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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 6055 | . . 3 ⊢ (◡◡𝐴 ↾ 𝐵) = (𝐴 ↾ 𝐵) | |
2 | 1 | rneqi 5801 | . 2 ⊢ ran (◡◡𝐴 ↾ 𝐵) = ran (𝐴 ↾ 𝐵) |
3 | df-ima 5562 | . 2 ⊢ (◡◡𝐴 “ 𝐵) = ran (◡◡𝐴 ↾ 𝐵) | |
4 | df-ima 5562 | . 2 ⊢ (𝐴 “ 𝐵) = ran (𝐴 ↾ 𝐵) | |
5 | 2, 3, 4 | 3eqtr4i 2854 | 1 ⊢ (◡◡𝐴 “ 𝐵) = (𝐴 “ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1528 ◡ccnv 5548 ran crn 5550 ↾ cres 5551 “ cima 5552 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2793 ax-sep 5195 ax-nul 5202 ax-pr 5321 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-rab 3147 df-v 3497 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4466 df-sn 4560 df-pr 4562 df-op 4566 df-br 5059 df-opab 5121 df-xp 5555 df-rel 5556 df-cnv 5557 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 |
This theorem is referenced by: curry1 7790 curry2 7793 fnwelem 7816 fpwwe2lem6 10046 fpwwe2lem9 10049 eqglact 18271 hmeoima 22303 hmeocld 22305 hmeocls 22306 hmeontr 22307 reghmph 22331 qtopf1 22354 tgpconncompeqg 22649 imasf1obl 23027 mbfimaopnlem 24185 hmeoclda 33579 |
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