f1ococnv1 - Metamath Proof Explorer

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Theorem f1ococnv1 6643

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 6618	. . . 4 ⊢ (𝐹:𝐴–1-1-onto→𝐵 → Rel 𝐹)
2		dfrel2 6046	. . . 4 ⊢ (Rel 𝐹 ↔ ^◡^◡𝐹 = 𝐹)
3	1, 2	sylib 220	. . 3 ⊢ (𝐹:𝐴–1-1-onto→𝐵 → ^◡^◡𝐹 = 𝐹)
4	3	coeq2d 5733	. 2 ⊢ (𝐹:𝐴–1-1-onto→𝐵 → (^◡𝐹 ∘ ^◡^◡𝐹) = (^◡𝐹 ∘ 𝐹))
5		f1ocnv 6627	. . 3 ⊢ (𝐹:𝐴–1-1-onto→𝐵 → ^◡𝐹:𝐵–1-1-onto→𝐴)
6		f1ococnv2 6641	. . 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 2858	1 ⊢ (𝐹:𝐴–1-1-onto→𝐵 → (^◡𝐹 ∘ 𝐹) = ( I ↾ 𝐴))

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1537 I cid 5459 ^◡ccnv 5554 ↾ cres 5557 ∘ ccom 5559 Rel wrel 5560 –1-1-onto→wf1o 6354

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 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pr 5330

This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-sn 4568 df-pr 4570 df-op 4574 df-br 5067 df-opab 5129 df-id 5460 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362

This theorem is referenced by: f1cocnv1 6644 f1ocnvfv1 7033 fcof1oinvd 7049 mapen 8681 mapfien 8871 hashfacen 13813 setcinv 17350 catcisolem 17366 symggrp 18528 f1omvdco2 18576 pf1mpf 20515 ufldom 22570 motgrp 26329 fmptco1f1o 30378 fcobij 30458 symgfcoeu 30726 pmtrcnel2 30734 cycpmconjslem1 30796 cycpmconjslem2 30797 reprpmtf1o 31897 subfacp1lem5 32431 ltrncoidN 37279 trlcoabs2N 37873 trlcoat 37874 trlcone 37879 cdlemg47 37887 tgrpgrplem 37900 tendoipl 37948 cdlemi2 37970 cdlemk2 37983 cdlemk4 37985 cdlemk8 37989 tendocnv 38172 dvhgrp 38258 cdlemn8 38355 dihopelvalcpre 38399 dssmap2d 40388 rngcinv 44272 rngcinvALTV 44284 ringcinv 44323 ringcinvALTV 44347

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