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Theorem dmmrnm 4864
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 4654 . . . . 5 dom 𝐴 = {𝑥 ∣ ∃𝑧 𝑥𝐴𝑧}
21eleq2i 2256 . . . 4 (𝑥 ∈ dom 𝐴𝑥 ∈ {𝑥 ∣ ∃𝑧 𝑥𝐴𝑧})
32exbii 1616 . . 3 (∃𝑥 𝑥 ∈ dom 𝐴 ↔ ∃𝑥 𝑥 ∈ {𝑥 ∣ ∃𝑧 𝑥𝐴𝑧})
4 abid 2177 . . . 4 (𝑥 ∈ {𝑥 ∣ ∃𝑧 𝑥𝐴𝑧} ↔ ∃𝑧 𝑥𝐴𝑧)
54exbii 1616 . . 3 (∃𝑥 𝑥 ∈ {𝑥 ∣ ∃𝑧 𝑥𝐴𝑧} ↔ ∃𝑥𝑧 𝑥𝐴𝑧)
63, 5bitri 184 . 2 (∃𝑥 𝑥 ∈ dom 𝐴 ↔ ∃𝑥𝑧 𝑥𝐴𝑧)
7 dfrn2 4833 . . . . 5 ran 𝐴 = {𝑧 ∣ ∃𝑥 𝑥𝐴𝑧}
87eleq2i 2256 . . . 4 (𝑧 ∈ ran 𝐴𝑧 ∈ {𝑧 ∣ ∃𝑥 𝑥𝐴𝑧})
98exbii 1616 . . 3 (∃𝑧 𝑧 ∈ ran 𝐴 ↔ ∃𝑧 𝑧 ∈ {𝑧 ∣ ∃𝑥 𝑥𝐴𝑧})
10 abid 2177 . . . . 5 (𝑧 ∈ {𝑧 ∣ ∃𝑥 𝑥𝐴𝑧} ↔ ∃𝑥 𝑥𝐴𝑧)
1110exbii 1616 . . . 4 (∃𝑧 𝑧 ∈ {𝑧 ∣ ∃𝑥 𝑥𝐴𝑧} ↔ ∃𝑧𝑥 𝑥𝐴𝑧)
12 excom 1675 . . . 4 (∃𝑧𝑥 𝑥𝐴𝑧 ↔ ∃𝑥𝑧 𝑥𝐴𝑧)
1311, 12bitri 184 . . 3 (∃𝑧 𝑧 ∈ {𝑧 ∣ ∃𝑥 𝑥𝐴𝑧} ↔ ∃𝑥𝑧 𝑥𝐴𝑧)
149, 13bitri 184 . 2 (∃𝑧 𝑧 ∈ ran 𝐴 ↔ ∃𝑥𝑧 𝑥𝐴𝑧)
15 eleq1 2252 . . 3 (𝑧 = 𝑦 → (𝑧 ∈ ran 𝐴𝑦 ∈ ran 𝐴))
1615cbvexv 1930 . 2 (∃𝑧 𝑧 ∈ ran 𝐴 ↔ ∃𝑦 𝑦 ∈ ran 𝐴)
176, 14, 163bitr2i 208 1 (∃𝑥 𝑥 ∈ dom 𝐴 ↔ ∃𝑦 𝑦 ∈ ran 𝐴)
Colors of variables: wff set class
Syntax hints:  wb 105  wex 1503  wcel 2160  {cab 2175   class class class wbr 4018  dom cdm 4644  ran crn 4645
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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-pow 4192  ax-pr 4227
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-v 2754  df-un 3148  df-in 3150  df-ss 3157  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-br 4019  df-opab 4080  df-cnv 4652  df-dm 4654  df-rn 4655
This theorem is referenced by:  rnsnm  5113  ghmrn  13213  nninfall  15237
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