funcnvsn - Metamath Proof Explorer

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Theorem funcnvsn 5936

Description: The converse singleton of an ordered pair is a function. This is equivalent to funsn 5939 via cnvsn 5618, but stating it this way allows us to skip the sethood assumptions on 𝐴 and 𝐵. (Contributed by NM, 30-Apr-2015.)

**Assertion**
Ref	Expression
funcnvsn	⊢ Fun ^◡{⟨𝐴, 𝐵⟩}

Proof of Theorem funcnvsn Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		relcnv 5503	. 2 ⊢ Rel ^◡{⟨𝐴, 𝐵⟩}
2		moeq 3382	. . . 4 ⊢ ∃*𝑦 𝑦 = 𝐴
3		vex 3203	. . . . . . . 8 ⊢ 𝑥 ∈ V
4		vex 3203	. . . . . . . 8 ⊢ 𝑦 ∈ V
5	3, 4	brcnv 5305	. . . . . . 7 ⊢ (𝑥^◡{⟨𝐴, 𝐵⟩}𝑦 ↔ 𝑦{⟨𝐴, 𝐵⟩}𝑥)
6		df-br 4654	. . . . . . 7 ⊢ (𝑦{⟨𝐴, 𝐵⟩}𝑥 ↔ ⟨𝑦, 𝑥⟩ ∈ {⟨𝐴, 𝐵⟩})
7	5, 6	bitri 264	. . . . . 6 ⊢ (𝑥^◡{⟨𝐴, 𝐵⟩}𝑦 ↔ ⟨𝑦, 𝑥⟩ ∈ {⟨𝐴, 𝐵⟩})
8		elsni 4194	. . . . . . 7 ⊢ (⟨𝑦, 𝑥⟩ ∈ {⟨𝐴, 𝐵⟩} → ⟨𝑦, 𝑥⟩ = ⟨𝐴, 𝐵⟩)
9	4, 3	opth1 4944	. . . . . . 7 ⊢ (⟨𝑦, 𝑥⟩ = ⟨𝐴, 𝐵⟩ → 𝑦 = 𝐴)
10	8, 9	syl 17	. . . . . 6 ⊢ (⟨𝑦, 𝑥⟩ ∈ {⟨𝐴, 𝐵⟩} → 𝑦 = 𝐴)
11	7, 10	sylbi 207	. . . . 5 ⊢ (𝑥^◡{⟨𝐴, 𝐵⟩}𝑦 → 𝑦 = 𝐴)
12	11	moimi 2520	. . . 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 1722	. 2 ⊢ ∀𝑥∃*𝑦 𝑥^◡{⟨𝐴, 𝐵⟩}𝑦
15		dffun6 5903	. 2 ⊢ (Fun ^◡{⟨𝐴, 𝐵⟩} ↔ (Rel ^◡{⟨𝐴, 𝐵⟩} ∧ ∀𝑥∃*𝑦 𝑥^◡{⟨𝐴, 𝐵⟩}𝑦))
16	1, 14, 15	mpbir2an 955	1 ⊢ Fun ^◡{⟨𝐴, 𝐵⟩}

Colors of variables: wff setvar class

Syntax hints: ∀wal 1481 = wceq 1483 ∈ wcel 1990 ∃*wmo 2471 {csn 4177 ⟨cop 4183 class class class wbr 4653 ^◡ccnv 5113 Rel wrel 5119 Fun wfun 5882

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pr 4906

This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ral 2917 df-rab 2921 df-v 3202 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-br 4654 df-opab 4713 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-fun 5890

This theorem is referenced by: funsng 5937 funcnvpr 5950 funcnvtp 5951 funcnvs1 13657 strlemor1OLD 15969 0spth 26987

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