Theorem cnvimarndm 6087

Description: The preimage of the range of a class is the domain of the class. (Contributed by Jeff Hankins, 15-Jul-2009.)

Assertion

Ref	Expression
cnvimarndm	⊢ (◡𝐴 “ ran 𝐴) = dom 𝐴

Proof of Theorem cnvimarndm

Step	Hyp	Ref	Expression
1		imadmrn 6074	. 2 ⊢ (◡𝐴 “ dom ◡𝐴) = ran ◡𝐴
2		df-rn 5674	. . 3 ⊢ ran 𝐴 = dom ◡𝐴
3	2	imaeq2i 6062	. 2 ⊢ (◡𝐴 “ ran 𝐴) = (◡𝐴 “ dom ◡𝐴)
4		dfdm4 5887	. 2 ⊢ dom 𝐴 = ran ◡𝐴
5	1, 3, 4	3eqtr4i 2798	1 ⊢ (◡𝐴 “ ran 𝐴) = dom 𝐴

This proof depends on syntax axioms:   = wceq 1570  ^◡ccnv 5662  dom cdm 5663  ran crn 5664   “ cima 5666

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-ext 2737  ax-sep 5259  ax-pr 5406

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2744  df-cleq 2757  df-clel 2840  df-ral 3082  df-rex 3092  df-rab 3419  df-v 3459  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4287  df-if 4490  df-sn 4592  df-pr 4594  df-op 4598  df-br 5112  df-opab 5176  df-xp 5669  df-cnv 5671  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676

This theorem is used by:  cnvimassrndm  6151  focnvimacdmdm  6808  cnvimainrn  7066  cnrest2  23495  mbfconstlem  25839  i1fima  25890  i1fima2  25891  i1fd  25893  i1f0rn  25894  itg1addlem5  25912  fcoinver  33022  supppreima  33109  sibfof  34797  itg2addnclem  38381  itg2addnclem2  38382  ftc1anclem6  38408  f1cof1blem  47871  f1cof1b  47874  fnfocofob  47876