Theorem cnvimarndm 5950

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 5939	. 2 ⊢ (◡𝐴 “ dom ◡𝐴) = ran ◡𝐴
2		df-rn 5566	. . 3 ⊢ ran 𝐴 = dom ◡𝐴
3	2	imaeq2i 5927	. 2 ⊢ (◡𝐴 “ ran 𝐴) = (◡𝐴 “ dom ◡𝐴)
4		dfdm4 5764	. 2 ⊢ dom 𝐴 = ran ◡𝐴
5	1, 3, 4	3eqtr4i 2854	1 ⊢ (◡𝐴 “ ran 𝐴) = dom 𝐴

Syntax hints:   = wceq 1537  ^◡ccnv 5554  dom cdm 5555  ran crn 5556   “ cima 5558

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2793  ax-sep 5203  ax-nul 5210  ax-pr 5330

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3496  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-nul 4292  df-if 4468  df-sn 4568  df-pr 4570  df-op 4574  df-br 5067  df-opab 5129  df-xp 5561  df-cnv 5563  df-dm 5565  df-rn 5566  df-res 5567  df-ima 5568

This theorem is referenced by:  cnrest2  21894  mbfconstlem  24228  i1fima  24279  i1fima2  24280  i1fd  24282  i1f0rn  24283  itg1addlem5  24301  fcoinver  30357  sibfof  31598  itg2addnclem  34958  itg2addnclem2  34959  ftc1anclem6  34987