Theorem dmmrnm 4976

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 4759	. . . . 5 ⊢ dom 𝐴 = {𝑥 ∣ ∃𝑧 𝑥𝐴𝑧}
2	1	eleq2i 2299	. . . 4 ⊢ (𝑥 ∈ dom 𝐴 ↔ 𝑥 ∈ {𝑥 ∣ ∃𝑧 𝑥𝐴𝑧})
3	2	exbii 1654	. . 3 ⊢ (∃𝑥 𝑥 ∈ dom 𝐴 ↔ ∃𝑥 𝑥 ∈ {𝑥 ∣ ∃𝑧 𝑥𝐴𝑧})
4		abid 2220	. . . 4 ⊢ (𝑥 ∈ {𝑥 ∣ ∃𝑧 𝑥𝐴𝑧} ↔ ∃𝑧 𝑥𝐴𝑧)
5	4	exbii 1654	. . 3 ⊢ (∃𝑥 𝑥 ∈ {𝑥 ∣ ∃𝑧 𝑥𝐴𝑧} ↔ ∃𝑥∃𝑧 𝑥𝐴𝑧)
6	3, 5	bitri 184	. 2 ⊢ (∃𝑥 𝑥 ∈ dom 𝐴 ↔ ∃𝑥∃𝑧 𝑥𝐴𝑧)
7		dfrn2 4943	. . . . 5 ⊢ ran 𝐴 = {𝑧 ∣ ∃𝑥 𝑥𝐴𝑧}
8	7	eleq2i 2299	. . . 4 ⊢ (𝑧 ∈ ran 𝐴 ↔ 𝑧 ∈ {𝑧 ∣ ∃𝑥 𝑥𝐴𝑧})
9	8	exbii 1654	. . 3 ⊢ (∃𝑧 𝑧 ∈ ran 𝐴 ↔ ∃𝑧 𝑧 ∈ {𝑧 ∣ ∃𝑥 𝑥𝐴𝑧})
10		abid 2220	. . . . 5 ⊢ (𝑧 ∈ {𝑧 ∣ ∃𝑥 𝑥𝐴𝑧} ↔ ∃𝑥 𝑥𝐴𝑧)
11	10	exbii 1654	. . . 4 ⊢ (∃𝑧 𝑧 ∈ {𝑧 ∣ ∃𝑥 𝑥𝐴𝑧} ↔ ∃𝑧∃𝑥 𝑥𝐴𝑧)
12		excom 1712	. . . 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 2295	. . 3 ⊢ (𝑧 = 𝑦 → (𝑧 ∈ ran 𝐴 ↔ 𝑦 ∈ ran 𝐴))
16	15	cbvexv 1968	. 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 1541   ∈ wcel 2203  {cab 2218   class class class wbr 4109  dom cdm 4749  ran crn 4750

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2206  ax-ext 2214  ax-sep 4228  ax-pow 4287  ax-pr 4322

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-v 2815  df-un 3215  df-in 3217  df-ss 3224  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-br 4110  df-opab 4172  df-cnv 4757  df-dm 4759  df-rn 4760

This theorem is referenced by:  rnsnm  5229  ghmrn  13974  nninfall  16787