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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 6550 via cnvsn 6175, 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 6053 | . 2 ⊢ Rel ◡{〈𝐴, 𝐵〉} | |
2 | moeq 3664 | . . . 4 ⊢ ∃*𝑦 𝑦 = 𝐴 | |
3 | vex 3448 | . . . . . . . 8 ⊢ 𝑥 ∈ V | |
4 | vex 3448 | . . . . . . . 8 ⊢ 𝑦 ∈ V | |
5 | 3, 4 | brcnv 5835 | . . . . . . 7 ⊢ (𝑥◡{〈𝐴, 𝐵〉}𝑦 ↔ 𝑦{〈𝐴, 𝐵〉}𝑥) |
6 | df-br 5105 | . . . . . . 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 4602 | . . . . . . 7 ⊢ (〈𝑦, 𝑥〉 ∈ {〈𝐴, 𝐵〉} → 〈𝑦, 𝑥〉 = 〈𝐴, 𝐵〉) | |
9 | 4, 3 | opth1 5431 | . . . . . . 7 ⊢ (〈𝑦, 𝑥〉 = 〈𝐴, 𝐵〉 → 𝑦 = 𝐴) |
10 | 8, 9 | syl 17 | . . . . . 6 ⊢ (〈𝑦, 𝑥〉 ∈ {〈𝐴, 𝐵〉} → 𝑦 = 𝐴) |
11 | 7, 10 | sylbi 216 | . . . . 5 ⊢ (𝑥◡{〈𝐴, 𝐵〉}𝑦 → 𝑦 = 𝐴) |
12 | 11 | moimi 2545 | . . . 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 1798 | . 2 ⊢ ∀𝑥∃*𝑦 𝑥◡{〈𝐴, 𝐵〉}𝑦 |
15 | dffun6 6505 | . 2 ⊢ (Fun ◡{〈𝐴, 𝐵〉} ↔ (Rel ◡{〈𝐴, 𝐵〉} ∧ ∀𝑥∃*𝑦 𝑥◡{〈𝐴, 𝐵〉}𝑦)) | |
16 | 1, 14, 15 | mpbir2an 710 | 1 ⊢ Fun ◡{〈𝐴, 𝐵〉} |
Colors of variables: wff setvar class |
Syntax hints: ∀wal 1540 = wceq 1542 ∈ wcel 2107 ∃*wmo 2538 {csn 4585 〈cop 4591 class class class wbr 5104 ◡ccnv 5630 Rel wrel 5636 Fun wfun 6486 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-ext 2709 ax-sep 5255 ax-nul 5262 ax-pr 5383 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-sb 2069 df-mo 2540 df-clab 2716 df-cleq 2730 df-clel 2816 df-ral 3064 df-rex 3073 df-rab 3407 df-v 3446 df-dif 3912 df-un 3914 df-in 3916 df-ss 3926 df-nul 4282 df-if 4486 df-sn 4586 df-pr 4588 df-op 4592 df-br 5105 df-opab 5167 df-id 5529 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-fun 6494 |
This theorem is referenced by: funsng 6548 funcnvpr 6559 funcnvtp 6560 funcnvs1 14734 0spth 28875 funen1cnv 33472 |
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