funcnveq - Intuitionistic Logic Explorer

	Intuitionistic Logic Explorer		< Previous Next > Nearby theorems
Mirrors > Home > ILE Home > Th. List > funcnveq		GIF version

Theorem funcnveq 5380

Description: Another way of expressing that a class is single-rooted. Counterpart to dffun2 5324. (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 5102	. . 3 ⊢ Rel ^◡𝐴
2		dffun2 5324	. . 3 ⊢ (Fun ^◡𝐴 ↔ (Rel ^◡𝐴 ∧ ∀𝑦∀𝑥∀𝑧((𝑦^◡𝐴𝑥 ∧ 𝑦^◡𝐴𝑧) → 𝑥 = 𝑧)))
3	1, 2	mpbiran 946	. 2 ⊢ (Fun ^◡𝐴 ↔ ∀𝑦∀𝑥∀𝑧((𝑦^◡𝐴𝑥 ∧ 𝑦^◡𝐴𝑧) → 𝑥 = 𝑧))
4		alcom 1524	. 2 ⊢ (∀𝑦∀𝑥∀𝑧((𝑦^◡𝐴𝑥 ∧ 𝑦^◡𝐴𝑧) → 𝑥 = 𝑧) ↔ ∀𝑥∀𝑦∀𝑧((𝑦^◡𝐴𝑥 ∧ 𝑦^◡𝐴𝑧) → 𝑥 = 𝑧))
5		vex 2802	. . . . . . 7 ⊢ 𝑦 ∈ V
6		vex 2802	. . . . . . 7 ⊢ 𝑥 ∈ V
7	5, 6	brcnv 4902	. . . . . 6 ⊢ (𝑦^◡𝐴𝑥 ↔ 𝑥𝐴𝑦)
8		vex 2802	. . . . . . 7 ⊢ 𝑧 ∈ V
9	5, 8	brcnv 4902	. . . . . 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 1517	. . 3 ⊢ (∀𝑦∀𝑧((𝑦^◡𝐴𝑥 ∧ 𝑦^◡𝐴𝑧) → 𝑥 = 𝑧) ↔ ∀𝑦∀𝑧((𝑥𝐴𝑦 ∧ 𝑧𝐴𝑦) → 𝑥 = 𝑧))
13	12	albii 1516	. 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 1393 class class class wbr 4082 ^◡ccnv 4715 Rel wrel 4721 Fun wfun 5308

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 714 ax-5 1493 ax-7 1494 ax-gen 1495 ax-ie1 1539 ax-ie2 1540 ax-8 1550 ax-10 1551 ax-11 1552 ax-i12 1553 ax-bndl 1555 ax-4 1556 ax-17 1572 ax-i9 1576 ax-ial 1580 ax-i5r 1581 ax-14 2203 ax-ext 2211 ax-sep 4201 ax-pow 4257 ax-pr 4292

This theorem depends on definitions: df-bi 117 df-3an 1004 df-tru 1398 df-nf 1507 df-sb 1809 df-eu 2080 df-mo 2081 df-clab 2216 df-cleq 2222 df-clel 2225 df-nfc 2361 df-ral 2513 df-rex 2514 df-v 2801 df-un 3201 df-in 3203 df-ss 3210 df-pw 3651 df-sn 3672 df-pr 3673 df-op 3675 df-br 4083 df-opab 4145 df-id 4381 df-xp 4722 df-rel 4723 df-cnv 4724 df-co 4725 df-fun 5316

This theorem is referenced by: imain 5399