Theorem relelrnb 5856

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

Assertion

Ref	Expression
relelrnb	⊢ (Rel 𝑅 → (𝐴 ∈ ran 𝑅 ↔ ∃𝑥 𝑥𝑅𝐴))

Distinct variable groups:   𝑥,𝐴   𝑥,𝑅

Proof of Theorem relelrnb

Step	Hyp	Ref	Expression
1		elrng 5800	. . 3 ⊢ (𝐴 ∈ ran 𝑅 → (𝐴 ∈ ran 𝑅 ↔ ∃𝑥 𝑥𝑅𝐴))
2	1	ibi 266	. 2 ⊢ (𝐴 ∈ ran 𝑅 → ∃𝑥 𝑥𝑅𝐴)
3		relelrn 5854	. . . 4 ⊢ ((Rel 𝑅 ∧ 𝑥𝑅𝐴) → 𝐴 ∈ ran 𝑅)
4	3	ex 413	. . 3 ⊢ (Rel 𝑅 → (𝑥𝑅𝐴 → 𝐴 ∈ ran 𝑅))
5	4	exlimdv 1936	. 2 ⊢ (Rel 𝑅 → (∃𝑥 𝑥𝑅𝐴 → 𝐴 ∈ ran 𝑅))
6	2, 5	impbid2 225	1 ⊢ (Rel 𝑅 → (𝐴 ∈ ran 𝑅 ↔ ∃𝑥 𝑥𝑅𝐴))

Syntax hints:   → wi 4   ↔ wb 205  ∃wex 1782   ∈ wcel 2106   class class class wbr 5074  ran crn 5590  Rel wrel 5594

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-br 5075  df-opab 5137  df-xp 5595  df-rel 5596  df-cnv 5597  df-dm 5599  df-rn 5600

This theorem is referenced by:  iscard4  41140