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Mirrors > Home > ILE Home > Th. List > funcnvsn | GIF version |
Description: The converse singleton of an ordered pair is a function. This is equivalent to funsn 5233 via cnvsn 5083, 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 4979 | . 2 ⊢ Rel ◡{〈𝐴, 𝐵〉} | |
2 | moeq 2899 | . . . 4 ⊢ ∃*𝑦 𝑦 = 𝐴 | |
3 | vex 2727 | . . . . . . . 8 ⊢ 𝑥 ∈ V | |
4 | vex 2727 | . . . . . . . 8 ⊢ 𝑦 ∈ V | |
5 | 3, 4 | brcnv 4784 | . . . . . . 7 ⊢ (𝑥◡{〈𝐴, 𝐵〉}𝑦 ↔ 𝑦{〈𝐴, 𝐵〉}𝑥) |
6 | df-br 3980 | . . . . . . 7 ⊢ (𝑦{〈𝐴, 𝐵〉}𝑥 ↔ 〈𝑦, 𝑥〉 ∈ {〈𝐴, 𝐵〉}) | |
7 | 5, 6 | bitri 183 | . . . . . 6 ⊢ (𝑥◡{〈𝐴, 𝐵〉}𝑦 ↔ 〈𝑦, 𝑥〉 ∈ {〈𝐴, 𝐵〉}) |
8 | elsni 3591 | . . . . . . 7 ⊢ (〈𝑦, 𝑥〉 ∈ {〈𝐴, 𝐵〉} → 〈𝑦, 𝑥〉 = 〈𝐴, 𝐵〉) | |
9 | 4, 3 | opth1 4211 | . . . . . . 7 ⊢ (〈𝑦, 𝑥〉 = 〈𝐴, 𝐵〉 → 𝑦 = 𝐴) |
10 | 8, 9 | syl 14 | . . . . . 6 ⊢ (〈𝑦, 𝑥〉 ∈ {〈𝐴, 𝐵〉} → 𝑦 = 𝐴) |
11 | 7, 10 | sylbi 120 | . . . . 5 ⊢ (𝑥◡{〈𝐴, 𝐵〉}𝑦 → 𝑦 = 𝐴) |
12 | 11 | moimi 2078 | . . . 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 1436 | . 2 ⊢ ∀𝑥∃*𝑦 𝑥◡{〈𝐴, 𝐵〉}𝑦 |
15 | dffun6 5199 | . 2 ⊢ (Fun ◡{〈𝐴, 𝐵〉} ↔ (Rel ◡{〈𝐴, 𝐵〉} ∧ ∀𝑥∃*𝑦 𝑥◡{〈𝐴, 𝐵〉}𝑦)) | |
16 | 1, 14, 15 | mpbir2an 931 | 1 ⊢ Fun ◡{〈𝐴, 𝐵〉} |
Colors of variables: wff set class |
Syntax hints: ∀wal 1340 = wceq 1342 ∃*wmo 2014 ∈ wcel 2135 {csn 3573 〈cop 3576 class class class wbr 3979 ◡ccnv 4600 Rel wrel 4606 Fun wfun 5179 |
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 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-14 2138 ax-ext 2146 ax-sep 4097 ax-pow 4150 ax-pr 4184 |
This theorem depends on definitions: df-bi 116 df-3an 969 df-tru 1345 df-nf 1448 df-sb 1750 df-eu 2016 df-mo 2017 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ral 2447 df-rex 2448 df-v 2726 df-un 3118 df-in 3120 df-ss 3127 df-pw 3558 df-sn 3579 df-pr 3580 df-op 3582 df-br 3980 df-opab 4041 df-id 4268 df-xp 4607 df-rel 4608 df-cnv 4609 df-co 4610 df-fun 5187 |
This theorem is referenced by: funsng 5231 |
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