Theorem imadmrn 6074

Description: The image of the domain of a class is the range of the class. (Contributed by NM, 14-Aug-1994.)

Assertion

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

Proof of Theorem imadmrn

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		vex 3459	. . . . . . 7 ⊢ 𝑥 ∈ V
2		vex 3459	. . . . . . 7 ⊢ 𝑦 ∈ V
3	1, 2	opeldm 5899	. . . . . 6 ⊢ (⟨𝑥, 𝑦⟩ ∈ 𝐴 → 𝑥 ∈ dom 𝐴)
4	3	pm4.71i 568	. . . . 5 ⊢ (⟨𝑥, 𝑦⟩ ∈ 𝐴 ↔ (⟨𝑥, 𝑦⟩ ∈ 𝐴 ∧ 𝑥 ∈ dom 𝐴))
5		ancom 465	. . . . 5 ⊢ ((⟨𝑥, 𝑦⟩ ∈ 𝐴 ∧ 𝑥 ∈ dom 𝐴) ↔ (𝑥 ∈ dom 𝐴 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐴))
6	4, 5	bitr2i 279	. . . 4 ⊢ ((𝑥 ∈ dom 𝐴 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐴) ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐴)
7	6	exbii 1878	. . 3 ⊢ (∃𝑥(𝑥 ∈ dom 𝐴 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐴) ↔ ∃𝑥⟨𝑥, 𝑦⟩ ∈ 𝐴)
8	7	abbii 2830	. 2 ⊢ {𝑦 ∣ ∃𝑥(𝑥 ∈ dom 𝐴 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐴)} = {𝑦 ∣ ∃𝑥⟨𝑥, 𝑦⟩ ∈ 𝐴}
9		dfima3 6067	. 2 ⊢ (𝐴 “ dom 𝐴) = {𝑦 ∣ ∃𝑥(𝑥 ∈ dom 𝐴 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐴)}
10		dfrn3 5881	. 2 ⊢ ran 𝐴 = {𝑦 ∣ ∃𝑥⟨𝑥, 𝑦⟩ ∈ 𝐴}
11	8, 9, 10	3eqtr4i 2796	1 ⊢ (𝐴 “ dom 𝐴) = ran 𝐴

Syntax hints:   ∧ wa 400   = wceq 1570  ∃wex 1809   ∈ wcel 2143  {cab 2741  ⟨cop 4596  dom cdm 5663  ran crn 5664   “ cima 5666

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-ext 2735  ax-sep 5258  ax-pr 5406

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-sb 2097  df-clab 2742  df-cleq 2755  df-clel 2838  df-ral 3080  df-rex 3090  df-rab 3417  df-v 3457  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4288  df-if 4489  df-sn 4591  df-pr 4593  df-op 4597  df-br 5111  df-opab 5175  df-xp 5669  df-cnv 5671  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676

This theorem is referenced by:  cnvimarndm  6087  foima  6799  fimadmfo  6803  f1imacnv  6839  fsn2  7134  resfunexg  7215  elunirnALT  7252  fnexALT  7949  uniqs2  8775  mapsnd  8885  phplem2  9190  php3  9194  pwfilem  9278  jech9.3  9787  fin4en1  10294  retopbas  24898  plyeq0  26349  bday0  27982  rnelshi  32389  s2rnOLD  33242  s3rnOLD  33244  rndrhmcl  33595  qusrn  33696  rhmimaidl  33718  ply1degltdimlem  33990  poimirlem3  38252  poimirlem30  38279  cycl3grtri  48689