Theorem relelrnb 5941

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 5885	. . 3 ⊢ (𝐴 ∈ ran 𝑅 → (𝐴 ∈ ran 𝑅 ↔ ∃𝑥 𝑥𝑅𝐴))
2	1	ibi 270	. 2 ⊢ (𝐴 ∈ ran 𝑅 → ∃𝑥 𝑥𝑅𝐴)
3		relelrn 5939	. . . 4 ⊢ ((Rel 𝑅 ∧ 𝑥𝑅𝐴) → 𝐴 ∈ ran 𝑅)
4	3	ex 418	. . 3 ⊢ (Rel 𝑅 → (𝑥𝑅𝐴 → 𝐴 ∈ ran 𝑅))
5	4	exlimdv 1968	. 2 ⊢ (Rel 𝑅 → (∃𝑥 𝑥𝑅𝐴 → 𝐴 ∈ ran 𝑅))
6	2, 5	impbid2 229	1 ⊢ (Rel 𝑅 → (𝐴 ∈ ran 𝑅 ↔ ∃𝑥 𝑥𝑅𝐴))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209  ∃wex 1812   ∈ wcel 2147   class class class wbr 5108  ran crn 5659  Rel wrel 5663

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735  ax-sep 5255  ax-pr 5398

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4841  df-br 5109  df-opab 5174  df-xp 5664  df-rel 5665  df-cnv 5666  df-dm 5668  df-rn 5669

This theorem is used by:  iscard4  44795