Theorem fnimaeq0 4963

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 4630	. 2 ⊢ ((𝐹 “ B) = ∅ ↔ (dom 𝐹 ∩ B) = ∅)
2		incom 3123	. . . 4 ⊢ (dom 𝐹 ∩ B) = (B ∩ dom 𝐹)
3		fndm 4941	. . . . . . 7 ⊢ (𝐹 Fn A → dom 𝐹 = A)
4	3	sseq2d 2967	. . . . . 6 ⊢ (𝐹 Fn A → (B ⊆ dom 𝐹 ↔ B ⊆ A))
5	4	biimpar 281	. . . . 5 ⊢ ((𝐹 Fn A ∧ B ⊆ A) → B ⊆ dom 𝐹)
6		df-ss 2925	. . . . 5 ⊢ (B ⊆ dom 𝐹 ↔ (B ∩ dom 𝐹) = B)
7	5, 6	sylib 127	. . . 4 ⊢ ((𝐹 Fn A ∧ B ⊆ A) → (B ∩ dom 𝐹) = B)
8	2, 7	syl5eq 2081	. . 3 ⊢ ((𝐹 Fn A ∧ B ⊆ A) → (dom 𝐹 ∩ B) = B)
9	8	eqeq1d 2045	. 2 ⊢ ((𝐹 Fn A ∧ B ⊆ A) → ((dom 𝐹 ∩ B) = ∅ ↔ B = ∅))
10	1, 9	syl5bb 181	1 ⊢ ((𝐹 Fn A ∧ B ⊆ A) → ((𝐹 “ B) = ∅ ↔ B = ∅))

Syntax hints:   → wi 4   ∧ wa 97   ↔ wb 98   = wceq 1242   ∩ cin 2910   ⊆ wss 2911  ∅c0 3218  dom cdm 4288   “ cima 4291   Fn wfn 4840

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935

This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-fal 1248  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-dif 2914  df-un 2916  df-in 2918  df-ss 2925  df-nul 3219  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-br 3756  df-opab 3810  df-xp 4294  df-cnv 4296  df-dm 4298  df-rn 4299  df-res 4300  df-ima 4301  df-fn 4848

This theorem is referenced by: (None)