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Mirrors > Home > ILE Home > Th. List > fnsng | GIF version |
Description: Functionality and domain of the singleton of an ordered pair. (Contributed by Mario Carneiro, 30-Apr-2015.) |
Ref | Expression |
---|---|
fnsng | ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → {〈𝐴, 𝐵〉} Fn {𝐴}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | funsng 5281 | . 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → Fun {〈𝐴, 𝐵〉}) | |
2 | dmsnopg 5118 | . . 3 ⊢ (𝐵 ∈ 𝑊 → dom {〈𝐴, 𝐵〉} = {𝐴}) | |
3 | 2 | adantl 277 | . 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → dom {〈𝐴, 𝐵〉} = {𝐴}) |
4 | df-fn 5238 | . 2 ⊢ ({〈𝐴, 𝐵〉} Fn {𝐴} ↔ (Fun {〈𝐴, 𝐵〉} ∧ dom {〈𝐴, 𝐵〉} = {𝐴})) | |
5 | 1, 3, 4 | sylanbrc 417 | 1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → {〈𝐴, 𝐵〉} Fn {𝐴}) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 = wceq 1364 ∈ wcel 2160 {csn 3607 〈cop 3610 dom cdm 4644 Fun wfun 5229 Fn wfn 5230 |
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-ral 2473 df-rex 2474 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-id 4311 df-xp 4650 df-rel 4651 df-cnv 4652 df-co 4653 df-dm 4654 df-fun 5237 df-fn 5238 |
This theorem is referenced by: fnsn 5289 fnunsn 5342 fsnunfv 5737 tfr0dm 6346 ennnfonelemhom 12465 |
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