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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 5169 | . 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → Fun {〈𝐴, 𝐵〉}) | |
2 | dmsnopg 5010 | . . 3 ⊢ (𝐵 ∈ 𝑊 → dom {〈𝐴, 𝐵〉} = {𝐴}) | |
3 | 2 | adantl 275 | . 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → dom {〈𝐴, 𝐵〉} = {𝐴}) |
4 | df-fn 5126 | . 2 ⊢ ({〈𝐴, 𝐵〉} Fn {𝐴} ↔ (Fun {〈𝐴, 𝐵〉} ∧ dom {〈𝐴, 𝐵〉} = {𝐴})) | |
5 | 1, 3, 4 | sylanbrc 413 | 1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → {〈𝐴, 𝐵〉} Fn {𝐴}) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 = wceq 1331 ∈ wcel 1480 {csn 3527 〈cop 3530 dom cdm 4539 Fun wfun 5117 Fn wfn 5118 |
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 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-sep 4046 ax-pow 4098 ax-pr 4131 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ral 2421 df-rex 2422 df-v 2688 df-un 3075 df-in 3077 df-ss 3084 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-br 3930 df-opab 3990 df-id 4215 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-fun 5125 df-fn 5126 |
This theorem is referenced by: fnsn 5177 fnunsn 5230 fsnunfv 5621 tfr0dm 6219 ennnfonelemhom 11928 |
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