Theorem fnimaeq0 6665

Description: Images under a function never map nonempty sets to empty sets. EDITORIAL: usable in fnwe2lem2 43892. (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 6076	. 2 ⊢ ((𝐹 “ 𝐵) = ∅ ↔ (dom 𝐹 ∩ 𝐵) = ∅)
2		incom 4155	. . . 4 ⊢ (dom 𝐹 ∩ 𝐵) = (𝐵 ∩ dom 𝐹)
3		fndm 6635	. . . . . . 7 ⊢ (𝐹 Fn 𝐴 → dom 𝐹 = 𝐴)
4	3	sseq2d 3963	. . . . . 6 ⊢ (𝐹 Fn 𝐴 → (𝐵 ⊆ dom 𝐹 ↔ 𝐵 ⊆ 𝐴))
5	4	biimpar 483	. . . . 5 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ⊆ 𝐴) → 𝐵 ⊆ dom 𝐹)
6		dfss2 3917	. . . . 5 ⊢ (𝐵 ⊆ dom 𝐹 ↔ (𝐵 ∩ dom 𝐹) = 𝐵)
7	5, 6	sylib 221	. . . 4 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ⊆ 𝐴) → (𝐵 ∩ dom 𝐹) = 𝐵)
8	2, 7	eqtrid 2807	. . 3 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ⊆ 𝐴) → (dom 𝐹 ∩ 𝐵) = 𝐵)
9	8	eqeq1d 2762	. 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ⊆ 𝐴) → ((dom 𝐹 ∩ 𝐵) = ∅ ↔ 𝐵 = ∅))
10	1, 9	bitrid 286	1 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ⊆ 𝐴) → ((𝐹 “ 𝐵) = ∅ ↔ 𝐵 = ∅))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∩ cin 3898   ⊆ wss 3899  ∅c0 4279  dom cdm 5655   “ cima 5658   Fn wfn 6528

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2732  ax-sep 5251  ax-pr 5398

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2739  df-cleq 2752  df-clel 2835  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-br 5104  df-opab 5168  df-xp 5661  df-cnv 5663  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-fn 6536

This theorem is used by:  ipodrsima  18629  mdegldg  26291  ig1peu  26400  ig1pdvds  26405  dimval  34111  dimvalfi  34112  nummin  35598  noinfepfnregs  35658  regsfromunir1  37159  aks6d1c6lem3  43038  kelac1  43904