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Theorem imadmrn 4705
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.
StepHypRef Expression
1 vex 2577 . . . . . . 7 𝑥 ∈ V
2 vex 2577 . . . . . . 7 𝑦 ∈ V
31, 2opeldm 4565 . . . . . 6 (⟨𝑥, 𝑦⟩ ∈ 𝐴𝑥 ∈ dom 𝐴)
43pm4.71i 377 . . . . 5 (⟨𝑥, 𝑦⟩ ∈ 𝐴 ↔ (⟨𝑥, 𝑦⟩ ∈ 𝐴𝑥 ∈ dom 𝐴))
5 ancom 257 . . . . 5 ((⟨𝑥, 𝑦⟩ ∈ 𝐴𝑥 ∈ dom 𝐴) ↔ (𝑥 ∈ dom 𝐴 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐴))
64, 5bitr2i 178 . . . 4 ((𝑥 ∈ dom 𝐴 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐴) ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐴)
76exbii 1512 . . 3 (∃𝑥(𝑥 ∈ dom 𝐴 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐴) ↔ ∃𝑥𝑥, 𝑦⟩ ∈ 𝐴)
87abbii 2169 . 2 {𝑦 ∣ ∃𝑥(𝑥 ∈ dom 𝐴 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐴)} = {𝑦 ∣ ∃𝑥𝑥, 𝑦⟩ ∈ 𝐴}
9 dfima3 4698 . 2 (𝐴 “ dom 𝐴) = {𝑦 ∣ ∃𝑥(𝑥 ∈ dom 𝐴 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐴)}
10 dfrn3 4551 . 2 ran 𝐴 = {𝑦 ∣ ∃𝑥𝑥, 𝑦⟩ ∈ 𝐴}
118, 9, 103eqtr4i 2086 1 (𝐴 “ dom 𝐴) = ran 𝐴
Colors of variables: wff set class
Syntax hints:  wa 101   = wceq 1259  wex 1397  wcel 1409  {cab 2042  cop 3405  dom cdm 4372  ran crn 4373  cima 4375
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-sep 3902  ax-pow 3954  ax-pr 3971
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-rex 2329  df-v 2576  df-un 2949  df-in 2951  df-ss 2958  df-pw 3388  df-sn 3408  df-pr 3409  df-op 3411  df-br 3792  df-opab 3846  df-xp 4378  df-cnv 4380  df-dm 4382  df-rn 4383  df-res 4384  df-ima 4385
This theorem is referenced by:  cnvimarndm  4716  foima  5138  f1imacnv  5170  fsn2  5364  resfunexg  5409  funiunfvdm  5429  fnexALT  5767  uniqs2  6196  phplem4  6348  phplem4on  6359
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