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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 5701 | . . 3 ⊢ (𝐴 “ 𝐵) = ran (𝐴 ↾ 𝐵) | |
2 | 1 | eqeq1i 2739 | . 2 ⊢ ((𝐴 “ 𝐵) = ∅ ↔ ran (𝐴 ↾ 𝐵) = ∅) |
3 | dm0rn0 5937 | . 2 ⊢ (dom (𝐴 ↾ 𝐵) = ∅ ↔ ran (𝐴 ↾ 𝐵) = ∅) | |
4 | dmres 6031 | . . . 4 ⊢ dom (𝐴 ↾ 𝐵) = (𝐵 ∩ dom 𝐴) | |
5 | incom 4216 | . . . 4 ⊢ (𝐵 ∩ dom 𝐴) = (dom 𝐴 ∩ 𝐵) | |
6 | 4, 5 | eqtri 2762 | . . 3 ⊢ dom (𝐴 ↾ 𝐵) = (dom 𝐴 ∩ 𝐵) |
7 | 6 | eqeq1i 2739 | . 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 1536 ∩ cin 3961 ∅c0 4338 dom cdm 5688 ran crn 5689 ↾ cres 5690 “ cima 5691 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-sep 5301 ax-nul 5311 ax-pr 5437 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-clab 2712 df-cleq 2726 df-clel 2813 df-ral 3059 df-rex 3068 df-rab 3433 df-v 3479 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-nul 4339 df-if 4531 df-sn 4631 df-pr 4633 df-op 4637 df-br 5148 df-opab 5210 df-xp 5694 df-cnv 5696 df-dm 5698 df-rn 5699 df-res 5700 df-ima 5701 |
This theorem is referenced by: imadisjlnd 6100 ndmima 6123 fnimadisj 6700 fnimaeq0 6701 fimacnvdisj 6786 frxp2 8167 frxp3 8174 acndom2 10091 isf34lem5 10415 isf34lem7 10416 isf34lem6 10417 limsupgre 15513 isercolllem3 15699 pf1rcl 22368 cnconn 23445 1stcfb 23468 xkohaus 23676 qtopeu 23739 fbasrn 23907 mbflimsup 25714 preiman0 32724 eulerpartlemt 34352 erdszelem5 35179 fnwe2lem2 43039 imadisjld 44149 |
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