Theorem fnbr 5434

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 5428	. . 3 ⊢ (𝐹 Fn 𝐴 → Rel 𝐹)
2		releldm 4967	. . 3 ⊢ ((Rel 𝐹 ∧ 𝐵𝐹𝐶) → 𝐵 ∈ dom 𝐹)
3	1, 2	sylan 283	. 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵𝐹𝐶) → 𝐵 ∈ dom 𝐹)
4		fndm 5429	. . . 4 ⊢ (𝐹 Fn 𝐴 → dom 𝐹 = 𝐴)
5	4	eleq2d 2301	. . 3 ⊢ (𝐹 Fn 𝐴 → (𝐵 ∈ dom 𝐹 ↔ 𝐵 ∈ 𝐴))
6	5	biimpa 296	. 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ∈ dom 𝐹) → 𝐵 ∈ 𝐴)
7	3, 6	syldan 282	1 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵𝐹𝐶) → 𝐵 ∈ 𝐴)

Syntax hints:   → wi 4   ∧ wa 104   ∈ wcel 2202   class class class wbr 4088  dom cdm 4725  Rel wrel 4730   Fn wfn 5321

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ral 2515  df-rex 2516  df-v 2804  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-br 4089  df-opab 4151  df-xp 4731  df-rel 4732  df-dm 4735  df-fun 5328  df-fn 5329

This theorem is referenced by:  fnop  5435  dffn5im  5691  dffo4  5795  dffo5  5796  tfrlem5  6479