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Mirrors > Home > MPE Home > Th. List > funcnvsn | Structured version Visualization version GIF version |
Description: The converse singleton of an ordered pair is a function. This is equivalent to funsn 5853 via cnvsn 5536, but stating it this way allows us 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 5422 | . 2 ⊢ Rel ◡{〈𝐴, 𝐵〉} | |
2 | moeq 3349 | . . . 4 ⊢ ∃*𝑦 𝑦 = 𝐴 | |
3 | vex 3176 | . . . . . . . 8 ⊢ 𝑥 ∈ V | |
4 | vex 3176 | . . . . . . . 8 ⊢ 𝑦 ∈ V | |
5 | 3, 4 | brcnv 5227 | . . . . . . 7 ⊢ (𝑥◡{〈𝐴, 𝐵〉}𝑦 ↔ 𝑦{〈𝐴, 𝐵〉}𝑥) |
6 | df-br 4584 | . . . . . . 7 ⊢ (𝑦{〈𝐴, 𝐵〉}𝑥 ↔ 〈𝑦, 𝑥〉 ∈ {〈𝐴, 𝐵〉}) | |
7 | 5, 6 | bitri 263 | . . . . . 6 ⊢ (𝑥◡{〈𝐴, 𝐵〉}𝑦 ↔ 〈𝑦, 𝑥〉 ∈ {〈𝐴, 𝐵〉}) |
8 | elsni 4142 | . . . . . . 7 ⊢ (〈𝑦, 𝑥〉 ∈ {〈𝐴, 𝐵〉} → 〈𝑦, 𝑥〉 = 〈𝐴, 𝐵〉) | |
9 | 4, 3 | opth1 4870 | . . . . . . 7 ⊢ (〈𝑦, 𝑥〉 = 〈𝐴, 𝐵〉 → 𝑦 = 𝐴) |
10 | 8, 9 | syl 17 | . . . . . 6 ⊢ (〈𝑦, 𝑥〉 ∈ {〈𝐴, 𝐵〉} → 𝑦 = 𝐴) |
11 | 7, 10 | sylbi 206 | . . . . 5 ⊢ (𝑥◡{〈𝐴, 𝐵〉}𝑦 → 𝑦 = 𝐴) |
12 | 11 | moimi 2508 | . . . 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 1713 | . 2 ⊢ ∀𝑥∃*𝑦 𝑥◡{〈𝐴, 𝐵〉}𝑦 |
15 | dffun6 5819 | . 2 ⊢ (Fun ◡{〈𝐴, 𝐵〉} ↔ (Rel ◡{〈𝐴, 𝐵〉} ∧ ∀𝑥∃*𝑦 𝑥◡{〈𝐴, 𝐵〉}𝑦)) | |
16 | 1, 14, 15 | mpbir2an 957 | 1 ⊢ Fun ◡{〈𝐴, 𝐵〉} |
Colors of variables: wff setvar class |
Syntax hints: ∀wal 1473 = wceq 1475 ∈ wcel 1977 ∃*wmo 2459 {csn 4125 〈cop 4131 class class class wbr 4583 ◡ccnv 5037 Rel wrel 5043 Fun wfun 5798 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pr 4833 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-rab 2905 df-v 3175 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-br 4584 df-opab 4644 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-fun 5806 |
This theorem is referenced by: funsng 5851 funcnvpr 5864 funcnvtp 5865 funcnvs1 13507 strlemor1 15796 0spth 26101 2pthlem1 26125 0spth-av 41294 |
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