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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 6600 via cnvsn 6224, 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 6102 | . 2 ⊢ Rel ◡{⟨𝐴, 𝐵⟩} | |
2 | moeq 3702 | . . . 4 ⊢ ∃*𝑦 𝑦 = 𝐴 | |
3 | vex 3476 | . . . . . . . 8 ⊢ 𝑥 ∈ V | |
4 | vex 3476 | . . . . . . . 8 ⊢ 𝑦 ∈ V | |
5 | 3, 4 | brcnv 5881 | . . . . . . 7 ⊢ (𝑥◡{⟨𝐴, 𝐵⟩}𝑦 ↔ 𝑦{⟨𝐴, 𝐵⟩}𝑥) |
6 | df-br 5148 | . . . . . . 7 ⊢ (𝑦{⟨𝐴, 𝐵⟩}𝑥 ↔ ⟨𝑦, 𝑥⟩ ∈ {⟨𝐴, 𝐵⟩}) | |
7 | 5, 6 | bitri 274 | . . . . . 6 ⊢ (𝑥◡{⟨𝐴, 𝐵⟩}𝑦 ↔ ⟨𝑦, 𝑥⟩ ∈ {⟨𝐴, 𝐵⟩}) |
8 | elsni 4644 | . . . . . . 7 ⊢ (⟨𝑦, 𝑥⟩ ∈ {⟨𝐴, 𝐵⟩} → ⟨𝑦, 𝑥⟩ = ⟨𝐴, 𝐵⟩) | |
9 | 4, 3 | opth1 5474 | . . . . . . 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 2537 | . . . 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 1795 | . 2 ⊢ ∀𝑥∃*𝑦 𝑥◡{⟨𝐴, 𝐵⟩}𝑦 |
15 | dffun6 6555 | . 2 ⊢ (Fun ◡{⟨𝐴, 𝐵⟩} ↔ (Rel ◡{⟨𝐴, 𝐵⟩} ∧ ∀𝑥∃*𝑦 𝑥◡{⟨𝐴, 𝐵⟩}𝑦)) | |
16 | 1, 14, 15 | mpbir2an 707 | 1 ⊢ Fun ◡{⟨𝐴, 𝐵⟩} |
Colors of variables: wff setvar class |
Syntax hints: ∀wal 1537 = wceq 1539 ∈ wcel 2104 ∃*wmo 2530 {csn 4627 ⟨cop 4633 class class class wbr 5147 ◡ccnv 5674 Rel wrel 5680 Fun wfun 6536 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-ext 2701 ax-sep 5298 ax-nul 5305 ax-pr 5426 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 844 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1780 df-sb 2066 df-mo 2532 df-clab 2708 df-cleq 2722 df-clel 2808 df-ral 3060 df-rex 3069 df-rab 3431 df-v 3474 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4322 df-if 4528 df-sn 4628 df-pr 4630 df-op 4634 df-br 5148 df-opab 5210 df-id 5573 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-fun 6544 |
This theorem is referenced by: funsng 6598 funcnvpr 6609 funcnvtp 6610 funcnvs1 14867 0spth 29646 funen1cnv 34389 |
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