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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 5685 | . . . . 5 ⊢ (Rel 𝑅 ↔ 𝑅 ⊆ (V × V)) | |
2 | 1 | biimpi 215 | . . . 4 ⊢ (Rel 𝑅 → 𝑅 ⊆ (V × V)) |
3 | 2 | sselda 3980 | . . 3 ⊢ ((Rel 𝑅 ∧ 𝐴 ∈ 𝑅) → 𝐴 ∈ (V × V)) |
4 | 1stval2 8010 | . . 3 ⊢ (𝐴 ∈ (V × V) → (1st ‘𝐴) = ∩ ∩ 𝐴) | |
5 | 3, 4 | syl 17 | . 2 ⊢ ((Rel 𝑅 ∧ 𝐴 ∈ 𝑅) → (1st ‘𝐴) = ∩ ∩ 𝐴) |
6 | elreldm 5937 | . 2 ⊢ ((Rel 𝑅 ∧ 𝐴 ∈ 𝑅) → ∩ ∩ 𝐴 ∈ dom 𝑅) | |
7 | 5, 6 | eqeltrd 2829 | 1 ⊢ ((Rel 𝑅 ∧ 𝐴 ∈ 𝑅) → (1st ‘𝐴) ∈ dom 𝑅) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1534 ∈ wcel 2099 Vcvv 3471 ⊆ wss 3947 ∩ cint 4949 × cxp 5676 dom cdm 5678 Rel wrel 5683 ‘cfv 6548 1st c1st 7991 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-sep 5299 ax-nul 5306 ax-pr 5429 ax-un 7740 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ral 3059 df-rex 3068 df-rab 3430 df-v 3473 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4324 df-if 4530 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4909 df-int 4950 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5576 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-iota 6500 df-fun 6550 df-fv 6556 df-1st 7993 |
This theorem is referenced by: releldmdifi 8049 funeldmdif 8052 frxp 8131 dprd2dlem2 19997 dprd2da 19999 gsummpt2d 32776 gsumhashmul 32783 satfdmlem 34978 satffunlem1lem2 35013 satffunlem2lem2 35016 |
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