Theorem fnimaeq0 5020

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 4687	. 2 ⊢ ((𝐹 “ 𝐵) = ∅ ↔ (dom 𝐹 ∩ 𝐵) = ∅)
2		incom 3129	. . . 4 ⊢ (dom 𝐹 ∩ 𝐵) = (𝐵 ∩ dom 𝐹)
3		fndm 4998	. . . . . . 7 ⊢ (𝐹 Fn 𝐴 → dom 𝐹 = 𝐴)
4	3	sseq2d 2973	. . . . . 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 2931	. . . . 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 2084	. . 3 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ⊆ 𝐴) → (dom 𝐹 ∩ 𝐵) = 𝐵)
9	8	eqeq1d 2048	. 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 1243   ∩ cin 2916   ⊆ wss 2917  ∅c0 3224  dom cdm 4345   “ cima 4348   Fn wfn 4897

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 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944

This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-fal 1249  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-v 2559  df-dif 2920  df-un 2922  df-in 2924  df-ss 2931  df-nul 3225  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-br 3765  df-opab 3819  df-xp 4351  df-cnv 4353  df-dm 4355  df-rn 4356  df-res 4357  df-ima 4358  df-fn 4905

This theorem is referenced by: (None)