Theorem fnimadisj 5406

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 5382	. . . . 5 ⊢ (𝐹 Fn 𝐴 → dom 𝐹 = 𝐴)
2	1	ineq1d 3377	. . . 4 ⊢ (𝐹 Fn 𝐴 → (dom 𝐹 ∩ 𝐶) = (𝐴 ∩ 𝐶))
3	2	eqeq1d 2215	. . 3 ⊢ (𝐹 Fn 𝐴 → ((dom 𝐹 ∩ 𝐶) = ∅ ↔ (𝐴 ∩ 𝐶) = ∅))
4	3	biimpar 297	. 2 ⊢ ((𝐹 Fn 𝐴 ∧ (𝐴 ∩ 𝐶) = ∅) → (dom 𝐹 ∩ 𝐶) = ∅)
5		imadisj 5053	. 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 1373   ∩ cin 3169  ∅c0 3464  dom cdm 4683   “ cima 4686   Fn wfn 5275

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-14 2180  ax-ext 2188  ax-sep 4170  ax-pow 4226  ax-pr 4261

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-ral 2490  df-rex 2491  df-v 2775  df-dif 3172  df-un 3174  df-in 3176  df-ss 3183  df-nul 3465  df-pw 3623  df-sn 3644  df-pr 3645  df-op 3647  df-br 4052  df-opab 4114  df-xp 4689  df-cnv 4691  df-dm 4693  df-rn 4694  df-res 4695  df-ima 4696  df-fn 5283

This theorem is referenced by: (None)