Theorem dmmrnm 4943

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.

Step	Hyp	Ref	Expression
1		df-dm 4729	. . . . 5 ⊢ dom 𝐴 = {𝑥 ∣ ∃𝑧 𝑥𝐴𝑧}
2	1	eleq2i 2296	. . . 4 ⊢ (𝑥 ∈ dom 𝐴 ↔ 𝑥 ∈ {𝑥 ∣ ∃𝑧 𝑥𝐴𝑧})
3	2	exbii 1651	. . 3 ⊢ (∃𝑥 𝑥 ∈ dom 𝐴 ↔ ∃𝑥 𝑥 ∈ {𝑥 ∣ ∃𝑧 𝑥𝐴𝑧})
4		abid 2217	. . . 4 ⊢ (𝑥 ∈ {𝑥 ∣ ∃𝑧 𝑥𝐴𝑧} ↔ ∃𝑧 𝑥𝐴𝑧)
5	4	exbii 1651	. . 3 ⊢ (∃𝑥 𝑥 ∈ {𝑥 ∣ ∃𝑧 𝑥𝐴𝑧} ↔ ∃𝑥∃𝑧 𝑥𝐴𝑧)
6	3, 5	bitri 184	. 2 ⊢ (∃𝑥 𝑥 ∈ dom 𝐴 ↔ ∃𝑥∃𝑧 𝑥𝐴𝑧)
7		dfrn2 4910	. . . . 5 ⊢ ran 𝐴 = {𝑧 ∣ ∃𝑥 𝑥𝐴𝑧}
8	7	eleq2i 2296	. . . 4 ⊢ (𝑧 ∈ ran 𝐴 ↔ 𝑧 ∈ {𝑧 ∣ ∃𝑥 𝑥𝐴𝑧})
9	8	exbii 1651	. . 3 ⊢ (∃𝑧 𝑧 ∈ ran 𝐴 ↔ ∃𝑧 𝑧 ∈ {𝑧 ∣ ∃𝑥 𝑥𝐴𝑧})
10		abid 2217	. . . . 5 ⊢ (𝑧 ∈ {𝑧 ∣ ∃𝑥 𝑥𝐴𝑧} ↔ ∃𝑥 𝑥𝐴𝑧)
11	10	exbii 1651	. . . 4 ⊢ (∃𝑧 𝑧 ∈ {𝑧 ∣ ∃𝑥 𝑥𝐴𝑧} ↔ ∃𝑧∃𝑥 𝑥𝐴𝑧)
12		excom 1710	. . . 4 ⊢ (∃𝑧∃𝑥 𝑥𝐴𝑧 ↔ ∃𝑥∃𝑧 𝑥𝐴𝑧)
13	11, 12	bitri 184	. . 3 ⊢ (∃𝑧 𝑧 ∈ {𝑧 ∣ ∃𝑥 𝑥𝐴𝑧} ↔ ∃𝑥∃𝑧 𝑥𝐴𝑧)
14	9, 13	bitri 184	. 2 ⊢ (∃𝑧 𝑧 ∈ ran 𝐴 ↔ ∃𝑥∃𝑧 𝑥𝐴𝑧)
15		eleq1 2292	. . 3 ⊢ (𝑧 = 𝑦 → (𝑧 ∈ ran 𝐴 ↔ 𝑦 ∈ ran 𝐴))
16	15	cbvexv 1965	. 2 ⊢ (∃𝑧 𝑧 ∈ ran 𝐴 ↔ ∃𝑦 𝑦 ∈ ran 𝐴)
17	6, 14, 16	3bitr2i 208	1 ⊢ (∃𝑥 𝑥 ∈ dom 𝐴 ↔ ∃𝑦 𝑦 ∈ ran 𝐴)

Syntax hints:   ↔ wb 105  ∃wex 1538   ∈ wcel 2200  {cab 2215   class class class wbr 4083  dom cdm 4719  ran crn 4720

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-14 2203  ax-ext 2211  ax-sep 4202  ax-pow 4258  ax-pr 4293

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-v 2801  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-br 4084  df-opab 4146  df-cnv 4727  df-dm 4729  df-rn 4730

This theorem is referenced by:  rnsnm  5195  ghmrn  13794  nninfall  16375