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Mirrors > Home > MPE Home > Th. List > cnvimarndm | Structured version Visualization version GIF version |
Description: The preimage of the range of a class is the domain of the class. (Contributed by Jeff Hankins, 15-Jul-2009.) |
Ref | Expression |
---|---|
cnvimarndm | ⊢ (◡𝐴 “ ran 𝐴) = dom 𝐴 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | imadmrn 5395 | . 2 ⊢ (◡𝐴 “ dom ◡𝐴) = ran ◡𝐴 | |
2 | df-rn 5049 | . . 3 ⊢ ran 𝐴 = dom ◡𝐴 | |
3 | 2 | imaeq2i 5383 | . 2 ⊢ (◡𝐴 “ ran 𝐴) = (◡𝐴 “ dom ◡𝐴) |
4 | dfdm4 5238 | . 2 ⊢ dom 𝐴 = ran ◡𝐴 | |
5 | 1, 3, 4 | 3eqtr4i 2642 | 1 ⊢ (◡𝐴 “ ran 𝐴) = dom 𝐴 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1475 ◡ccnv 5037 dom cdm 5038 ran crn 5039 “ cima 5041 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pr 4833 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-br 4584 df-opab 4644 df-xp 5044 df-cnv 5046 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 |
This theorem is referenced by: cnrest2 20900 mbfconstlem 23202 i1fima 23251 i1fima2 23252 i1fd 23254 i1f0rn 23255 itg1addlem5 23273 fcoinver 28798 sibfof 29729 itg2addnclem 32631 itg2addnclem2 32632 ftc1anclem6 32660 |
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