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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 5166 via cnvsn 5016, 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 4912 | . 2 ⊢ Rel ◡{〈𝐴, 𝐵〉} | |
2 | moeq 2854 | . . . 4 ⊢ ∃*𝑦 𝑦 = 𝐴 | |
3 | vex 2684 | . . . . . . . 8 ⊢ 𝑥 ∈ V | |
4 | vex 2684 | . . . . . . . 8 ⊢ 𝑦 ∈ V | |
5 | 3, 4 | brcnv 4717 | . . . . . . 7 ⊢ (𝑥◡{〈𝐴, 𝐵〉}𝑦 ↔ 𝑦{〈𝐴, 𝐵〉}𝑥) |
6 | df-br 3925 | . . . . . . 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 3540 | . . . . . . 7 ⊢ (〈𝑦, 𝑥〉 ∈ {〈𝐴, 𝐵〉} → 〈𝑦, 𝑥〉 = 〈𝐴, 𝐵〉) | |
9 | 4, 3 | opth1 4153 | . . . . . . 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 2062 | . . . 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 1425 | . 2 ⊢ ∀𝑥∃*𝑦 𝑥◡{〈𝐴, 𝐵〉}𝑦 |
15 | dffun6 5132 | . 2 ⊢ (Fun ◡{〈𝐴, 𝐵〉} ↔ (Rel ◡{〈𝐴, 𝐵〉} ∧ ∀𝑥∃*𝑦 𝑥◡{〈𝐴, 𝐵〉}𝑦)) | |
16 | 1, 14, 15 | mpbir2an 926 | 1 ⊢ Fun ◡{〈𝐴, 𝐵〉} |
Colors of variables: wff set class |
Syntax hints: ∀wal 1329 = wceq 1331 ∈ wcel 1480 ∃*wmo 1998 {csn 3522 〈cop 3525 class class class wbr 3924 ◡ccnv 4533 Rel wrel 4539 Fun wfun 5112 |
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 2119 ax-sep 4041 ax-pow 4093 ax-pr 4126 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-eu 2000 df-mo 2001 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-ral 2419 df-rex 2420 df-v 2683 df-un 3070 df-in 3072 df-ss 3079 df-pw 3507 df-sn 3528 df-pr 3529 df-op 3531 df-br 3925 df-opab 3985 df-id 4210 df-xp 4540 df-rel 4541 df-cnv 4542 df-co 4543 df-fun 5120 |
This theorem is referenced by: funsng 5164 |
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