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| Mirrors > Home > MPE Home > Th. List > imacnvcnv | Structured version Visualization version GIF version | ||
| Description: The image of the double converse of a class. (Contributed by NM, 8-Apr-2007.) | 
| Ref | Expression | 
|---|---|
| imacnvcnv | ⊢ (◡◡𝐴 “ 𝐵) = (𝐴 “ 𝐵) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | rescnvcnv 6223 | . . 3 ⊢ (◡◡𝐴 ↾ 𝐵) = (𝐴 ↾ 𝐵) | |
| 2 | 1 | rneqi 5947 | . 2 ⊢ ran (◡◡𝐴 ↾ 𝐵) = ran (𝐴 ↾ 𝐵) | 
| 3 | df-ima 5697 | . 2 ⊢ (◡◡𝐴 “ 𝐵) = ran (◡◡𝐴 ↾ 𝐵) | |
| 4 | df-ima 5697 | . 2 ⊢ (𝐴 “ 𝐵) = ran (𝐴 ↾ 𝐵) | |
| 5 | 2, 3, 4 | 3eqtr4i 2774 | 1 ⊢ (◡◡𝐴 “ 𝐵) = (𝐴 “ 𝐵) | 
| Colors of variables: wff setvar class | 
| Syntax hints: = wceq 1539 ◡ccnv 5683 ran crn 5685 ↾ cres 5686 “ cima 5687 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-ext 2707 ax-sep 5295 ax-nul 5305 ax-pr 5431 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-sb 2064 df-clab 2714 df-cleq 2728 df-clel 2815 df-ral 3061 df-rex 3070 df-rab 3436 df-v 3481 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-nul 4333 df-if 4525 df-sn 4626 df-pr 4628 df-op 4632 df-br 5143 df-opab 5205 df-xp 5690 df-rel 5691 df-cnv 5692 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 | 
| This theorem is referenced by: curry1 8130 curry2 8133 fnwelem 8157 fpwwe2lem5 10676 fpwwe2lem8 10679 eqglact 19198 hmeoima 23774 hmeocld 23776 hmeocls 23777 hmeontr 23778 reghmph 23802 qtopf1 23825 tgpconncompeqg 24121 imasf1obl 24502 mbfimaopnlem 25691 hmeoclda 36335 | 
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