Theorem breldmg 5864

Description: Membership of first of a binary relation in a domain. (Contributed by NM, 21-Mar-2007.)

Assertion

Ref	Expression
breldmg	⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷 ∧ 𝐴𝑅𝐵) → 𝐴 ∈ dom 𝑅)

Proof of Theorem breldmg

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		breq2 5106	. . . . 5 ⊢ (𝑥 = 𝐵 → (𝐴𝑅𝑥 ↔ 𝐴𝑅𝐵))
2	1	spcegv 3560	. . . 4 ⊢ (𝐵 ∈ 𝐷 → (𝐴𝑅𝐵 → ∃𝑥 𝐴𝑅𝑥))
3	2	imp 406	. . 3 ⊢ ((𝐵 ∈ 𝐷 ∧ 𝐴𝑅𝐵) → ∃𝑥 𝐴𝑅𝑥)
4		eldmg 5853	. . 3 ⊢ (𝐴 ∈ 𝐶 → (𝐴 ∈ dom 𝑅 ↔ ∃𝑥 𝐴𝑅𝑥))
5	3, 4	imbitrrid 246	. 2 ⊢ (𝐴 ∈ 𝐶 → ((𝐵 ∈ 𝐷 ∧ 𝐴𝑅𝐵) → 𝐴 ∈ dom 𝑅))
6	5	3impib 1116	1 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷 ∧ 𝐴𝑅𝐵) → 𝐴 ∈ dom 𝑅)

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1086  ∃wex 1779   ∈ wcel 2109   class class class wbr 5102  dom cdm 5631

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 2008  ax-8 2111  ax-9 2119  ax-ext 2701

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-rab 3403  df-v 3446  df-dif 3914  df-un 3916  df-ss 3928  df-nul 4293  df-if 4485  df-sn 4586  df-pr 4588  df-op 4592  df-br 5103  df-dm 5641

This theorem is referenced by:  breldmd  5867  brelrng  5895  releldm  5898  sossfld  6148  brtpos  8192  fprresex  8267  tfrlem9a  8332  perpln1  28692  lmdvg  33938  esumcvgsum  34073  climeldmeq  45658  climfv  45684  climxlim2  45839  sge0isum  46420  smflimsuplem6  46818  eubrdm  47032  funressneu  47043  tz6.12-afv  47169  rlimdmafv  47173  tz6.12-afv2  47236  rlimdmafv2  47254