dfrel2 - Metamath Proof Explorer

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Theorem dfrel2 5583

Description: Alternate definition of relation. Exercise 2 of [TakeutiZaring] p. 25. (Contributed by NM, 29-Dec-1996.)

**Assertion**
Ref	Expression
dfrel2	⊢ (Rel 𝑅 ↔ ^◡^◡𝑅 = 𝑅)

Proof of Theorem dfrel2 Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		relcnv 5503	. . 3 ⊢ Rel ^◡^◡𝑅
2		vex 3203	. . . . . 6 ⊢ 𝑥 ∈ V
3		vex 3203	. . . . . 6 ⊢ 𝑦 ∈ V
4	2, 3	opelcnv 5304	. . . . 5 ⊢ (⟨𝑥, 𝑦⟩ ∈ ^◡^◡𝑅 ↔ ⟨𝑦, 𝑥⟩ ∈ ^◡𝑅)
5	3, 2	opelcnv 5304	. . . . 5 ⊢ (⟨𝑦, 𝑥⟩ ∈ ^◡𝑅 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝑅)
6	4, 5	bitri 264	. . . 4 ⊢ (⟨𝑥, 𝑦⟩ ∈ ^◡^◡𝑅 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝑅)
7	6	eqrelriv 5213	. . 3 ⊢ ((Rel ^◡^◡𝑅 ∧ Rel 𝑅) → ^◡^◡𝑅 = 𝑅)
8	1, 7	mpan 706	. 2 ⊢ (Rel 𝑅 → ^◡^◡𝑅 = 𝑅)
9		releq 5201	. . 3 ⊢ (^◡^◡𝑅 = 𝑅 → (Rel ^◡^◡𝑅 ↔ Rel 𝑅))
10	1, 9	mpbii 223	. 2 ⊢ (^◡^◡𝑅 = 𝑅 → Rel 𝑅)
11	8, 10	impbii 199	1 ⊢ (Rel 𝑅 ↔ ^◡^◡𝑅 = 𝑅)

Colors of variables: wff setvar class

Syntax hints: ↔ wb 196 = wceq 1483 ∈ wcel 1990 ⟨cop 4183 ^◡ccnv 5113 Rel wrel 5119

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-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-xp 5120 df-rel 5121 df-cnv 5122

This theorem is referenced by: dfrel4v 5584 cnvcnv 5586 cnvcnvOLD 5587 cnveqb 5590 dfrel3 5592 cnvcnvres 5598 cnvsn 5618 cores2 5648 co01 5650 coi2 5652 relcnvtr 5655 funcnvres2 5969 f1cnvcnv 6109 f1ocnv 6149 f1ocnvb 6150 f1ococnv1 6165 fimacnvinrn 6348 isores1 6584 relcnvexb 7114 cnvf1o 7276 fnwelem 7292 tposf12 7377 ssenen 8134 cantnffval2 8592 fsumcnv 14504 fprodcnv 14713 structcnvcnv 15871 imasless 16200 oppcinv 16440 cnvps 17212 cnvpsb 17213 cnvtsr 17222 gimcnv 17709 lmimcnv 19067 hmeocnv 21565 hmeocnvb 21577 cmphaushmeo 21603 ustexsym 22019 pi1xfrcnv 22857 dvlog 24397 efopnlem2 24403 gtiso 29478 f1ocan2fv 33522 relcnveq3 34092 relcnveq2 34094 dfrel5 34114 ltrncnvnid 35413 relintab 37889 cnvssb 37892 relnonrel 37893 cononrel1 37900 cononrel2 37901 clrellem 37929 clcnvlem 37930 relexpaddss 38010