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| 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 6087 | . 2 ⊢ (◡𝐴 “ dom ◡𝐴) = ran ◡𝐴 | |
| 2 | df-rn 5695 | . . 3 ⊢ ran 𝐴 = dom ◡𝐴 | |
| 3 | 2 | imaeq2i 6075 | . 2 ⊢ (◡𝐴 “ ran 𝐴) = (◡𝐴 “ dom ◡𝐴) | 
| 4 | dfdm4 5905 | . 2 ⊢ dom 𝐴 = ran ◡𝐴 | |
| 5 | 1, 3, 4 | 3eqtr4i 2774 | 1 ⊢ (◡𝐴 “ ran 𝐴) = dom 𝐴 | 
| Colors of variables: wff setvar class | 
| Syntax hints: = wceq 1539 ◡ccnv 5683 dom cdm 5684 ran crn 5685 “ cima 5687 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-ext 2707 ax-sep 5295 ax-nul 5305 ax-pr 5431 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-sb 2064 df-clab 2714 df-cleq 2728 df-clel 2815 df-ral 3061 df-rex 3070 df-rab 3436 df-v 3481 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-nul 4333 df-if 4525 df-sn 4626 df-pr 4628 df-op 4632 df-br 5143 df-opab 5205 df-xp 5690 df-cnv 5692 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 | 
| This theorem is referenced by: cnvimassrndm 6171 focnvimacdmdm 6831 cnvimainrn 7086 cnrest2 23295 mbfconstlem 25663 i1fima 25714 i1fima2 25715 i1fd 25717 i1f0rn 25718 itg1addlem5 25736 fcoinver 32618 supppreima 32701 sibfof 34343 itg2addnclem 37679 itg2addnclem2 37680 ftc1anclem6 37706 f1cof1blem 47091 f1cof1b 47094 fnfocofob 47096 | 
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