|   | Metamath Proof Explorer | < Previous  
      Next > Nearby theorems | |
| Mirrors > Home > MPE Home > Th. List > fnsng | Structured version Visualization version 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 6616 | . 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → Fun {〈𝐴, 𝐵〉}) | |
| 2 | dmsnopg 6232 | . . 3 ⊢ (𝐵 ∈ 𝑊 → dom {〈𝐴, 𝐵〉} = {𝐴}) | |
| 3 | 2 | adantl 481 | . 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → dom {〈𝐴, 𝐵〉} = {𝐴}) | 
| 4 | df-fn 6563 | . 2 ⊢ ({〈𝐴, 𝐵〉} Fn {𝐴} ↔ (Fun {〈𝐴, 𝐵〉} ∧ dom {〈𝐴, 𝐵〉} = {𝐴})) | |
| 5 | 1, 3, 4 | sylanbrc 583 | 1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → {〈𝐴, 𝐵〉} Fn {𝐴}) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1539 ∈ wcel 2107 {csn 4625 〈cop 4631 dom cdm 5684 Fun wfun 6554 Fn wfn 6555 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-ext 2707 ax-sep 5295 ax-nul 5305 ax-pr 5431 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-sb 2064 df-mo 2539 df-clab 2714 df-cleq 2728 df-clel 2815 df-ral 3061 df-rex 3070 df-rab 3436 df-v 3481 df-dif 3953 df-un 3955 df-ss 3967 df-nul 4333 df-if 4525 df-sn 4626 df-pr 4628 df-op 4632 df-br 5143 df-opab 5205 df-id 5577 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-fun 6562 df-fn 6563 | 
| This theorem is referenced by: fnsn 6623 fnunop 6683 fnsnbg 7185 fvsnun2 7204 fsnunfv 7208 mat1dimscm 22482 m1detdiag 22604 noextenddif 27714 noextendlt 27715 noextendgt 27716 actfunsnf1o 34620 actfunsnrndisj 34621 breprexplema 34646 sticksstones11 42158 metakunt19 42225 | 
| Copyright terms: Public domain | W3C validator |