Theorem imadmrn 4886

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 2684	. . . . . . 7 ⊢ 𝑥 ∈ V
2		vex 2684	. . . . . . 7 ⊢ 𝑦 ∈ V
3	1, 2	opeldm 4737	. . . . . 6 ⊢ (⟨𝑥, 𝑦⟩ ∈ 𝐴 → 𝑥 ∈ dom 𝐴)
4	3	pm4.71i 388	. . . . 5 ⊢ (⟨𝑥, 𝑦⟩ ∈ 𝐴 ↔ (⟨𝑥, 𝑦⟩ ∈ 𝐴 ∧ 𝑥 ∈ dom 𝐴))
5		ancom 264	. . . . 5 ⊢ ((⟨𝑥, 𝑦⟩ ∈ 𝐴 ∧ 𝑥 ∈ dom 𝐴) ↔ (𝑥 ∈ dom 𝐴 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐴))
6	4, 5	bitr2i 184	. . . 4 ⊢ ((𝑥 ∈ dom 𝐴 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐴) ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐴)
7	6	exbii 1584	. . 3 ⊢ (∃𝑥(𝑥 ∈ dom 𝐴 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐴) ↔ ∃𝑥⟨𝑥, 𝑦⟩ ∈ 𝐴)
8	7	abbii 2253	. 2 ⊢ {𝑦 ∣ ∃𝑥(𝑥 ∈ dom 𝐴 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐴)} = {𝑦 ∣ ∃𝑥⟨𝑥, 𝑦⟩ ∈ 𝐴}
9		dfima3 4879	. 2 ⊢ (𝐴 “ dom 𝐴) = {𝑦 ∣ ∃𝑥(𝑥 ∈ dom 𝐴 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐴)}
10		dfrn3 4723	. 2 ⊢ ran 𝐴 = {𝑦 ∣ ∃𝑥⟨𝑥, 𝑦⟩ ∈ 𝐴}
11	8, 9, 10	3eqtr4i 2168	1 ⊢ (𝐴 “ dom 𝐴) = ran 𝐴

Syntax hints:   ∧ wa 103   = wceq 1331  ∃wex 1468   ∈ wcel 1480  {cab 2123  ⟨cop 3525  dom cdm 4534  ran crn 4535   “ cima 4537

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-sep 4041  ax-pow 4093  ax-pr 4126

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2000  df-mo 2001  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ral 2419  df-rex 2420  df-v 2683  df-un 3070  df-in 3072  df-ss 3079  df-pw 3507  df-sn 3528  df-pr 3529  df-op 3531  df-br 3925  df-opab 3985  df-xp 4540  df-cnv 4542  df-dm 4544  df-rn 4545  df-res 4546  df-ima 4547

This theorem is referenced by:  cnvimarndm  4898  foima  5345  f1imacnv  5377  fsn2  5587  resfunexg  5634  funiunfvdm  5657  fnexALT  6004  uniqs2  6482  mapsn  6577  phplem4  6742  phplem4on  6754  retopbas  12681