Theorem 1stdm 8017

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 5652	. . . . 5 ⊢ (Rel 𝑅 ↔ 𝑅 ⊆ (V × V))
2	1	biimpi 218	. . . 4 ⊢ (Rel 𝑅 → 𝑅 ⊆ (V × V))
3	2	sselda 3936	. . 3 ⊢ ((Rel 𝑅 ∧ 𝐴 ∈ 𝑅) → 𝐴 ∈ (V × V))
4		1stval2 7983	. . 3 ⊢ (𝐴 ∈ (V × V) → (1st ‘𝐴) = ∩ ∩ 𝐴)
5	3, 4	syl 17	. 2 ⊢ ((Rel 𝑅 ∧ 𝐴 ∈ 𝑅) → (1st ‘𝐴) = ∩ ∩ 𝐴)
6		elreldm 5909	. 2 ⊢ ((Rel 𝑅 ∧ 𝐴 ∈ 𝑅) → ∩ ∩ 𝐴 ∈ dom 𝑅)
7	5, 6	eqeltrd 2861	1 ⊢ ((Rel 𝑅 ∧ 𝐴 ∈ 𝑅) → (1st ‘𝐴) ∈ dom 𝑅)

Syntax hints:   → wi 4   ∧ wa 399   = wceq 1559   ∈ wcel 2141  Vcvv 3453   ⊆ wss 3904  ∩ cint 4904   × cxp 5643  dom cdm 5645  Rel wrel 5650  ‘cfv 6517  1^st c1st 7964

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-sep 5245  ax-nul 5255  ax-pr 5389  ax-un 7714

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-nf 1803  df-sb 2090  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3076  df-rex 3086  df-rab 3414  df-v 3455  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4480  df-sn 4582  df-pr 4584  df-op 4588  df-uni 4865  df-int 4905  df-br 5100  df-opab 5162  df-mpt 5181  df-id 5540  df-xp 5651  df-rel 5652  df-cnv 5653  df-co 5654  df-dm 5655  df-rn 5656  df-iota 6473  df-fun 6519  df-fv 6525  df-1st 7966

This theorem is referenced by:  releldmdifi  8022  funeldmdif  8025  frxp  8101  dprd2dlem2  20065  dprd2da  20067  gsummpt2d  33190  gsumhashmul  33208  gsumwrd2dccat  33219  satfdmlem  35682  satffunlem1lem2  35717  satffunlem2lem2  35720