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Theorem funcnveq 5186

Description: Another way of expressing that a class is single-rooted. Counterpart to dffun2 5133. (Contributed by Jim Kingdon, 24-Dec-2018.)

**Assertion**
Ref	Expression
funcnveq	⊢ (Fun ^◡𝐴 ↔ ∀𝑥∀𝑦∀𝑧((𝑥𝐴𝑦 ∧ 𝑧𝐴𝑦) → 𝑥 = 𝑧))

Distinct variable group: 𝑥,𝑦,𝑧,𝐴

**Proof of Theorem funcnveq**
Step	Hyp	Ref	Expression
1		relcnv 4917	. . 3 ⊢ Rel ^◡𝐴
2		dffun2 5133	. . 3 ⊢ (Fun ^◡𝐴 ↔ (Rel ^◡𝐴 ∧ ∀𝑦∀𝑥∀𝑧((𝑦^◡𝐴𝑥 ∧ 𝑦^◡𝐴𝑧) → 𝑥 = 𝑧)))
3	1, 2	mpbiran 924	. 2 ⊢ (Fun ^◡𝐴 ↔ ∀𝑦∀𝑥∀𝑧((𝑦^◡𝐴𝑥 ∧ 𝑦^◡𝐴𝑧) → 𝑥 = 𝑧))
4		alcom 1454	. 2 ⊢ (∀𝑦∀𝑥∀𝑧((𝑦^◡𝐴𝑥 ∧ 𝑦^◡𝐴𝑧) → 𝑥 = 𝑧) ↔ ∀𝑥∀𝑦∀𝑧((𝑦^◡𝐴𝑥 ∧ 𝑦^◡𝐴𝑧) → 𝑥 = 𝑧))
5		vex 2689	. . . . . . 7 ⊢ 𝑦 ∈ V
6		vex 2689	. . . . . . 7 ⊢ 𝑥 ∈ V
7	5, 6	brcnv 4722	. . . . . 6 ⊢ (𝑦^◡𝐴𝑥 ↔ 𝑥𝐴𝑦)
8		vex 2689	. . . . . . 7 ⊢ 𝑧 ∈ V
9	5, 8	brcnv 4722	. . . . . 6 ⊢ (𝑦^◡𝐴𝑧 ↔ 𝑧𝐴𝑦)
10	7, 9	anbi12i 455	. . . . 5 ⊢ ((𝑦^◡𝐴𝑥 ∧ 𝑦^◡𝐴𝑧) ↔ (𝑥𝐴𝑦 ∧ 𝑧𝐴𝑦))
11	10	imbi1i 237	. . . 4 ⊢ (((𝑦^◡𝐴𝑥 ∧ 𝑦^◡𝐴𝑧) → 𝑥 = 𝑧) ↔ ((𝑥𝐴𝑦 ∧ 𝑧𝐴𝑦) → 𝑥 = 𝑧))
12	11	2albii 1447	. . 3 ⊢ (∀𝑦∀𝑧((𝑦^◡𝐴𝑥 ∧ 𝑦^◡𝐴𝑧) → 𝑥 = 𝑧) ↔ ∀𝑦∀𝑧((𝑥𝐴𝑦 ∧ 𝑧𝐴𝑦) → 𝑥 = 𝑧))
13	12	albii 1446	. 2 ⊢ (∀𝑥∀𝑦∀𝑧((𝑦^◡𝐴𝑥 ∧ 𝑦^◡𝐴𝑧) → 𝑥 = 𝑧) ↔ ∀𝑥∀𝑦∀𝑧((𝑥𝐴𝑦 ∧ 𝑧𝐴𝑦) → 𝑥 = 𝑧))
14	3, 4, 13	3bitri 205	1 ⊢ (Fun ^◡𝐴 ↔ ∀𝑥∀𝑦∀𝑧((𝑥𝐴𝑦 ∧ 𝑧𝐴𝑦) → 𝑥 = 𝑧))

Colors of variables: wff set class

Syntax hints: → wi 4 ∧ wa 103 ↔ wb 104 ∀wal 1329 class class class wbr 3929 ^◡ccnv 4538 Rel wrel 4544 Fun wfun 5117

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-sep 4046 ax-pow 4098 ax-pr 4131

This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ral 2421 df-rex 2422 df-v 2688 df-un 3075 df-in 3077 df-ss 3084 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-br 3930 df-opab 3990 df-id 4215 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-fun 5125

This theorem is referenced by: imain 5205