Theorem dmmrnm 4554

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 4355	. . . . 5 ⊢ dom 𝐴 = {𝑥 ∣ ∃𝑧 𝑥𝐴𝑧}
2	1	eleq2i 2104	. . . 4 ⊢ (𝑥 ∈ dom 𝐴 ↔ 𝑥 ∈ {𝑥 ∣ ∃𝑧 𝑥𝐴𝑧})
3	2	exbii 1496	. . 3 ⊢ (∃𝑥 𝑥 ∈ dom 𝐴 ↔ ∃𝑥 𝑥 ∈ {𝑥 ∣ ∃𝑧 𝑥𝐴𝑧})
4		abid 2028	. . . 4 ⊢ (𝑥 ∈ {𝑥 ∣ ∃𝑧 𝑥𝐴𝑧} ↔ ∃𝑧 𝑥𝐴𝑧)
5	4	exbii 1496	. . 3 ⊢ (∃𝑥 𝑥 ∈ {𝑥 ∣ ∃𝑧 𝑥𝐴𝑧} ↔ ∃𝑥∃𝑧 𝑥𝐴𝑧)
6	3, 5	bitri 173	. 2 ⊢ (∃𝑥 𝑥 ∈ dom 𝐴 ↔ ∃𝑥∃𝑧 𝑥𝐴𝑧)
7		dfrn2 4523	. . . . 5 ⊢ ran 𝐴 = {𝑧 ∣ ∃𝑥 𝑥𝐴𝑧}
8	7	eleq2i 2104	. . . 4 ⊢ (𝑧 ∈ ran 𝐴 ↔ 𝑧 ∈ {𝑧 ∣ ∃𝑥 𝑥𝐴𝑧})
9	8	exbii 1496	. . 3 ⊢ (∃𝑧 𝑧 ∈ ran 𝐴 ↔ ∃𝑧 𝑧 ∈ {𝑧 ∣ ∃𝑥 𝑥𝐴𝑧})
10		abid 2028	. . . . 5 ⊢ (𝑧 ∈ {𝑧 ∣ ∃𝑥 𝑥𝐴𝑧} ↔ ∃𝑥 𝑥𝐴𝑧)
11	10	exbii 1496	. . . 4 ⊢ (∃𝑧 𝑧 ∈ {𝑧 ∣ ∃𝑥 𝑥𝐴𝑧} ↔ ∃𝑧∃𝑥 𝑥𝐴𝑧)
12		excom 1554	. . . 4 ⊢ (∃𝑧∃𝑥 𝑥𝐴𝑧 ↔ ∃𝑥∃𝑧 𝑥𝐴𝑧)
13	11, 12	bitri 173	. . 3 ⊢ (∃𝑧 𝑧 ∈ {𝑧 ∣ ∃𝑥 𝑥𝐴𝑧} ↔ ∃𝑥∃𝑧 𝑥𝐴𝑧)
14	9, 13	bitri 173	. 2 ⊢ (∃𝑧 𝑧 ∈ ran 𝐴 ↔ ∃𝑥∃𝑧 𝑥𝐴𝑧)
15		eleq1 2100	. . 3 ⊢ (𝑧 = 𝑦 → (𝑧 ∈ ran 𝐴 ↔ 𝑦 ∈ ran 𝐴))
16	15	cbvexv 1795	. 2 ⊢ (∃𝑧 𝑧 ∈ ran 𝐴 ↔ ∃𝑦 𝑦 ∈ ran 𝐴)
17	6, 14, 16	3bitr2i 197	1 ⊢ (∃𝑥 𝑥 ∈ dom 𝐴 ↔ ∃𝑦 𝑦 ∈ ran 𝐴)

Syntax hints:   ↔ wb 98  ∃wex 1381   ∈ wcel 1393  {cab 2026   class class class wbr 3764  dom cdm 4345  ran crn 4346

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944

This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-v 2559  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-br 3765  df-opab 3819  df-cnv 4353  df-dm 4355  df-rn 4356

This theorem is referenced by:  rnsnm  4787