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Theorem dmmrnm 4726
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 4517 . . . . 5 dom 𝐴 = {𝑥 ∣ ∃𝑧 𝑥𝐴𝑧}
21eleq2i 2182 . . . 4 (𝑥 ∈ dom 𝐴𝑥 ∈ {𝑥 ∣ ∃𝑧 𝑥𝐴𝑧})
32exbii 1567 . . 3 (∃𝑥 𝑥 ∈ dom 𝐴 ↔ ∃𝑥 𝑥 ∈ {𝑥 ∣ ∃𝑧 𝑥𝐴𝑧})
4 abid 2103 . . . 4 (𝑥 ∈ {𝑥 ∣ ∃𝑧 𝑥𝐴𝑧} ↔ ∃𝑧 𝑥𝐴𝑧)
54exbii 1567 . . 3 (∃𝑥 𝑥 ∈ {𝑥 ∣ ∃𝑧 𝑥𝐴𝑧} ↔ ∃𝑥𝑧 𝑥𝐴𝑧)
63, 5bitri 183 . 2 (∃𝑥 𝑥 ∈ dom 𝐴 ↔ ∃𝑥𝑧 𝑥𝐴𝑧)
7 dfrn2 4695 . . . . 5 ran 𝐴 = {𝑧 ∣ ∃𝑥 𝑥𝐴𝑧}
87eleq2i 2182 . . . 4 (𝑧 ∈ ran 𝐴𝑧 ∈ {𝑧 ∣ ∃𝑥 𝑥𝐴𝑧})
98exbii 1567 . . 3 (∃𝑧 𝑧 ∈ ran 𝐴 ↔ ∃𝑧 𝑧 ∈ {𝑧 ∣ ∃𝑥 𝑥𝐴𝑧})
10 abid 2103 . . . . 5 (𝑧 ∈ {𝑧 ∣ ∃𝑥 𝑥𝐴𝑧} ↔ ∃𝑥 𝑥𝐴𝑧)
1110exbii 1567 . . . 4 (∃𝑧 𝑧 ∈ {𝑧 ∣ ∃𝑥 𝑥𝐴𝑧} ↔ ∃𝑧𝑥 𝑥𝐴𝑧)
12 excom 1625 . . . 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 2178 . . 3 (𝑧 = 𝑦 → (𝑧 ∈ ran 𝐴𝑦 ∈ ran 𝐴))
1615cbvexv 1870 . 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 1451  wcel 1463  {cab 2101   class class class wbr 3897  dom cdm 4507  ran crn 4508
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 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-sep 4014  ax-pow 4066  ax-pr 4099
This theorem depends on definitions:  df-bi 116  df-3an 947  df-tru 1317  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-v 2660  df-un 3043  df-in 3045  df-ss 3052  df-pw 3480  df-sn 3501  df-pr 3502  df-op 3504  df-br 3898  df-opab 3958  df-cnv 4515  df-dm 4517  df-rn 4518
This theorem is referenced by:  rnsnm  4973  nninfall  13038
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