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Theorem funcnveq 5424

Description: Another way of expressing that a class is single-rooted. Counterpart to dffun2 5367. (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 5145	. . 3 ⊢ Rel ^◡𝐴
2		dffun2 5367	. . 3 ⊢ (Fun ^◡𝐴 ↔ (Rel ^◡𝐴 ∧ ∀𝑦∀𝑥∀𝑧((𝑦^◡𝐴𝑥 ∧ 𝑦^◡𝐴𝑧) → 𝑥 = 𝑧)))
3	1, 2	mpbiran 949	. 2 ⊢ (Fun ^◡𝐴 ↔ ∀𝑦∀𝑥∀𝑧((𝑦^◡𝐴𝑥 ∧ 𝑦^◡𝐴𝑧) → 𝑥 = 𝑧))
4		alcom 1527	. 2 ⊢ (∀𝑦∀𝑥∀𝑧((𝑦^◡𝐴𝑥 ∧ 𝑦^◡𝐴𝑧) → 𝑥 = 𝑧) ↔ ∀𝑥∀𝑦∀𝑧((𝑦^◡𝐴𝑥 ∧ 𝑦^◡𝐴𝑧) → 𝑥 = 𝑧))
5		vex 2818	. . . . . . 7 ⊢ 𝑦 ∈ V
6		vex 2818	. . . . . . 7 ⊢ 𝑥 ∈ V
7	5, 6	brcnv 4943	. . . . . 6 ⊢ (𝑦^◡𝐴𝑥 ↔ 𝑥𝐴𝑦)
8		vex 2818	. . . . . . 7 ⊢ 𝑧 ∈ V
9	5, 8	brcnv 4943	. . . . . 6 ⊢ (𝑦^◡𝐴𝑧 ↔ 𝑧𝐴𝑦)
10	7, 9	anbi12i 460	. . . . 5 ⊢ ((𝑦^◡𝐴𝑥 ∧ 𝑦^◡𝐴𝑧) ↔ (𝑥𝐴𝑦 ∧ 𝑧𝐴𝑦))
11	10	imbi1i 238	. . . 4 ⊢ (((𝑦^◡𝐴𝑥 ∧ 𝑦^◡𝐴𝑧) → 𝑥 = 𝑧) ↔ ((𝑥𝐴𝑦 ∧ 𝑧𝐴𝑦) → 𝑥 = 𝑧))
12	11	2albii 1520	. . 3 ⊢ (∀𝑦∀𝑧((𝑦^◡𝐴𝑥 ∧ 𝑦^◡𝐴𝑧) → 𝑥 = 𝑧) ↔ ∀𝑦∀𝑧((𝑥𝐴𝑦 ∧ 𝑧𝐴𝑦) → 𝑥 = 𝑧))
13	12	albii 1519	. 2 ⊢ (∀𝑥∀𝑦∀𝑧((𝑦^◡𝐴𝑥 ∧ 𝑦^◡𝐴𝑧) → 𝑥 = 𝑧) ↔ ∀𝑥∀𝑦∀𝑧((𝑥𝐴𝑦 ∧ 𝑧𝐴𝑦) → 𝑥 = 𝑧))
14	3, 4, 13	3bitri 206	1 ⊢ (Fun ^◡𝐴 ↔ ∀𝑥∀𝑦∀𝑧((𝑥𝐴𝑦 ∧ 𝑧𝐴𝑦) → 𝑥 = 𝑧))

Colors of variables: wff set class

Syntax hints: → wi 4 ∧ wa 104 ↔ wb 105 ∀wal 1396 class class class wbr 4114 ^◡ccnv 4753 Rel wrel 4759 Fun wfun 5351

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 717 ax-5 1496 ax-7 1497 ax-gen 1498 ax-ie1 1542 ax-ie2 1543 ax-8 1553 ax-10 1554 ax-11 1555 ax-i12 1556 ax-bndl 1558 ax-4 1559 ax-17 1575 ax-i9 1579 ax-ial 1583 ax-i5r 1584 ax-14 2208 ax-ext 2216 ax-sep 4233 ax-pow 4292 ax-pr 4327

This theorem depends on definitions: df-bi 117 df-3an 1007 df-tru 1401 df-nf 1510 df-sb 1812 df-eu 2085 df-mo 2086 df-clab 2221 df-cleq 2227 df-clel 2230 df-nfc 2375 df-ral 2527 df-rex 2528 df-v 2817 df-un 3218 df-in 3220 df-ss 3227 df-pw 3676 df-sn 3700 df-pr 3701 df-op 3703 df-br 4115 df-opab 4177 df-id 4419 df-xp 4760 df-rel 4761 df-cnv 4762 df-co 4763 df-fun 5359

This theorem is referenced by: imain 5443