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Theorem dmmrnm 4581
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 4382 . . . . 5 dom 𝐴 = {𝑥 ∣ ∃𝑧 𝑥𝐴𝑧}
21eleq2i 2120 . . . 4 (𝑥 ∈ dom 𝐴𝑥 ∈ {𝑥 ∣ ∃𝑧 𝑥𝐴𝑧})
32exbii 1512 . . 3 (∃𝑥 𝑥 ∈ dom 𝐴 ↔ ∃𝑥 𝑥 ∈ {𝑥 ∣ ∃𝑧 𝑥𝐴𝑧})
4 abid 2044 . . . 4 (𝑥 ∈ {𝑥 ∣ ∃𝑧 𝑥𝐴𝑧} ↔ ∃𝑧 𝑥𝐴𝑧)
54exbii 1512 . . 3 (∃𝑥 𝑥 ∈ {𝑥 ∣ ∃𝑧 𝑥𝐴𝑧} ↔ ∃𝑥𝑧 𝑥𝐴𝑧)
63, 5bitri 177 . 2 (∃𝑥 𝑥 ∈ dom 𝐴 ↔ ∃𝑥𝑧 𝑥𝐴𝑧)
7 dfrn2 4550 . . . . 5 ran 𝐴 = {𝑧 ∣ ∃𝑥 𝑥𝐴𝑧}
87eleq2i 2120 . . . 4 (𝑧 ∈ ran 𝐴𝑧 ∈ {𝑧 ∣ ∃𝑥 𝑥𝐴𝑧})
98exbii 1512 . . 3 (∃𝑧 𝑧 ∈ ran 𝐴 ↔ ∃𝑧 𝑧 ∈ {𝑧 ∣ ∃𝑥 𝑥𝐴𝑧})
10 abid 2044 . . . . 5 (𝑧 ∈ {𝑧 ∣ ∃𝑥 𝑥𝐴𝑧} ↔ ∃𝑥 𝑥𝐴𝑧)
1110exbii 1512 . . . 4 (∃𝑧 𝑧 ∈ {𝑧 ∣ ∃𝑥 𝑥𝐴𝑧} ↔ ∃𝑧𝑥 𝑥𝐴𝑧)
12 excom 1570 . . . 4 (∃𝑧𝑥 𝑥𝐴𝑧 ↔ ∃𝑥𝑧 𝑥𝐴𝑧)
1311, 12bitri 177 . . 3 (∃𝑧 𝑧 ∈ {𝑧 ∣ ∃𝑥 𝑥𝐴𝑧} ↔ ∃𝑥𝑧 𝑥𝐴𝑧)
149, 13bitri 177 . 2 (∃𝑧 𝑧 ∈ ran 𝐴 ↔ ∃𝑥𝑧 𝑥𝐴𝑧)
15 eleq1 2116 . . 3 (𝑧 = 𝑦 → (𝑧 ∈ ran 𝐴𝑦 ∈ ran 𝐴))
1615cbvexv 1811 . 2 (∃𝑧 𝑧 ∈ ran 𝐴 ↔ ∃𝑦 𝑦 ∈ ran 𝐴)
176, 14, 163bitr2i 201 1 (∃𝑥 𝑥 ∈ dom 𝐴 ↔ ∃𝑦 𝑦 ∈ ran 𝐴)
Colors of variables: wff set class
Syntax hints:  wb 102  wex 1397  wcel 1409  {cab 2042   class class class wbr 3791  dom cdm 4372  ran crn 4373
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-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-cnv 4380  df-dm 4382  df-rn 4383
This theorem is referenced by:  rnsnm  4814
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