Theorem brelrng 5887

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 5826	. . . . 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 5856	. . . 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 5634	. 2 ⊢ ran 𝐶 = dom ◡𝐶
8	6, 7	eleqtrrdi 2839	1 ⊢ ((𝐴 ∈ 𝐹 ∧ 𝐵 ∈ 𝐺 ∧ 𝐴𝐶𝐵) → 𝐵 ∈ ran 𝐶)

Syntax hints:   → wi 4   ↔ wb 206   ∧ w3a 1086   ∈ wcel 2109   class class class wbr 5095  ^◡ccnv 5622  dom cdm 5623  ran crn 5624

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  ax-sep 5238  ax-nul 5248  ax-pr 5374

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 3397  df-v 3440  df-dif 3908  df-un 3910  df-ss 3922  df-nul 4287  df-if 4479  df-sn 4580  df-pr 4582  df-op 4586  df-br 5096  df-opab 5158  df-cnv 5631  df-dm 5633  df-rn 5634

This theorem is referenced by:  brelrn  5888  relelrn  5891  sossfld  6139  fvrn0  6854  pgpfaclem1  19980  perpln2  28674