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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 6028 | . . 3 ⊢ (◡◡𝐴 ↾ 𝐵) = (𝐴 ↾ 𝐵) | |
2 | 1 | rneqi 5771 | . 2 ⊢ ran (◡◡𝐴 ↾ 𝐵) = ran (𝐴 ↾ 𝐵) |
3 | df-ima 5532 | . 2 ⊢ (◡◡𝐴 “ 𝐵) = ran (◡◡𝐴 ↾ 𝐵) | |
4 | df-ima 5532 | . 2 ⊢ (𝐴 “ 𝐵) = ran (𝐴 ↾ 𝐵) | |
5 | 2, 3, 4 | 3eqtr4i 2831 | 1 ⊢ (◡◡𝐴 “ 𝐵) = (𝐴 “ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1538 ◡ccnv 5518 ran crn 5520 ↾ cres 5521 “ cima 5522 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pr 5295 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-rab 3115 df-v 3443 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-sn 4526 df-pr 4528 df-op 4532 df-br 5031 df-opab 5093 df-xp 5525 df-rel 5526 df-cnv 5527 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 |
This theorem is referenced by: curry1 7782 curry2 7785 fnwelem 7808 fpwwe2lem6 10046 fpwwe2lem9 10049 eqglact 18323 hmeoima 22370 hmeocld 22372 hmeocls 22373 hmeontr 22374 reghmph 22398 qtopf1 22421 tgpconncompeqg 22717 imasf1obl 23095 mbfimaopnlem 24259 hmeoclda 33794 |
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