f1ococnv1 - Metamath Proof Explorer

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Theorem f1ococnv1 6799

Description: The composition of a one-to-one onto function's converse and itself equals the identity relation restricted to the function's domain. (Contributed by NM, 13-Dec-2003.)

**Assertion**
Ref	Expression
f1ococnv1	⊢ (𝐹:𝐴–1-1-onto→𝐵 → (^◡𝐹 ∘ 𝐹) = ( I ↾ 𝐴))

**Proof of Theorem f1ococnv1**
Step	Hyp	Ref	Expression
1		f1orel 6773	. . . 4 ⊢ (𝐹:𝐴–1-1-onto→𝐵 → Rel 𝐹)
2		dfrel2 6143	. . . 4 ⊢ (Rel 𝐹 ↔ ^◡^◡𝐹 = 𝐹)
3	1, 2	sylib 218	. . 3 ⊢ (𝐹:𝐴–1-1-onto→𝐵 → ^◡^◡𝐹 = 𝐹)
4	3	coeq2d 5808	. 2 ⊢ (𝐹:𝐴–1-1-onto→𝐵 → (^◡𝐹 ∘ ^◡^◡𝐹) = (^◡𝐹 ∘ 𝐹))
5		f1ocnv 6782	. . 3 ⊢ (𝐹:𝐴–1-1-onto→𝐵 → ^◡𝐹:𝐵–1-1-onto→𝐴)
6		f1ococnv2 6797	. . 3 ⊢ (^◡𝐹:𝐵–1-1-onto→𝐴 → (^◡𝐹 ∘ ^◡^◡𝐹) = ( I ↾ 𝐴))
7	5, 6	syl 17	. 2 ⊢ (𝐹:𝐴–1-1-onto→𝐵 → (^◡𝐹 ∘ ^◡^◡𝐹) = ( I ↾ 𝐴))
8	4, 7	eqtr3d 2770	1 ⊢ (𝐹:𝐴–1-1-onto→𝐵 → (^◡𝐹 ∘ 𝐹) = ( I ↾ 𝐴))

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1541 I cid 5515 ^◡ccnv 5620 ↾ cres 5623 ∘ ccom 5625 Rel wrel 5626 –1-1-onto→wf1o 6487

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2115 ax-9 2123 ax-ext 2705 ax-sep 5238 ax-nul 5248 ax-pr 5374

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-sb 2068 df-clab 2712 df-cleq 2725 df-clel 2808 df-ral 3049 df-rex 3058 df-rab 3397 df-v 3439 df-dif 3901 df-un 3903 df-in 3905 df-ss 3915 df-nul 4283 df-if 4477 df-sn 4578 df-pr 4580 df-op 4584 df-br 5096 df-opab 5158 df-id 5516 df-xp 5627 df-rel 5628 df-cnv 5629 df-co 5630 df-dm 5631 df-rn 5632 df-res 5633 df-fun 6490 df-fn 6491 df-f 6492 df-f1 6493 df-fo 6494 df-f1o 6495

This theorem is referenced by: f1cocnv1 6800 f1ocnvfv1 7218 fcof1oinvd 7235 mapen 9063 mapfien 9301 hashfacen 14365 setcinv 18001 catcisolem 18021 symggrp 19316 f1omvdco2 19364 rngcinv 20556 ringcinv 20590 pf1mpf 22270 ufldom 23880 motgrp 28524 fmptco1f1o 32619 fcobij 32709 cocnvf1o 32718 symgfcoeu 33060 pmtrcnel2 33068 cycpmconjslem1 33132 cycpmconjslem2 33133 reprpmtf1o 34662 subfacp1lem5 35251 ltrncoidN 40250 trlcoabs2N 40844 trlcoat 40845 trlcone 40850 cdlemg47 40858 tgrpgrplem 40871 tendoipl 40919 cdlemi2 40941 cdlemk2 40954 cdlemk4 40956 cdlemk8 40960 tendocnv 41143 dvhgrp 41229 cdlemn8 41326 dihopelvalcpre 41370 aks6d1c6lem5 42293 dssmap2d 44142 rngcinvALTV 48403 ringcinvALTV 48437

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