Theorem cnvimarndm 6080

Description: The preimage of the range of a class is equal to the domain of the class. (Contributed by Jeff Hankins, 15-Jul-2009.) (Proof shortened by BJ, 29-Sep-2026.)

Assertion

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

Proof of Theorem cnvimarndm

Step	Hyp	Ref	Expression
1		ssid 3953	. 2 ⊢ ran 𝐴 ⊆ ran 𝐴
2		cnvimassrndm 6079	. 2 ⊢ (ran 𝐴 ⊆ ran 𝐴 → (◡𝐴 “ ran 𝐴) = dom 𝐴)
3	1, 2	ax-mp 5	1 ⊢ (◡𝐴 “ ran 𝐴) = dom 𝐴

This proof depends on syntax axioms:   = wceq 1570   ⊆ wss 3899  ^◡ccnv 5650  dom cdm 5651  ran crn 5652   “ cima 5654

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 2147  ax-9 2155  ax-ext 2733  ax-sep 5249  ax-pr 5391

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 2740  df-cleq 2753  df-clel 2836  df-ral 3078  df-rex 3088  df-rab 3414  df-v 3453  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-br 5104  df-opab 5168  df-xp 5657  df-cnv 5659  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664

This theorem is used by:  cnvimassrndmOLD  6199  focnvimacdmdm  6808  cnvimainrn  7066  cnrest2  23604  mbfconstlem  25948  i1fima  25999  i1fima2  26000  i1fd  26002  i1f0rn  26003  itg1addlem5  26021  rnplynfin  26630  fcoinver  33198  supppreima  33284  sibfof  34972  itg2addnclem  38589  itg2addnclem2  38590  ftc1anclem6  38616  f1cof1blem  48143  f1cof1b  48146  fnfocofob  48148