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

Theorem dmmrnm 4554
 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 4355 . . . . 5 dom 𝐴 = {𝑥 ∣ ∃𝑧 𝑥𝐴𝑧}
21eleq2i 2104 . . . 4 (𝑥 ∈ dom 𝐴𝑥 ∈ {𝑥 ∣ ∃𝑧 𝑥𝐴𝑧})
32exbii 1496 . . 3 (∃𝑥 𝑥 ∈ dom 𝐴 ↔ ∃𝑥 𝑥 ∈ {𝑥 ∣ ∃𝑧 𝑥𝐴𝑧})
4 abid 2028 . . . 4 (𝑥 ∈ {𝑥 ∣ ∃𝑧 𝑥𝐴𝑧} ↔ ∃𝑧 𝑥𝐴𝑧)
54exbii 1496 . . 3 (∃𝑥 𝑥 ∈ {𝑥 ∣ ∃𝑧 𝑥𝐴𝑧} ↔ ∃𝑥𝑧 𝑥𝐴𝑧)
63, 5bitri 173 . 2 (∃𝑥 𝑥 ∈ dom 𝐴 ↔ ∃𝑥𝑧 𝑥𝐴𝑧)
7 dfrn2 4523 . . . . 5 ran 𝐴 = {𝑧 ∣ ∃𝑥 𝑥𝐴𝑧}
87eleq2i 2104 . . . 4 (𝑧 ∈ ran 𝐴𝑧 ∈ {𝑧 ∣ ∃𝑥 𝑥𝐴𝑧})
98exbii 1496 . . 3 (∃𝑧 𝑧 ∈ ran 𝐴 ↔ ∃𝑧 𝑧 ∈ {𝑧 ∣ ∃𝑥 𝑥𝐴𝑧})
10 abid 2028 . . . . 5 (𝑧 ∈ {𝑧 ∣ ∃𝑥 𝑥𝐴𝑧} ↔ ∃𝑥 𝑥𝐴𝑧)
1110exbii 1496 . . . 4 (∃𝑧 𝑧 ∈ {𝑧 ∣ ∃𝑥 𝑥𝐴𝑧} ↔ ∃𝑧𝑥 𝑥𝐴𝑧)
12 excom 1554 . . . 4 (∃𝑧𝑥 𝑥𝐴𝑧 ↔ ∃𝑥𝑧 𝑥𝐴𝑧)
1311, 12bitri 173 . . 3 (∃𝑧 𝑧 ∈ {𝑧 ∣ ∃𝑥 𝑥𝐴𝑧} ↔ ∃𝑥𝑧 𝑥𝐴𝑧)
149, 13bitri 173 . 2 (∃𝑧 𝑧 ∈ ran 𝐴 ↔ ∃𝑥𝑧 𝑥𝐴𝑧)
15 eleq1 2100 . . 3 (𝑧 = 𝑦 → (𝑧 ∈ ran 𝐴𝑦 ∈ ran 𝐴))
1615cbvexv 1795 . 2 (∃𝑧 𝑧 ∈ ran 𝐴 ↔ ∃𝑦 𝑦 ∈ ran 𝐴)
176, 14, 163bitr2i 197 1 (∃𝑥 𝑥 ∈ dom 𝐴 ↔ ∃𝑦 𝑦 ∈ ran 𝐴)
 Colors of variables: wff set class Syntax hints:   ↔ wb 98  ∃wex 1381   ∈ wcel 1393  {cab 2026   class class class wbr 3764  dom cdm 4345  ran crn 4346 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944 This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-v 2559  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-br 3765  df-opab 3819  df-cnv 4353  df-dm 4355  df-rn 4356 This theorem is referenced by:  rnsnm  4787
 Copyright terms: Public domain W3C validator