Theorem brelrng 4835

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 4785	. . . . 5 ⊢ ((𝐵 ∈ 𝐺 ∧ 𝐴 ∈ 𝐹) → (𝐵◡𝐶𝐴 ↔ 𝐴𝐶𝐵))
2	1	ancoms 266	. . . 4 ⊢ ((𝐴 ∈ 𝐹 ∧ 𝐵 ∈ 𝐺) → (𝐵◡𝐶𝐴 ↔ 𝐴𝐶𝐵))
3	2	biimp3ar 1336	. . 3 ⊢ ((𝐴 ∈ 𝐹 ∧ 𝐵 ∈ 𝐺 ∧ 𝐴𝐶𝐵) → 𝐵◡𝐶𝐴)
4		breldmg 4810	. . . 4 ⊢ ((𝐵 ∈ 𝐺 ∧ 𝐴 ∈ 𝐹 ∧ 𝐵◡𝐶𝐴) → 𝐵 ∈ dom ◡𝐶)
5	4	3com12 1197	. . 3 ⊢ ((𝐴 ∈ 𝐹 ∧ 𝐵 ∈ 𝐺 ∧ 𝐵◡𝐶𝐴) → 𝐵 ∈ dom ◡𝐶)
6	3, 5	syld3an3 1273	. 2 ⊢ ((𝐴 ∈ 𝐹 ∧ 𝐵 ∈ 𝐺 ∧ 𝐴𝐶𝐵) → 𝐵 ∈ dom ◡𝐶)
7		df-rn 4615	. 2 ⊢ ran 𝐶 = dom ◡𝐶
8	6, 7	eleqtrrdi 2260	1 ⊢ ((𝐴 ∈ 𝐹 ∧ 𝐵 ∈ 𝐺 ∧ 𝐴𝐶𝐵) → 𝐵 ∈ ran 𝐶)

Syntax hints:   → wi 4   ↔ wb 104   ∧ w3a 968   ∈ wcel 2136   class class class wbr 3982  ^◡ccnv 4603  dom cdm 4604  ran crn 4605

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-pow 4153  ax-pr 4187

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-v 2728  df-un 3120  df-in 3122  df-ss 3129  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-br 3983  df-opab 4044  df-cnv 4612  df-dm 4614  df-rn 4615

This theorem is referenced by:  opelrng  4836  brelrn  4837  relelrn  4840  fvssunirng  5501  shftfvalg  10760  ovshftex  10761  shftfval  10763