f1ococnv1 - Metamath Proof Explorer

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Theorem f1ococnv1 6410

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 6385	. . . 4 ⊢ (𝐹:𝐴–1-1-onto→𝐵 → Rel 𝐹)
2		dfrel2 5828	. . . 4 ⊢ (Rel 𝐹 ↔ ^◡^◡𝐹 = 𝐹)
3	1, 2	sylib 210	. . 3 ⊢ (𝐹:𝐴–1-1-onto→𝐵 → ^◡^◡𝐹 = 𝐹)
4	3	coeq2d 5521	. 2 ⊢ (𝐹:𝐴–1-1-onto→𝐵 → (^◡𝐹 ∘ ^◡^◡𝐹) = (^◡𝐹 ∘ 𝐹))
5		f1ocnv 6394	. . 3 ⊢ (𝐹:𝐴–1-1-onto→𝐵 → ^◡𝐹:𝐵–1-1-onto→𝐴)
6		f1ococnv2 6408	. . 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 2863	1 ⊢ (𝐹:𝐴–1-1-onto→𝐵 → (^◡𝐹 ∘ 𝐹) = ( I ↾ 𝐴))

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1656 I cid 5251 ^◡ccnv 5345 ↾ cres 5348 ∘ ccom 5350 Rel wrel 5351 –1-1-onto→wf1o 6126

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1894 ax-4 1908 ax-5 2009 ax-6 2075 ax-7 2112 ax-9 2173 ax-10 2192 ax-11 2207 ax-12 2220 ax-13 2389 ax-ext 2803 ax-sep 5007 ax-nul 5015 ax-pr 5129

This theorem depends on definitions: df-bi 199 df-an 387 df-or 879 df-3an 1113 df-tru 1660 df-ex 1879 df-nf 1883 df-sb 2068 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ral 3122 df-rex 3123 df-rab 3126 df-v 3416 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-nul 4147 df-if 4309 df-sn 4400 df-pr 4402 df-op 4406 df-br 4876 df-opab 4938 df-id 5252 df-xp 5352 df-rel 5353 df-cnv 5354 df-co 5355 df-dm 5356 df-rn 5357 df-res 5358 df-fun 6129 df-fn 6130 df-f 6131 df-f1 6132 df-fo 6133 df-f1o 6134

This theorem is referenced by: f1cocnv1 6411 f1ocnvfv1 6792 fcof1oinvd 6808 mapen 8399 mapfien 8588 hashfacen 13534 setcinv 17099 catcisolem 17115 symggrp 18177 f1omvdco2 18225 pf1mpf 20083 ufldom 22143 motgrp 25862 fmptco1f1o 29979 fcobij 30044 symgfcoeu 30386 reprpmtf1o 31249 subfacp1lem5 31708 ltrncoidN 36202 trlcoabs2N 36796 trlcoat 36797 trlcone 36802 cdlemg47 36810 tgrpgrplem 36823 tendoipl 36871 cdlemi2 36893 cdlemk2 36906 cdlemk4 36908 cdlemk8 36912 tendocnv 37095 dvhgrp 37181 cdlemn8 37278 dihopelvalcpre 37322 dssmap2d 39155 rngcinv 42846 rngcinvALTV 42858 ringcinv 42897 ringcinvALTV 42921

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