Theorem imadmrn 5110

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 2815	. . . . . . 7 ⊢ 𝑥 ∈ V
2		vex 2815	. . . . . . 7 ⊢ 𝑦 ∈ V
3	1, 2	opeldm 4958	. . . . . 6 ⊢ (⟨𝑥, 𝑦⟩ ∈ 𝐴 → 𝑥 ∈ dom 𝐴)
4	3	pm4.71i 391	. . . . 5 ⊢ (⟨𝑥, 𝑦⟩ ∈ 𝐴 ↔ (⟨𝑥, 𝑦⟩ ∈ 𝐴 ∧ 𝑥 ∈ dom 𝐴))
5		ancom 266	. . . . 5 ⊢ ((⟨𝑥, 𝑦⟩ ∈ 𝐴 ∧ 𝑥 ∈ dom 𝐴) ↔ (𝑥 ∈ dom 𝐴 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐴))
6	4, 5	bitr2i 185	. . . 4 ⊢ ((𝑥 ∈ dom 𝐴 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐴) ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐴)
7	6	exbii 1654	. . 3 ⊢ (∃𝑥(𝑥 ∈ dom 𝐴 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐴) ↔ ∃𝑥⟨𝑥, 𝑦⟩ ∈ 𝐴)
8	7	abbii 2348	. 2 ⊢ {𝑦 ∣ ∃𝑥(𝑥 ∈ dom 𝐴 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐴)} = {𝑦 ∣ ∃𝑥⟨𝑥, 𝑦⟩ ∈ 𝐴}
9		dfima3 5103	. 2 ⊢ (𝐴 “ dom 𝐴) = {𝑦 ∣ ∃𝑥(𝑥 ∈ dom 𝐴 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐴)}
10		dfrn3 4943	. 2 ⊢ ran 𝐴 = {𝑦 ∣ ∃𝑥⟨𝑥, 𝑦⟩ ∈ 𝐴}
11	8, 9, 10	3eqtr4i 2263	1 ⊢ (𝐴 “ dom 𝐴) = ran 𝐴

Syntax hints:   ∧ wa 104   = wceq 1398  ∃wex 1541   ∈ wcel 2203  {cab 2218  ⟨cop 3691  dom cdm 4748  ran crn 4749   “ cima 4751

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2206  ax-ext 2214  ax-sep 4227  ax-pow 4286  ax-pr 4321

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ral 2525  df-rex 2526  df-v 2814  df-un 3214  df-in 3216  df-ss 3223  df-pw 3670  df-sn 3694  df-pr 3695  df-op 3697  df-br 4109  df-opab 4171  df-xp 4754  df-cnv 4756  df-dm 4758  df-rn 4759  df-res 4760  df-ima 4761

This theorem is referenced by:  cnvimarndm  5125  foima  5594  fimadmfo  5598  f1imacnv  5630  fsn2  5850  resfunexg  5904  funiunfvdm  5935  fnexALT  6303  uniqs2  6828  mapsnd  6922  mapsn  6924  phplem4  7108  phplem4on  7121  retopbas  15380