Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > ILE Home > Th. List > fnsn | GIF version |
Description: Functionality and domain of the singleton of an ordered pair. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) |
Ref | Expression |
---|---|
fnsn.1 | ⊢ 𝐴 ∈ V |
fnsn.2 | ⊢ 𝐵 ∈ V |
Ref | Expression |
---|---|
fnsn | ⊢ {〈𝐴, 𝐵〉} Fn {𝐴} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fnsn.1 | . 2 ⊢ 𝐴 ∈ V | |
2 | fnsn.2 | . 2 ⊢ 𝐵 ∈ V | |
3 | fnsng 5243 | . 2 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → {〈𝐴, 𝐵〉} Fn {𝐴}) | |
4 | 1, 2, 3 | mp2an 424 | 1 ⊢ {〈𝐴, 𝐵〉} Fn {𝐴} |
Colors of variables: wff set class |
Syntax hints: ∈ wcel 2141 Vcvv 2730 {csn 3581 〈cop 3584 Fn wfn 5191 |
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 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-14 2144 ax-ext 2152 ax-sep 4105 ax-pow 4158 ax-pr 4192 |
This theorem depends on definitions: df-bi 116 df-3an 975 df-tru 1351 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ral 2453 df-rex 2454 df-v 2732 df-un 3125 df-in 3127 df-ss 3134 df-pw 3566 df-sn 3587 df-pr 3588 df-op 3590 df-br 3988 df-opab 4049 df-id 4276 df-xp 4615 df-rel 4616 df-cnv 4617 df-co 4618 df-dm 4619 df-fun 5198 df-fn 5199 |
This theorem is referenced by: f1osn 5480 fvsnun2 5691 elixpsn 6709 |
Copyright terms: Public domain | W3C validator |