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Mirrors > Home > MPE Home > Th. List > fnimaeq0 | Structured version Visualization version GIF version |
Description: Images under a function never map nonempty sets to empty sets. EDITORIAL: usable in fnwe2lem2 42258. (Contributed by Stefan O'Rear, 21-Jan-2015.) |
Ref | Expression |
---|---|
fnimaeq0 | ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ⊆ 𝐴) → ((𝐹 “ 𝐵) = ∅ ↔ 𝐵 = ∅)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | imadisj 6079 | . 2 ⊢ ((𝐹 “ 𝐵) = ∅ ↔ (dom 𝐹 ∩ 𝐵) = ∅) | |
2 | incom 4201 | . . . 4 ⊢ (dom 𝐹 ∩ 𝐵) = (𝐵 ∩ dom 𝐹) | |
3 | fndm 6652 | . . . . . . 7 ⊢ (𝐹 Fn 𝐴 → dom 𝐹 = 𝐴) | |
4 | 3 | sseq2d 4014 | . . . . . 6 ⊢ (𝐹 Fn 𝐴 → (𝐵 ⊆ dom 𝐹 ↔ 𝐵 ⊆ 𝐴)) |
5 | 4 | biimpar 477 | . . . . 5 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ⊆ 𝐴) → 𝐵 ⊆ dom 𝐹) |
6 | df-ss 3965 | . . . . 5 ⊢ (𝐵 ⊆ dom 𝐹 ↔ (𝐵 ∩ dom 𝐹) = 𝐵) | |
7 | 5, 6 | sylib 217 | . . . 4 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ⊆ 𝐴) → (𝐵 ∩ dom 𝐹) = 𝐵) |
8 | 2, 7 | eqtrid 2783 | . . 3 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ⊆ 𝐴) → (dom 𝐹 ∩ 𝐵) = 𝐵) |
9 | 8 | eqeq1d 2733 | . 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ⊆ 𝐴) → ((dom 𝐹 ∩ 𝐵) = ∅ ↔ 𝐵 = ∅)) |
10 | 1, 9 | bitrid 283 | 1 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ⊆ 𝐴) → ((𝐹 “ 𝐵) = ∅ ↔ 𝐵 = ∅)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 = wceq 1540 ∩ cin 3947 ⊆ wss 3948 ∅c0 4322 dom cdm 5676 “ cima 5679 Fn wfn 6538 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-sep 5299 ax-nul 5306 ax-pr 5427 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-clab 2709 df-cleq 2723 df-clel 2809 df-ral 3061 df-rex 3070 df-rab 3432 df-v 3475 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-sn 4629 df-pr 4631 df-op 4635 df-br 5149 df-opab 5211 df-xp 5682 df-cnv 5684 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-fn 6546 |
This theorem is referenced by: ipodrsima 18504 mdegldg 25922 ig1peu 26027 ig1pdvds 26032 dimval 33140 dimvalfi 33141 nummin 34559 kelac1 42270 |
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