Theorem releldmb 4903

Description: Membership in a domain. (Contributed by Mario Carneiro, 5-Nov-2015.)

Assertion

Ref	Expression
releldmb	⊢ (Rel 𝑅 → (𝐴 ∈ dom 𝑅 ↔ ∃𝑥 𝐴𝑅𝑥))

Distinct variable groups:   𝑥,𝐴   𝑥,𝑅

Proof of Theorem releldmb

Step	Hyp	Ref	Expression
1		eldmg 4861	. . 3 ⊢ (𝐴 ∈ dom 𝑅 → (𝐴 ∈ dom 𝑅 ↔ ∃𝑥 𝐴𝑅𝑥))
2	1	ibi 176	. 2 ⊢ (𝐴 ∈ dom 𝑅 → ∃𝑥 𝐴𝑅𝑥)
3		releldm 4901	. . . 4 ⊢ ((Rel 𝑅 ∧ 𝐴𝑅𝑥) → 𝐴 ∈ dom 𝑅)
4	3	ex 115	. . 3 ⊢ (Rel 𝑅 → (𝐴𝑅𝑥 → 𝐴 ∈ dom 𝑅))
5	4	exlimdv 1833	. 2 ⊢ (Rel 𝑅 → (∃𝑥 𝐴𝑅𝑥 → 𝐴 ∈ dom 𝑅))
6	2, 5	impbid2 143	1 ⊢ (Rel 𝑅 → (𝐴 ∈ dom 𝑅 ↔ ∃𝑥 𝐴𝑅𝑥))

Syntax hints:   → wi 4   ↔ wb 105  ∃wex 1506   ∈ wcel 2167   class class class wbr 4033  dom cdm 4663  Rel wrel 4668

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-14 2170  ax-ext 2178  ax-sep 4151  ax-pow 4207  ax-pr 4242

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480  df-rex 2481  df-v 2765  df-un 3161  df-in 3163  df-ss 3170  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-br 4034  df-opab 4095  df-xp 4669  df-rel 4670  df-dm 4673

This theorem is referenced by: (None)