Theorem 1stdm 8025

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.)

Assertion

Ref	Expression
1stdm	⊢ ((Rel 𝑅 ∧ 𝐴 ∈ 𝑅) → (1st ‘𝐴) ∈ dom 𝑅)

Proof of Theorem 1stdm

Step	Hyp	Ref	Expression
1		df-rel 5659	. . . . 5 ⊢ (Rel 𝑅 ↔ 𝑅 ⊆ (V × V))
2	1	biimpi 219	. . . 4 ⊢ (Rel 𝑅 → 𝑅 ⊆ (V × V))
3	2	sselda 3939	. . 3 ⊢ ((Rel 𝑅 ∧ 𝐴 ∈ 𝑅) → 𝐴 ∈ (V × V))
4		1stval2 7991	. . 3 ⊢ (𝐴 ∈ (V × V) → (1st ‘𝐴) = ∩ ∩ 𝐴)
5	3, 4	syl 18	. 2 ⊢ ((Rel 𝑅 ∧ 𝐴 ∈ 𝑅) → (1st ‘𝐴) = ∩ ∩ 𝐴)
6		elreldm 5916	. 2 ⊢ ((Rel 𝑅 ∧ 𝐴 ∈ 𝑅) → ∩ ∩ 𝐴 ∈ dom 𝑅)
7	5, 6	eqeltrd 2865	1 ⊢ ((Rel 𝑅 ∧ 𝐴 ∈ 𝑅) → (1st ‘𝐴) ∈ dom 𝑅)

Syntax hints:   → wi 4   ∧ wa 400   = wceq 1563   ∈ wcel 2145  Vcvv 3457   ⊆ wss 3907  ∩ cint 4908   × cxp 5650  dom cdm 5652  Rel wrel 5657  ‘cfv 6525  1^st c1st 7972

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2737  ax-sep 5251  ax-nul 5261  ax-pr 5395  ax-un 7722

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-nf 1807  df-sb 2094  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3080  df-rex 3090  df-rab 3418  df-v 3459  df-dif 3910  df-un 3912  df-in 3914  df-ss 3924  df-nul 4289  df-if 4484  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4869  df-int 4909  df-br 5106  df-opab 5168  df-mpt 5187  df-id 5547  df-xp 5658  df-rel 5659  df-cnv 5660  df-co 5661  df-dm 5662  df-rn 5663  df-iota 6481  df-fun 6527  df-fv 6533  df-1st 7974

This theorem is referenced by:  releldmdifi  8030  funeldmdif  8033  frxp  8110  dprd2dlem2  20103  dprd2da  20105  gsummpt2d  33282  gsumhashmul  33300  gsumwrd2dccat  33311  satfdmlem  35731  satffunlem1lem2  35766  satffunlem2lem2  35769