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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 6107 | . . 3 ⊢ (◡◡𝐴 ↾ 𝐵) = (𝐴 ↾ 𝐵) | |
2 | 1 | rneqi 5846 | . 2 ⊢ ran (◡◡𝐴 ↾ 𝐵) = ran (𝐴 ↾ 𝐵) |
3 | df-ima 5602 | . 2 ⊢ (◡◡𝐴 “ 𝐵) = ran (◡◡𝐴 ↾ 𝐵) | |
4 | df-ima 5602 | . 2 ⊢ (𝐴 “ 𝐵) = ran (𝐴 ↾ 𝐵) | |
5 | 2, 3, 4 | 3eqtr4i 2776 | 1 ⊢ (◡◡𝐴 “ 𝐵) = (𝐴 “ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1539 ◡ccnv 5588 ran crn 5590 ↾ cres 5591 “ cima 5592 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-12 2171 ax-ext 2709 ax-sep 5223 ax-nul 5230 ax-pr 5352 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-clab 2716 df-cleq 2730 df-clel 2816 df-rab 3073 df-v 3434 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-nul 4257 df-if 4460 df-sn 4562 df-pr 4564 df-op 4568 df-br 5075 df-opab 5137 df-xp 5595 df-rel 5596 df-cnv 5597 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 |
This theorem is referenced by: curry1 7944 curry2 7947 fnwelem 7972 fpwwe2lem5 10391 fpwwe2lem8 10394 eqglact 18807 hmeoima 22916 hmeocld 22918 hmeocls 22919 hmeontr 22920 reghmph 22944 qtopf1 22967 tgpconncompeqg 23263 imasf1obl 23644 mbfimaopnlem 24819 hmeoclda 34522 |
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