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| Description: The converse singleton of an ordered pair is a function. This is equivalent to funsn 6618 via cnvsn 6245, but stating it this way allows to skip the sethood assumptions on 𝐴 and 𝐵. (Contributed by NM, 30-Apr-2015.) | 
| Ref | Expression | 
|---|---|
| funcnvsn | ⊢ Fun ◡{〈𝐴, 𝐵〉} | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | relcnv 6121 | . 2 ⊢ Rel ◡{〈𝐴, 𝐵〉} | |
| 2 | moeq 3712 | . . . 4 ⊢ ∃*𝑦 𝑦 = 𝐴 | |
| 3 | vex 3483 | . . . . . . . 8 ⊢ 𝑥 ∈ V | |
| 4 | vex 3483 | . . . . . . . 8 ⊢ 𝑦 ∈ V | |
| 5 | 3, 4 | brcnv 5892 | . . . . . . 7 ⊢ (𝑥◡{〈𝐴, 𝐵〉}𝑦 ↔ 𝑦{〈𝐴, 𝐵〉}𝑥) | 
| 6 | df-br 5143 | . . . . . . 7 ⊢ (𝑦{〈𝐴, 𝐵〉}𝑥 ↔ 〈𝑦, 𝑥〉 ∈ {〈𝐴, 𝐵〉}) | |
| 7 | 5, 6 | bitri 275 | . . . . . 6 ⊢ (𝑥◡{〈𝐴, 𝐵〉}𝑦 ↔ 〈𝑦, 𝑥〉 ∈ {〈𝐴, 𝐵〉}) | 
| 8 | elsni 4642 | . . . . . . 7 ⊢ (〈𝑦, 𝑥〉 ∈ {〈𝐴, 𝐵〉} → 〈𝑦, 𝑥〉 = 〈𝐴, 𝐵〉) | |
| 9 | 4, 3 | opth1 5479 | . . . . . . 7 ⊢ (〈𝑦, 𝑥〉 = 〈𝐴, 𝐵〉 → 𝑦 = 𝐴) | 
| 10 | 8, 9 | syl 17 | . . . . . 6 ⊢ (〈𝑦, 𝑥〉 ∈ {〈𝐴, 𝐵〉} → 𝑦 = 𝐴) | 
| 11 | 7, 10 | sylbi 217 | . . . . 5 ⊢ (𝑥◡{〈𝐴, 𝐵〉}𝑦 → 𝑦 = 𝐴) | 
| 12 | 11 | moimi 2544 | . . . 4 ⊢ (∃*𝑦 𝑦 = 𝐴 → ∃*𝑦 𝑥◡{〈𝐴, 𝐵〉}𝑦) | 
| 13 | 2, 12 | ax-mp 5 | . . 3 ⊢ ∃*𝑦 𝑥◡{〈𝐴, 𝐵〉}𝑦 | 
| 14 | 13 | ax-gen 1794 | . 2 ⊢ ∀𝑥∃*𝑦 𝑥◡{〈𝐴, 𝐵〉}𝑦 | 
| 15 | dffun6 6573 | . 2 ⊢ (Fun ◡{〈𝐴, 𝐵〉} ↔ (Rel ◡{〈𝐴, 𝐵〉} ∧ ∀𝑥∃*𝑦 𝑥◡{〈𝐴, 𝐵〉}𝑦)) | |
| 16 | 1, 14, 15 | mpbir2an 711 | 1 ⊢ Fun ◡{〈𝐴, 𝐵〉} | 
| Colors of variables: wff setvar class | 
| Syntax hints: ∀wal 1537 = wceq 1539 ∈ wcel 2107 ∃*wmo 2537 {csn 4625 〈cop 4631 class class class wbr 5142 ◡ccnv 5683 Rel wrel 5689 Fun wfun 6554 | 
| 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-fun 6562 | 
| This theorem is referenced by: funsng 6616 funcnvpr 6627 funcnvtp 6628 funcnvs1 14952 0spth 30146 funen1cnv 35103 | 
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