Theorem fnimaeq0 5375

Description: Images under a function never map nonempty sets to empty sets. (Contributed by Stefan O'Rear, 21-Jan-2015.)

Assertion

Ref	Expression
fnimaeq0	⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ⊆ 𝐴) → ((𝐹 “ 𝐵) = ∅ ↔ 𝐵 = ∅))

Proof of Theorem fnimaeq0

Step	Hyp	Ref	Expression
1		imadisj 5027	. 2 ⊢ ((𝐹 “ 𝐵) = ∅ ↔ (dom 𝐹 ∩ 𝐵) = ∅)
2		incom 3351	. . . 4 ⊢ (dom 𝐹 ∩ 𝐵) = (𝐵 ∩ dom 𝐹)
3		fndm 5353	. . . . . . 7 ⊢ (𝐹 Fn 𝐴 → dom 𝐹 = 𝐴)
4	3	sseq2d 3209	. . . . . 6 ⊢ (𝐹 Fn 𝐴 → (𝐵 ⊆ dom 𝐹 ↔ 𝐵 ⊆ 𝐴))
5	4	biimpar 297	. . . . 5 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ⊆ 𝐴) → 𝐵 ⊆ dom 𝐹)
6		df-ss 3166	. . . . 5 ⊢ (𝐵 ⊆ dom 𝐹 ↔ (𝐵 ∩ dom 𝐹) = 𝐵)
7	5, 6	sylib 122	. . . 4 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ⊆ 𝐴) → (𝐵 ∩ dom 𝐹) = 𝐵)
8	2, 7	eqtrid 2238	. . 3 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ⊆ 𝐴) → (dom 𝐹 ∩ 𝐵) = 𝐵)
9	8	eqeq1d 2202	. 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ⊆ 𝐴) → ((dom 𝐹 ∩ 𝐵) = ∅ ↔ 𝐵 = ∅))
10	1, 9	bitrid 192	1 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ⊆ 𝐴) → ((𝐹 “ 𝐵) = ∅ ↔ 𝐵 = ∅))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1364   ∩ cin 3152   ⊆ wss 3153  ∅c0 3446  dom cdm 4659   “ cima 4662   Fn wfn 5249

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 2167  ax-ext 2175  ax-sep 4147  ax-pow 4203  ax-pr 4238

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 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-v 2762  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3447  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-br 4030  df-opab 4091  df-xp 4665  df-cnv 4667  df-dm 4669  df-rn 4670  df-res 4671  df-ima 4672  df-fn 5257

This theorem is referenced by: (None)