Theorem fnimadisj 5318

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 5297	. . . . 5 ⊢ (𝐹 Fn 𝐴 → dom 𝐹 = 𝐴)
2	1	ineq1d 3327	. . . 4 ⊢ (𝐹 Fn 𝐴 → (dom 𝐹 ∩ 𝐶) = (𝐴 ∩ 𝐶))
3	2	eqeq1d 2179	. . 3 ⊢ (𝐹 Fn 𝐴 → ((dom 𝐹 ∩ 𝐶) = ∅ ↔ (𝐴 ∩ 𝐶) = ∅))
4	3	biimpar 295	. 2 ⊢ ((𝐹 Fn 𝐴 ∧ (𝐴 ∩ 𝐶) = ∅) → (dom 𝐹 ∩ 𝐶) = ∅)
5		imadisj 4973	. 2 ⊢ ((𝐹 “ 𝐶) = ∅ ↔ (dom 𝐹 ∩ 𝐶) = ∅)
6	4, 5	sylibr 133	1 ⊢ ((𝐹 Fn 𝐴 ∧ (𝐴 ∩ 𝐶) = ∅) → (𝐹 “ 𝐶) = ∅)

Syntax hints:   → wi 4   ∧ wa 103   = wceq 1348   ∩ cin 3120  ∅c0 3414  dom cdm 4611   “ cima 4614   Fn wfn 5193

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-v 2732  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-br 3990  df-opab 4051  df-xp 4617  df-cnv 4619  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-fn 5201

This theorem is referenced by: (None)