Theorem imadmrn 6066

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 3930	. 2 ⊢ dom 𝐴 ⊆ dom 𝐴
2		dfrn7 6065	. . 3 ⊢ (dom 𝐴 ⊆ dom 𝐴 → ran 𝐴 = (𝐴 “ dom 𝐴))
3	2	eqcomd 2766	. 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 1569   ⊆ wss 3895  dom cdm 5650  ran crn 5651   “ cima 5653

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-ext 2732  ax-sep 5248  ax-pr 5390

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-ss 3896  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-sn 4585  df-pr 4587  df-op 4835  df-br 5103  df-opab 5167  df-xp 5656  df-cnv 5658  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663

This theorem is used by:  cnvimarndmOLD  6080  foima  6801  fimadmfo  6805  f1imacnv  6841  fsn2  7137  resfunexg  7221  elunirnALT  7256  fnexALT  7963  uniqs2  8800  mapsnd  8917  phplem2  9223  php3  9227  pwfilem  9312  jech9.3OLD  9826  fin4en1  10390  retopbas  25084  plyeq0  26535  bday0  28202  rnelshi  32666  rndrhmcl  33904  qusrn  34006  rhmimaidl  34028  ply1degltdimlem  34300  poimirlem3  38634  poimirlem30  38661  rnstructfi  46021  cycl3grtri  49138