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Theorem funcnveq 5261

Description: Another way of expressing that a class is single-rooted. Counterpart to dffun2 5208. (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 4989	. . 3 ⊢ Rel ^◡𝐴
2		dffun2 5208	. . 3 ⊢ (Fun ^◡𝐴 ↔ (Rel ^◡𝐴 ∧ ∀𝑦∀𝑥∀𝑧((𝑦^◡𝐴𝑥 ∧ 𝑦^◡𝐴𝑧) → 𝑥 = 𝑧)))
3	1, 2	mpbiran 935	. 2 ⊢ (Fun ^◡𝐴 ↔ ∀𝑦∀𝑥∀𝑧((𝑦^◡𝐴𝑥 ∧ 𝑦^◡𝐴𝑧) → 𝑥 = 𝑧))
4		alcom 1471	. 2 ⊢ (∀𝑦∀𝑥∀𝑧((𝑦^◡𝐴𝑥 ∧ 𝑦^◡𝐴𝑧) → 𝑥 = 𝑧) ↔ ∀𝑥∀𝑦∀𝑧((𝑦^◡𝐴𝑥 ∧ 𝑦^◡𝐴𝑧) → 𝑥 = 𝑧))
5		vex 2733	. . . . . . 7 ⊢ 𝑦 ∈ V
6		vex 2733	. . . . . . 7 ⊢ 𝑥 ∈ V
7	5, 6	brcnv 4794	. . . . . 6 ⊢ (𝑦^◡𝐴𝑥 ↔ 𝑥𝐴𝑦)
8		vex 2733	. . . . . . 7 ⊢ 𝑧 ∈ V
9	5, 8	brcnv 4794	. . . . . 6 ⊢ (𝑦^◡𝐴𝑧 ↔ 𝑧𝐴𝑦)
10	7, 9	anbi12i 457	. . . . 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 1464	. . 3 ⊢ (∀𝑦∀𝑧((𝑦^◡𝐴𝑥 ∧ 𝑦^◡𝐴𝑧) → 𝑥 = 𝑧) ↔ ∀𝑦∀𝑧((𝑥𝐴𝑦 ∧ 𝑧𝐴𝑦) → 𝑥 = 𝑧))
13	12	albii 1463	. 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 1346 class class class wbr 3989 ^◡ccnv 4610 Rel wrel 4616 Fun wfun 5192

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 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-14 2144 ax-ext 2152 ax-sep 4107 ax-pow 4160 ax-pr 4194

This theorem depends on definitions: df-bi 116 df-3an 975 df-tru 1351 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ral 2453 df-rex 2454 df-v 2732 df-un 3125 df-in 3127 df-ss 3134 df-pw 3568 df-sn 3589 df-pr 3590 df-op 3592 df-br 3990 df-opab 4051 df-id 4278 df-xp 4617 df-rel 4618 df-cnv 4619 df-co 4620 df-fun 5200

This theorem is referenced by: imain 5280