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| Mirrors > Home > MPE Home > Th. List > imadisj | Structured version Visualization version GIF version | ||
| Description: A class whose image under another is empty is disjoint with the other's domain. (Contributed by FL, 24-Jan-2007.) |
| Ref | Expression |
|---|---|
| imadisj | ⊢ ((𝐴 “ 𝐵) = ∅ ↔ (dom 𝐴 ∩ 𝐵) = ∅) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | df-ima 5698 | . . 3 ⊢ (𝐴 “ 𝐵) = ran (𝐴 ↾ 𝐵) | |
| 2 | 1 | eqeq1i 2742 | . 2 ⊢ ((𝐴 “ 𝐵) = ∅ ↔ ran (𝐴 ↾ 𝐵) = ∅) |
| 3 | dm0rn0 5935 | . 2 ⊢ (dom (𝐴 ↾ 𝐵) = ∅ ↔ ran (𝐴 ↾ 𝐵) = ∅) | |
| 4 | dmres 6030 | . . . 4 ⊢ dom (𝐴 ↾ 𝐵) = (𝐵 ∩ dom 𝐴) | |
| 5 | incom 4209 | . . . 4 ⊢ (𝐵 ∩ dom 𝐴) = (dom 𝐴 ∩ 𝐵) | |
| 6 | 4, 5 | eqtri 2765 | . . 3 ⊢ dom (𝐴 ↾ 𝐵) = (dom 𝐴 ∩ 𝐵) |
| 7 | 6 | eqeq1i 2742 | . 2 ⊢ (dom (𝐴 ↾ 𝐵) = ∅ ↔ (dom 𝐴 ∩ 𝐵) = ∅) |
| 8 | 2, 3, 7 | 3bitr2i 299 | 1 ⊢ ((𝐴 “ 𝐵) = ∅ ↔ (dom 𝐴 ∩ 𝐵) = ∅) |
| Colors of variables: wff setvar class |
| Syntax hints: ↔ wb 206 = wceq 1540 ∩ cin 3950 ∅c0 4333 dom cdm 5685 ran crn 5686 ↾ cres 5687 “ cima 5688 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-br 5144 df-opab 5206 df-xp 5691 df-cnv 5693 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 |
| This theorem is referenced by: imadisjlnd 6099 ndmima 6121 fnimadisj 6700 fnimaeq0 6701 fimacnvdisj 6786 frxp2 8169 frxp3 8176 acndom2 10094 isf34lem5 10418 isf34lem7 10419 isf34lem6 10420 limsupgre 15517 isercolllem3 15703 pf1rcl 22353 cnconn 23430 1stcfb 23453 xkohaus 23661 qtopeu 23724 fbasrn 23892 mbflimsup 25701 preiman0 32719 eulerpartlemt 34373 erdszelem5 35200 fnwe2lem2 43063 imadisjld 44173 |
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