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| Mirrors > Home > ILE Home > Th. List > rnsnm | GIF version | ||
| Description: The range of a singleton is inhabited iff the singleton argument is an ordered pair. (Contributed by Jim Kingdon, 15-Dec-2018.) |
| Ref | Expression |
|---|---|
| rnsnm | ⊢ (𝐴 ∈ (V × V) ↔ ∃𝑥 𝑥 ∈ ran {𝐴}) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | dmsnm 5147 | . 2 ⊢ (𝐴 ∈ (V × V) ↔ ∃𝑥 𝑥 ∈ dom {𝐴}) | |
| 2 | dmmrnm 4896 | . 2 ⊢ (∃𝑥 𝑥 ∈ dom {𝐴} ↔ ∃𝑥 𝑥 ∈ ran {𝐴}) | |
| 3 | 1, 2 | bitri 184 | 1 ⊢ (𝐴 ∈ (V × V) ↔ ∃𝑥 𝑥 ∈ ran {𝐴}) |
| Colors of variables: wff set class |
| Syntax hints: ↔ wb 105 ∃wex 1514 ∈ wcel 2175 Vcvv 2771 {csn 3632 × cxp 4672 dom cdm 4674 ran crn 4675 |
| 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 1469 ax-7 1470 ax-gen 1471 ax-ie1 1515 ax-ie2 1516 ax-8 1526 ax-10 1527 ax-11 1528 ax-i12 1529 ax-bndl 1531 ax-4 1532 ax-17 1548 ax-i9 1552 ax-ial 1556 ax-i5r 1557 ax-14 2178 ax-ext 2186 ax-sep 4161 ax-pow 4217 ax-pr 4252 |
| This theorem depends on definitions: df-bi 117 df-3an 982 df-tru 1375 df-nf 1483 df-sb 1785 df-eu 2056 df-mo 2057 df-clab 2191 df-cleq 2197 df-clel 2200 df-nfc 2336 df-v 2773 df-un 3169 df-in 3171 df-ss 3178 df-pw 3617 df-sn 3638 df-pr 3639 df-op 3641 df-br 4044 df-opab 4105 df-xp 4680 df-cnv 4682 df-dm 4684 df-rn 4685 |
| This theorem is referenced by: (None) |
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