Theorem cnvimarndm 6043

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 6030	. 2 ⊢ (◡𝐴 “ dom ◡𝐴) = ran ◡𝐴
2		df-rn 5636	. . 3 ⊢ ran 𝐴 = dom ◡𝐴
3	2	imaeq2i 6018	. 2 ⊢ (◡𝐴 “ ran 𝐴) = (◡𝐴 “ dom ◡𝐴)
4		dfdm4 5845	. 2 ⊢ dom 𝐴 = ran ◡𝐴
5	1, 3, 4	3eqtr4i 2770	1 ⊢ (◡𝐴 “ ran 𝐴) = dom 𝐴

Syntax hints:   = wceq 1542  ^◡ccnv 5624  dom cdm 5625  ran crn 5626   “ cima 5628

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709  ax-sep 5232  ax-pr 5371

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-ral 3053  df-rex 3063  df-rab 3391  df-v 3432  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-sn 4569  df-pr 4571  df-op 4575  df-br 5087  df-opab 5149  df-xp 5631  df-cnv 5633  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638

This theorem is referenced by:  cnvimassrndm  6111  focnvimacdmdm  6759  cnvimainrn  7014  cnrest2  23264  mbfconstlem  25607  i1fima  25658  i1fima2  25659  i1fd  25661  i1f0rn  25662  itg1addlem5  25680  fcoinver  32692  supppreima  32782  sibfof  34503  itg2addnclem  38009  itg2addnclem2  38010  ftc1anclem6  38036  f1cof1blem  47537  f1cof1b  47540  fnfocofob  47542