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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 6046 | . . 3 ⊢ (◡◡𝐴 ↾ 𝐵) = (𝐴 ↾ 𝐵) | |
2 | 1 | rneqi 5790 | . 2 ⊢ ran (◡◡𝐴 ↾ 𝐵) = ran (𝐴 ↾ 𝐵) |
3 | df-ima 5548 | . 2 ⊢ (◡◡𝐴 “ 𝐵) = ran (◡◡𝐴 ↾ 𝐵) | |
4 | df-ima 5548 | . 2 ⊢ (𝐴 “ 𝐵) = ran (𝐴 ↾ 𝐵) | |
5 | 2, 3, 4 | 3eqtr4i 2772 | 1 ⊢ (◡◡𝐴 “ 𝐵) = (𝐴 “ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1542 ◡ccnv 5534 ran crn 5536 ↾ cres 5537 “ cima 5538 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-12 2179 ax-ext 2711 ax-sep 5177 ax-nul 5184 ax-pr 5306 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2075 df-clab 2718 df-cleq 2731 df-clel 2812 df-rab 3063 df-v 3402 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-nul 4222 df-if 4425 df-sn 4527 df-pr 4529 df-op 4533 df-br 5041 df-opab 5103 df-xp 5541 df-rel 5542 df-cnv 5543 df-dm 5545 df-rn 5546 df-res 5547 df-ima 5548 |
This theorem is referenced by: curry1 7838 curry2 7841 fnwelem 7864 fpwwe2lem5 10148 fpwwe2lem8 10151 eqglact 18462 hmeoima 22529 hmeocld 22531 hmeocls 22532 hmeontr 22533 reghmph 22557 qtopf1 22580 tgpconncompeqg 22876 imasf1obl 23254 mbfimaopnlem 24420 hmeoclda 34178 |
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