Theorem fnimadisj 5355

Description: A class that is disjoint with the domain of a function has an empty image under the function. (Contributed by FL, 24-Jan-2007.)

Assertion

Ref	Expression
fnimadisj	⊢ ((𝐹 Fn 𝐴 ∧ (𝐴 ∩ 𝐶) = ∅) → (𝐹 “ 𝐶) = ∅)

Proof of Theorem fnimadisj

Step	Hyp	Ref	Expression
1		fndm 5334	. . . . 5 ⊢ (𝐹 Fn 𝐴 → dom 𝐹 = 𝐴)
2	1	ineq1d 3350	. . . 4 ⊢ (𝐹 Fn 𝐴 → (dom 𝐹 ∩ 𝐶) = (𝐴 ∩ 𝐶))
3	2	eqeq1d 2198	. . 3 ⊢ (𝐹 Fn 𝐴 → ((dom 𝐹 ∩ 𝐶) = ∅ ↔ (𝐴 ∩ 𝐶) = ∅))
4	3	biimpar 297	. 2 ⊢ ((𝐹 Fn 𝐴 ∧ (𝐴 ∩ 𝐶) = ∅) → (dom 𝐹 ∩ 𝐶) = ∅)
5		imadisj 5008	. 2 ⊢ ((𝐹 “ 𝐶) = ∅ ↔ (dom 𝐹 ∩ 𝐶) = ∅)
6	4, 5	sylibr 134	1 ⊢ ((𝐹 Fn 𝐴 ∧ (𝐴 ∩ 𝐶) = ∅) → (𝐹 “ 𝐶) = ∅)

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1364   ∩ cin 3143  ∅c0 3437  dom cdm 4644   “ cima 4647   Fn wfn 5230

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-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-pow 4192  ax-pr 4227

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-rex 2474  df-v 2754  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-nul 3438  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-br 4019  df-opab 4080  df-xp 4650  df-cnv 4652  df-dm 4654  df-rn 4655  df-res 4656  df-ima 4657  df-fn 5238

This theorem is referenced by: (None)