑅 ∧ 𝐴 ∈ 𝑅) → ∩ ∩ 𝐴 ∈ dom 𝑅) | |
| 7 | 5, 6 | eqeltrd 2841 | 1 ⊢ ((Rel 𝑅 ∧ 𝐴 ∈ 𝑅) → (1st ‘𝐴) ∈ dom 𝑅) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2108 Vcvv 3480 ⊆ wss 3951 ∩ cint 4946 × cxp 5683 dom cdm 5685 Rel wrel 5690 ‘cfv 6561 1st c1st 8012 |
| 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 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 ax-un 7755 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-int 4947 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-iota 6514 df-fun 6563 df-fv 6569 df-1st 8014 |
| This theorem is referenced by: releldmdifi 8070 funeldmdif 8073 frxp 8151 dprd2dlem2 20060 dprd2da 20062 gsummpt2d 33052 gsumhashmul 33064 gsumwrd2dccat 33070 satfdmlem 35373 satffunlem1lem2 35408 satffunlem2lem2 35411 |
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