Theorem fnbr 6031

Description: The first argument of binary relation on a function belongs to the function's domain. (Contributed by NM, 7-May-2004.)

Assertion

Ref	Expression
fnbr	⊢ ((𝐹 Fn 𝐴 ∧ 𝐵𝐹𝐶) → 𝐵 ∈ 𝐴)

Proof of Theorem fnbr

Step	Hyp	Ref	Expression
1		fnrel 6027	. . 3 ⊢ (𝐹 Fn 𝐴 → Rel 𝐹)
2		releldm 5390	. . 3 ⊢ ((Rel 𝐹 ∧ 𝐵𝐹𝐶) → 𝐵 ∈ dom 𝐹)
3	1, 2	sylan 487	. 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵𝐹𝐶) → 𝐵 ∈ dom 𝐹)
4		fndm 6028	. . . 4 ⊢ (𝐹 Fn 𝐴 → dom 𝐹 = 𝐴)
5	4	eleq2d 2716	. . 3 ⊢ (𝐹 Fn 𝐴 → (𝐵 ∈ dom 𝐹 ↔ 𝐵 ∈ 𝐴))
6	5	biimpa 500	. 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ∈ dom 𝐹) → 𝐵 ∈ 𝐴)
7	3, 6	syldan 486	1 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵𝐹𝐶) → 𝐵 ∈ 𝐴)

Syntax hints:   → wi 4   ∧ wa 383   ∈ wcel 2030   class class class wbr 4685  dom cdm 5143  Rel wrel 5148   Fn wfn 5921

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pr 4936

This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-br 4686  df-opab 4746  df-xp 5149  df-rel 5150  df-dm 5153  df-fun 5928  df-fn 5929

This theorem is referenced by:  fnop  6032  dffn5  6280  feqmptdf  6290  dffo4  6415  dffo5  6416  tfrlem5  7521  occllem  28290  chscllem2  28625  brcoffn  38645  fvelima2  39789  dfafn5a  41561