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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 6601 via cnvsn 6225, 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 6103 | . 2 ⊢ Rel ◡{⟨𝐴, 𝐵⟩} | |
2 | moeq 3703 | . . . 4 ⊢ ∃*𝑦 𝑦 = 𝐴 | |
3 | vex 3478 | . . . . . . . 8 ⊢ 𝑥 ∈ V | |
4 | vex 3478 | . . . . . . . 8 ⊢ 𝑦 ∈ V | |
5 | 3, 4 | brcnv 5882 | . . . . . . 7 ⊢ (𝑥◡{⟨𝐴, 𝐵⟩}𝑦 ↔ 𝑦{⟨𝐴, 𝐵⟩}𝑥) |
6 | df-br 5149 | . . . . . . 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 4645 | . . . . . . 7 ⊢ (⟨𝑦, 𝑥⟩ ∈ {⟨𝐴, 𝐵⟩} → ⟨𝑦, 𝑥⟩ = ⟨𝐴, 𝐵⟩) | |
9 | 4, 3 | opth1 5475 | . . . . . . 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 2539 | . . . 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 1797 | . 2 ⊢ ∀𝑥∃*𝑦 𝑥◡{⟨𝐴, 𝐵⟩}𝑦 |
15 | dffun6 6556 | . 2 ⊢ (Fun ◡{⟨𝐴, 𝐵⟩} ↔ (Rel ◡{⟨𝐴, 𝐵⟩} ∧ ∀𝑥∃*𝑦 𝑥◡{⟨𝐴, 𝐵⟩}𝑦)) | |
16 | 1, 14, 15 | mpbir2an 709 | 1 ⊢ Fun ◡{⟨𝐴, 𝐵⟩} |
Colors of variables: wff setvar class |
Syntax hints: ∀wal 1539 = wceq 1541 ∈ wcel 2106 ∃*wmo 2532 {csn 4628 ⟨cop 4634 class class class wbr 5148 ◡ccnv 5675 Rel wrel 5681 Fun wfun 6537 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-ext 2703 ax-sep 5299 ax-nul 5306 ax-pr 5427 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-sb 2068 df-mo 2534 df-clab 2710 df-cleq 2724 df-clel 2810 df-ral 3062 df-rex 3071 df-rab 3433 df-v 3476 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-sn 4629 df-pr 4631 df-op 4635 df-br 5149 df-opab 5211 df-id 5574 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-fun 6545 |
This theorem is referenced by: funsng 6599 funcnvpr 6610 funcnvtp 6611 funcnvs1 14862 0spth 29376 funen1cnv 34086 |
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