Theorem imadmrn 6079

Description: The image of the domain of a class is equal to its range. (Contributed by NM, 14-Aug-1994.) (Proof shortened by BJ, 27-Sep-2026.)

Assertion

Ref	Expression
imadmrn	⊢ (𝐴 “ dom 𝐴) = ran 𝐴

Proof of Theorem imadmrn

Step	Hyp	Ref	Expression
1		ssid 3935	. 2 ⊢ dom 𝐴 ⊆ dom 𝐴
2		dfrn7 6078	. . 3 ⊢ (dom 𝐴 ⊆ dom 𝐴 → ran 𝐴 = (𝐴 “ dom 𝐴))
3	2	eqcomd 2769	. 2 ⊢ (dom 𝐴 ⊆ dom 𝐴 → (𝐴 “ dom 𝐴) = ran 𝐴)
4	1, 3	ax-mp 5	1 ⊢ (𝐴 “ dom 𝐴) = ran 𝐴

This proof depends on syntax axioms:   = wceq 1570   ⊆ wss 3900  dom cdm 5655  ran crn 5656   “ cima 5658

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 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735  ax-sep 5252  ax-pr 5395

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 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4839  df-br 5107  df-opab 5171  df-xp 5661  df-cnv 5663  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668

This theorem is used by:  cnvimarndmOLD  6093  foima  6823  fimadmfo  6827  f1imacnv  6863  fsn2  7161  resfunexg  7244  elunirnALT  7279  fnexALT  7987  uniqs2  8825  mapsnd  8942  phplem2  9248  php3  9252  pwfilem  9337  jech9.3OLD  9850  fin4en1  10414  retopbas  25109  plyeq0  26560  bday0  28227  rnelshi  32691  rndrhmcl  33928  qusrn  34030  rhmimaidl  34052  ply1degltdimlem  34324  poimirlem3  38733  poimirlem30  38760  rnstructfi  46121  cycl3grtri  49250