ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  dmmrnm GIF version

Theorem dmmrnm 4728
Description: A domain is inhabited if and only if the range is inhabited. (Contributed by Jim Kingdon, 15-Dec-2018.)
Assertion
Ref Expression
dmmrnm (∃𝑥 𝑥 ∈ dom 𝐴 ↔ ∃𝑦 𝑦 ∈ ran 𝐴)
Distinct variable groups:   𝑦,𝐴   𝑥,𝐴

Proof of Theorem dmmrnm
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 df-dm 4519 . . . . 5 dom 𝐴 = {𝑥 ∣ ∃𝑧 𝑥𝐴𝑧}
21eleq2i 2184 . . . 4 (𝑥 ∈ dom 𝐴𝑥 ∈ {𝑥 ∣ ∃𝑧 𝑥𝐴𝑧})
32exbii 1569 . . 3 (∃𝑥 𝑥 ∈ dom 𝐴 ↔ ∃𝑥 𝑥 ∈ {𝑥 ∣ ∃𝑧 𝑥𝐴𝑧})
4 abid 2105 . . . 4 (𝑥 ∈ {𝑥 ∣ ∃𝑧 𝑥𝐴𝑧} ↔ ∃𝑧 𝑥𝐴𝑧)
54exbii 1569 . . 3 (∃𝑥 𝑥 ∈ {𝑥 ∣ ∃𝑧 𝑥𝐴𝑧} ↔ ∃𝑥𝑧 𝑥𝐴𝑧)
63, 5bitri 183 . 2 (∃𝑥 𝑥 ∈ dom 𝐴 ↔ ∃𝑥𝑧 𝑥𝐴𝑧)
7 dfrn2 4697 . . . . 5 ran 𝐴 = {𝑧 ∣ ∃𝑥 𝑥𝐴𝑧}
87eleq2i 2184 . . . 4 (𝑧 ∈ ran 𝐴𝑧 ∈ {𝑧 ∣ ∃𝑥 𝑥𝐴𝑧})
98exbii 1569 . . 3 (∃𝑧 𝑧 ∈ ran 𝐴 ↔ ∃𝑧 𝑧 ∈ {𝑧 ∣ ∃𝑥 𝑥𝐴𝑧})
10 abid 2105 . . . . 5 (𝑧 ∈ {𝑧 ∣ ∃𝑥 𝑥𝐴𝑧} ↔ ∃𝑥 𝑥𝐴𝑧)
1110exbii 1569 . . . 4 (∃𝑧 𝑧 ∈ {𝑧 ∣ ∃𝑥 𝑥𝐴𝑧} ↔ ∃𝑧𝑥 𝑥𝐴𝑧)
12 excom 1627 . . . 4 (∃𝑧𝑥 𝑥𝐴𝑧 ↔ ∃𝑥𝑧 𝑥𝐴𝑧)
1311, 12bitri 183 . . 3 (∃𝑧 𝑧 ∈ {𝑧 ∣ ∃𝑥 𝑥𝐴𝑧} ↔ ∃𝑥𝑧 𝑥𝐴𝑧)
149, 13bitri 183 . 2 (∃𝑧 𝑧 ∈ ran 𝐴 ↔ ∃𝑥𝑧 𝑥𝐴𝑧)
15 eleq1 2180 . . 3 (𝑧 = 𝑦 → (𝑧 ∈ ran 𝐴𝑦 ∈ ran 𝐴))
1615cbvexv 1872 . 2 (∃𝑧 𝑧 ∈ ran 𝐴 ↔ ∃𝑦 𝑦 ∈ ran 𝐴)
176, 14, 163bitr2i 207 1 (∃𝑥 𝑥 ∈ dom 𝐴 ↔ ∃𝑦 𝑦 ∈ ran 𝐴)
Colors of variables: wff set class
Syntax hints:  wb 104  wex 1453  wcel 1465  {cab 2103   class class class wbr 3899  dom cdm 4509  ran crn 4510
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 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-sep 4016  ax-pow 4068  ax-pr 4101
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-v 2662  df-un 3045  df-in 3047  df-ss 3054  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-br 3900  df-opab 3960  df-cnv 4517  df-dm 4519  df-rn 4520
This theorem is referenced by:  rnsnm  4975  nninfall  13131
  Copyright terms: Public domain W3C validator