Theorem dmmrnm 4823

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 4614	. . . . 5 ⊢ dom 𝐴 = {𝑥 ∣ ∃𝑧 𝑥𝐴𝑧}
2	1	eleq2i 2233	. . . 4 ⊢ (𝑥 ∈ dom 𝐴 ↔ 𝑥 ∈ {𝑥 ∣ ∃𝑧 𝑥𝐴𝑧})
3	2	exbii 1593	. . 3 ⊢ (∃𝑥 𝑥 ∈ dom 𝐴 ↔ ∃𝑥 𝑥 ∈ {𝑥 ∣ ∃𝑧 𝑥𝐴𝑧})
4		abid 2153	. . . 4 ⊢ (𝑥 ∈ {𝑥 ∣ ∃𝑧 𝑥𝐴𝑧} ↔ ∃𝑧 𝑥𝐴𝑧)
5	4	exbii 1593	. . 3 ⊢ (∃𝑥 𝑥 ∈ {𝑥 ∣ ∃𝑧 𝑥𝐴𝑧} ↔ ∃𝑥∃𝑧 𝑥𝐴𝑧)
6	3, 5	bitri 183	. 2 ⊢ (∃𝑥 𝑥 ∈ dom 𝐴 ↔ ∃𝑥∃𝑧 𝑥𝐴𝑧)
7		dfrn2 4792	. . . . 5 ⊢ ran 𝐴 = {𝑧 ∣ ∃𝑥 𝑥𝐴𝑧}
8	7	eleq2i 2233	. . . 4 ⊢ (𝑧 ∈ ran 𝐴 ↔ 𝑧 ∈ {𝑧 ∣ ∃𝑥 𝑥𝐴𝑧})
9	8	exbii 1593	. . 3 ⊢ (∃𝑧 𝑧 ∈ ran 𝐴 ↔ ∃𝑧 𝑧 ∈ {𝑧 ∣ ∃𝑥 𝑥𝐴𝑧})
10		abid 2153	. . . . 5 ⊢ (𝑧 ∈ {𝑧 ∣ ∃𝑥 𝑥𝐴𝑧} ↔ ∃𝑥 𝑥𝐴𝑧)
11	10	exbii 1593	. . . 4 ⊢ (∃𝑧 𝑧 ∈ {𝑧 ∣ ∃𝑥 𝑥𝐴𝑧} ↔ ∃𝑧∃𝑥 𝑥𝐴𝑧)
12		excom 1652	. . . 4 ⊢ (∃𝑧∃𝑥 𝑥𝐴𝑧 ↔ ∃𝑥∃𝑧 𝑥𝐴𝑧)
13	11, 12	bitri 183	. . 3 ⊢ (∃𝑧 𝑧 ∈ {𝑧 ∣ ∃𝑥 𝑥𝐴𝑧} ↔ ∃𝑥∃𝑧 𝑥𝐴𝑧)
14	9, 13	bitri 183	. 2 ⊢ (∃𝑧 𝑧 ∈ ran 𝐴 ↔ ∃𝑥∃𝑧 𝑥𝐴𝑧)
15		eleq1 2229	. . 3 ⊢ (𝑧 = 𝑦 → (𝑧 ∈ ran 𝐴 ↔ 𝑦 ∈ ran 𝐴))
16	15	cbvexv 1906	. 2 ⊢ (∃𝑧 𝑧 ∈ ran 𝐴 ↔ ∃𝑦 𝑦 ∈ ran 𝐴)
17	6, 14, 16	3bitr2i 207	1 ⊢ (∃𝑥 𝑥 ∈ dom 𝐴 ↔ ∃𝑦 𝑦 ∈ ran 𝐴)

Syntax hints:   ↔ wb 104  ∃wex 1480   ∈ wcel 2136  {cab 2151   class class class wbr 3982  dom cdm 4604  ran crn 4605

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 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-pow 4153  ax-pr 4187

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-v 2728  df-un 3120  df-in 3122  df-ss 3129  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-br 3983  df-opab 4044  df-cnv 4612  df-dm 4614  df-rn 4615

This theorem is referenced by:  rnsnm  5070  nninfall  13889