Theorem brelrng 5890

Description: The second argument of a binary relation belongs to its range. (Contributed by NM, 29-Jun-2008.)

Assertion

Ref	Expression
brelrng	⊢ ((𝐴 ∈ 𝐹 ∧ 𝐵 ∈ 𝐺 ∧ 𝐴𝐶𝐵) → 𝐵 ∈ ran 𝐶)

Proof of Theorem brelrng

Step	Hyp	Ref	Expression
1		brcnvg 5828	. . . . 5 ⊢ ((𝐵 ∈ 𝐺 ∧ 𝐴 ∈ 𝐹) → (𝐵◡𝐶𝐴 ↔ 𝐴𝐶𝐵))
2	1	ancoms 458	. . . 4 ⊢ ((𝐴 ∈ 𝐹 ∧ 𝐵 ∈ 𝐺) → (𝐵◡𝐶𝐴 ↔ 𝐴𝐶𝐵))
3	2	biimp3ar 1472	. . 3 ⊢ ((𝐴 ∈ 𝐹 ∧ 𝐵 ∈ 𝐺 ∧ 𝐴𝐶𝐵) → 𝐵◡𝐶𝐴)
4		breldmg 5858	. . . 4 ⊢ ((𝐵 ∈ 𝐺 ∧ 𝐴 ∈ 𝐹 ∧ 𝐵◡𝐶𝐴) → 𝐵 ∈ dom ◡𝐶)
5	4	3com12 1123	. . 3 ⊢ ((𝐴 ∈ 𝐹 ∧ 𝐵 ∈ 𝐺 ∧ 𝐵◡𝐶𝐴) → 𝐵 ∈ dom ◡𝐶)
6	3, 5	syld3an3 1411	. 2 ⊢ ((𝐴 ∈ 𝐹 ∧ 𝐵 ∈ 𝐺 ∧ 𝐴𝐶𝐵) → 𝐵 ∈ dom ◡𝐶)
7		df-rn 5635	. 2 ⊢ ran 𝐶 = dom ◡𝐶
8	6, 7	eleqtrrdi 2847	1 ⊢ ((𝐴 ∈ 𝐹 ∧ 𝐵 ∈ 𝐺 ∧ 𝐴𝐶𝐵) → 𝐵 ∈ ran 𝐶)

Syntax hints:   → wi 4   ↔ wb 206   ∧ w3a 1086   ∈ wcel 2113   class class class wbr 5098  ^◡ccnv 5623  dom cdm 5624  ran crn 5625

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-ext 2708  ax-sep 5241  ax-nul 5251  ax-pr 5377

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2715  df-cleq 2728  df-clel 2811  df-rab 3400  df-v 3442  df-dif 3904  df-un 3906  df-ss 3918  df-nul 4286  df-if 4480  df-sn 4581  df-pr 4583  df-op 4587  df-br 5099  df-opab 5161  df-cnv 5632  df-dm 5634  df-rn 5635

This theorem is referenced by:  brelrn  5891  relelrn  5894  sossfld  6144  fvrn0  6862  pgpfaclem1  20012  perpln2  28783