Theorem fnbr 2726

Description: The first argument of binary relation on a function belongs to the function's domain.

Hypotheses

Ref	Expression
fnbr.1	⊢ B ∈ V
fnbr.2	⊢ C ∈ V

Assertion

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

Proof of Theorem fnbr

Step	Hyp	Ref	Expression
1		fndm 2723	. . . 4 ⊢ (F Fn A → dom F = A)
2	1	eleq2d 1156	. . 3 ⊢ (F Fn A → (B ∈ dom F ↔ B ∈ A))
3	2	biimpa 324	. 2 ⊢ ((F Fn A ∧ B ∈ dom F) → B ∈ A)
4		fnbr.1	. . 3 ⊢ B ∈ V
5	4	breldm 2535	. 2 ⊢ (BFC → B ∈ dom F)
6	3, 5	sylan2 346	1 ⊢ ((F Fn A ∧ BFC) → B ∈ A)

Syntax hints:   → wi 2   ∧ wa 196   ∈ wcel 1092  Vcvv 1348   class class class wbr 2054  dom cdm 2410   Fn wfn 2417

This theorem is referenced by:  fnop 2727  fvelrn 2883

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-16 922  ax-17 925  ax-ext 1074

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-v 1349  df-dif 1489  df-un 1490  df-nul 1708  df-sn 1811  df-pr 1812  df-op 1815  df-br 2063  df-dm 2428  df-fn 2433

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