![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 6555 via cnvsn 6179, 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 6057 | . 2 ⊢ Rel ◡{⟨𝐴, 𝐵⟩} | |
2 | moeq 3666 | . . . 4 ⊢ ∃*𝑦 𝑦 = 𝐴 | |
3 | vex 3448 | . . . . . . . 8 ⊢ 𝑥 ∈ V | |
4 | vex 3448 | . . . . . . . 8 ⊢ 𝑦 ∈ V | |
5 | 3, 4 | brcnv 5839 | . . . . . . 7 ⊢ (𝑥◡{⟨𝐴, 𝐵⟩}𝑦 ↔ 𝑦{⟨𝐴, 𝐵⟩}𝑥) |
6 | df-br 5107 | . . . . . . 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 4604 | . . . . . . 7 ⊢ (⟨𝑦, 𝑥⟩ ∈ {⟨𝐴, 𝐵⟩} → ⟨𝑦, 𝑥⟩ = ⟨𝐴, 𝐵⟩) | |
9 | 4, 3 | opth1 5433 | . . . . . . 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 2540 | . . . 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 6510 | . 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 2533 {csn 4587 ⟨cop 4593 class class class wbr 5106 ◡ccnv 5633 Rel wrel 5639 Fun wfun 6491 |
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 2704 ax-sep 5257 ax-nul 5264 ax-pr 5385 |
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 2535 df-clab 2711 df-cleq 2725 df-clel 2811 df-ral 3062 df-rex 3071 df-rab 3407 df-v 3446 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-nul 4284 df-if 4488 df-sn 4588 df-pr 4590 df-op 4594 df-br 5107 df-opab 5169 df-id 5532 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-fun 6499 |
This theorem is referenced by: funsng 6553 funcnvpr 6564 funcnvtp 6565 funcnvs1 14807 0spth 29112 funen1cnv 33749 |
Copyright terms: Public domain | W3C validator |