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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 5561 | . . 3 ⊢ (𝐴 “ 𝐵) = ran (𝐴 ↾ 𝐵) | |
2 | 1 | eqeq1i 2825 | . 2 ⊢ ((𝐴 “ 𝐵) = ∅ ↔ ran (𝐴 ↾ 𝐵) = ∅) |
3 | dm0rn0 5788 | . 2 ⊢ (dom (𝐴 ↾ 𝐵) = ∅ ↔ ran (𝐴 ↾ 𝐵) = ∅) | |
4 | dmres 5868 | . . . 4 ⊢ dom (𝐴 ↾ 𝐵) = (𝐵 ∩ dom 𝐴) | |
5 | incom 4171 | . . . 4 ⊢ (𝐵 ∩ dom 𝐴) = (dom 𝐴 ∩ 𝐵) | |
6 | 4, 5 | eqtri 2843 | . . 3 ⊢ dom (𝐴 ↾ 𝐵) = (dom 𝐴 ∩ 𝐵) |
7 | 6 | eqeq1i 2825 | . 2 ⊢ (dom (𝐴 ↾ 𝐵) = ∅ ↔ (dom 𝐴 ∩ 𝐵) = ∅) |
8 | 2, 3, 7 | 3bitr2i 301 | 1 ⊢ ((𝐴 “ 𝐵) = ∅ ↔ (dom 𝐴 ∩ 𝐵) = ∅) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 208 = wceq 1536 ∩ cin 3928 ∅c0 4284 dom cdm 5548 ran crn 5549 ↾ cres 5550 “ cima 5551 |
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 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2792 ax-sep 5196 ax-nul 5203 ax-pr 5323 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1084 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2799 df-cleq 2813 df-clel 2892 df-nfc 2962 df-ral 3142 df-rex 3143 df-rab 3146 df-v 3493 df-dif 3932 df-un 3934 df-in 3936 df-ss 3945 df-nul 4285 df-if 4461 df-sn 4561 df-pr 4563 df-op 4567 df-br 5060 df-opab 5122 df-xp 5554 df-cnv 5556 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 |
This theorem is referenced by: ndmima 5959 fnimadisj 6473 fnimaeq0 6474 fimacnvdisj 6550 acndom2 9473 isf34lem5 9793 isf34lem7 9794 isf34lem6 9795 limsupgre 14831 isercolllem3 15016 pf1rcl 20505 cnconn 22023 1stcfb 22046 xkohaus 22254 qtopeu 22317 fbasrn 22485 mbflimsup 24260 eulerpartlemt 31648 erdszelem5 32461 fnwe2lem2 39728 imadisjld 40585 imadisjlnd 40586 |
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