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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 5564 | . . 3 ⊢ (𝐴 “ 𝐵) = ran (𝐴 ↾ 𝐵) | |
2 | 1 | eqeq1i 2742 | . 2 ⊢ ((𝐴 “ 𝐵) = ∅ ↔ ran (𝐴 ↾ 𝐵) = ∅) |
3 | dm0rn0 5794 | . 2 ⊢ (dom (𝐴 ↾ 𝐵) = ∅ ↔ ran (𝐴 ↾ 𝐵) = ∅) | |
4 | dmres 5873 | . . . 4 ⊢ dom (𝐴 ↾ 𝐵) = (𝐵 ∩ dom 𝐴) | |
5 | incom 4115 | . . . 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 302 | 1 ⊢ ((𝐴 “ 𝐵) = ∅ ↔ (dom 𝐴 ∩ 𝐵) = ∅) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 209 = wceq 1543 ∩ cin 3865 ∅c0 4237 dom cdm 5551 ran crn 5552 ↾ cres 5553 “ cima 5554 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2708 ax-sep 5192 ax-nul 5199 ax-pr 5322 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-clab 2715 df-cleq 2729 df-clel 2816 df-ral 3066 df-rex 3067 df-rab 3070 df-v 3410 df-dif 3869 df-un 3871 df-in 3873 df-nul 4238 df-if 4440 df-sn 4542 df-pr 4544 df-op 4548 df-br 5054 df-opab 5116 df-xp 5557 df-cnv 5559 df-dm 5561 df-rn 5562 df-res 5563 df-ima 5564 |
This theorem is referenced by: ndmima 5971 fnimadisj 6510 fnimaeq0 6511 fimacnvdisj 6597 acndom2 9668 isf34lem5 9992 isf34lem7 9993 isf34lem6 9994 limsupgre 15042 isercolllem3 15230 pf1rcl 21265 cnconn 22319 1stcfb 22342 xkohaus 22550 qtopeu 22613 fbasrn 22781 mbflimsup 24563 preiman0 30762 eulerpartlemt 32050 erdszelem5 32870 frxp2 33528 frxp3 33534 fnwe2lem2 40579 imadisjld 41447 imadisjlnd 41448 |
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