Theorem fnbr 6459

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 6454	. . 3 ⊢ (𝐹 Fn 𝐴 → Rel 𝐹)
2		releldm 5814	. . 3 ⊢ ((Rel 𝐹 ∧ 𝐵𝐹𝐶) → 𝐵 ∈ dom 𝐹)
3	1, 2	sylan 582	. 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵𝐹𝐶) → 𝐵 ∈ dom 𝐹)
4		fndm 6455	. . . 4 ⊢ (𝐹 Fn 𝐴 → dom 𝐹 = 𝐴)
5	4	eleq2d 2898	. . 3 ⊢ (𝐹 Fn 𝐴 → (𝐵 ∈ dom 𝐹 ↔ 𝐵 ∈ 𝐴))
6	5	biimpa 479	. 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ∈ dom 𝐹) → 𝐵 ∈ 𝐴)
7	3, 6	syldan 593	1 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵𝐹𝐶) → 𝐵 ∈ 𝐴)

Syntax hints:   → wi 4   ∧ wa 398   ∈ wcel 2114   class class class wbr 5066  dom cdm 5555  Rel wrel 5560   Fn wfn 6350

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 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2793  ax-sep 5203  ax-nul 5210  ax-pr 5330

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3496  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-nul 4292  df-if 4468  df-sn 4568  df-pr 4570  df-op 4574  df-br 5067  df-opab 5129  df-xp 5561  df-rel 5562  df-dm 5565  df-fun 6357  df-fn 6358

This theorem is referenced by:  fnop  6460  dffn5  6724  feqmptdf  6735  dffo4  6869  dffo5  6870  tfrlem5  8016  occllem  29080  chscllem2  29415  brcoffn  40400  fvelima2  41552  dfafn5a  43379