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Mirrors > Home > MPE Home > Th. List > imadmres | Structured version Visualization version GIF version |
Description: The image of the domain of a restriction. (Contributed by NM, 8-Apr-2007.) |
Ref | Expression |
---|---|
imadmres | ⊢ (𝐴 “ dom (𝐴 ↾ 𝐵)) = (𝐴 “ 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | resdmres 6254 | . . 3 ⊢ (𝐴 ↾ dom (𝐴 ↾ 𝐵)) = (𝐴 ↾ 𝐵) | |
2 | 1 | rneqi 5951 | . 2 ⊢ ran (𝐴 ↾ dom (𝐴 ↾ 𝐵)) = ran (𝐴 ↾ 𝐵) |
3 | df-ima 5702 | . 2 ⊢ (𝐴 “ dom (𝐴 ↾ 𝐵)) = ran (𝐴 ↾ dom (𝐴 ↾ 𝐵)) | |
4 | df-ima 5702 | . 2 ⊢ (𝐴 “ 𝐵) = ran (𝐴 ↾ 𝐵) | |
5 | 2, 3, 4 | 3eqtr4i 2773 | 1 ⊢ (𝐴 “ dom (𝐴 ↾ 𝐵)) = (𝐴 “ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1537 dom cdm 5689 ran crn 5690 ↾ cres 5691 “ cima 5692 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-br 5149 df-opab 5211 df-xp 5695 df-rel 5696 df-cnv 5697 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 |
This theorem is referenced by: ssimaex 6994 fnwelem 8155 imafi 9351 r0weon 10050 limsupgle 15510 kqdisj 23756 isubgruhgr 47792 |
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