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Mirrors > Home > ILE Home > Th. List > dmmrnm | GIF version |
Description: A domain is inhabited if and only if the range is inhabited. (Contributed by Jim Kingdon, 15-Dec-2018.) |
Ref | Expression |
---|---|
dmmrnm | ⊢ (∃𝑥 𝑥 ∈ dom 𝐴 ↔ ∃𝑦 𝑦 ∈ ran 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-dm 4654 | . . . . 5 ⊢ dom 𝐴 = {𝑥 ∣ ∃𝑧 𝑥𝐴𝑧} | |
2 | 1 | eleq2i 2256 | . . . 4 ⊢ (𝑥 ∈ dom 𝐴 ↔ 𝑥 ∈ {𝑥 ∣ ∃𝑧 𝑥𝐴𝑧}) |
3 | 2 | exbii 1616 | . . 3 ⊢ (∃𝑥 𝑥 ∈ dom 𝐴 ↔ ∃𝑥 𝑥 ∈ {𝑥 ∣ ∃𝑧 𝑥𝐴𝑧}) |
4 | abid 2177 | . . . 4 ⊢ (𝑥 ∈ {𝑥 ∣ ∃𝑧 𝑥𝐴𝑧} ↔ ∃𝑧 𝑥𝐴𝑧) | |
5 | 4 | exbii 1616 | . . 3 ⊢ (∃𝑥 𝑥 ∈ {𝑥 ∣ ∃𝑧 𝑥𝐴𝑧} ↔ ∃𝑥∃𝑧 𝑥𝐴𝑧) |
6 | 3, 5 | bitri 184 | . 2 ⊢ (∃𝑥 𝑥 ∈ dom 𝐴 ↔ ∃𝑥∃𝑧 𝑥𝐴𝑧) |
7 | dfrn2 4833 | . . . . 5 ⊢ ran 𝐴 = {𝑧 ∣ ∃𝑥 𝑥𝐴𝑧} | |
8 | 7 | eleq2i 2256 | . . . 4 ⊢ (𝑧 ∈ ran 𝐴 ↔ 𝑧 ∈ {𝑧 ∣ ∃𝑥 𝑥𝐴𝑧}) |
9 | 8 | exbii 1616 | . . 3 ⊢ (∃𝑧 𝑧 ∈ ran 𝐴 ↔ ∃𝑧 𝑧 ∈ {𝑧 ∣ ∃𝑥 𝑥𝐴𝑧}) |
10 | abid 2177 | . . . . 5 ⊢ (𝑧 ∈ {𝑧 ∣ ∃𝑥 𝑥𝐴𝑧} ↔ ∃𝑥 𝑥𝐴𝑧) | |
11 | 10 | exbii 1616 | . . . 4 ⊢ (∃𝑧 𝑧 ∈ {𝑧 ∣ ∃𝑥 𝑥𝐴𝑧} ↔ ∃𝑧∃𝑥 𝑥𝐴𝑧) |
12 | excom 1675 | . . . 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 2252 | . . 3 ⊢ (𝑧 = 𝑦 → (𝑧 ∈ ran 𝐴 ↔ 𝑦 ∈ ran 𝐴)) | |
16 | 15 | cbvexv 1930 | . 2 ⊢ (∃𝑧 𝑧 ∈ ran 𝐴 ↔ ∃𝑦 𝑦 ∈ ran 𝐴) |
17 | 6, 14, 16 | 3bitr2i 208 | 1 ⊢ (∃𝑥 𝑥 ∈ dom 𝐴 ↔ ∃𝑦 𝑦 ∈ ran 𝐴) |
Colors of variables: wff set class |
Syntax hints: ↔ wb 105 ∃wex 1503 ∈ wcel 2160 {cab 2175 class class class wbr 4018 dom cdm 4644 ran crn 4645 |
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 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-14 2163 ax-ext 2171 ax-sep 4136 ax-pow 4192 ax-pr 4227 |
This theorem depends on definitions: df-bi 117 df-3an 982 df-tru 1367 df-nf 1472 df-sb 1774 df-eu 2041 df-mo 2042 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-v 2754 df-un 3148 df-in 3150 df-ss 3157 df-pw 3592 df-sn 3613 df-pr 3614 df-op 3616 df-br 4019 df-opab 4080 df-cnv 4652 df-dm 4654 df-rn 4655 |
This theorem is referenced by: rnsnm 5113 ghmrn 13213 nninfall 15237 |
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