Theorem f1ococnv1 6877

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 6851	. . . 4 ⊢ (𝐹:𝐴–1-1-onto→𝐵 → Rel 𝐹)
2		dfrel2 6209	. . . 4 ⊢ (Rel 𝐹 ↔ ◡◡𝐹 = 𝐹)
3	1, 2	sylib 218	. . 3 ⊢ (𝐹:𝐴–1-1-onto→𝐵 → ◡◡𝐹 = 𝐹)
4	3	coeq2d 5873	. 2 ⊢ (𝐹:𝐴–1-1-onto→𝐵 → (◡𝐹 ∘ ◡◡𝐹) = (◡𝐹 ∘ 𝐹))
5		f1ocnv 6860	. . 3 ⊢ (𝐹:𝐴–1-1-onto→𝐵 → ◡𝐹:𝐵–1-1-onto→𝐴)
6		f1ococnv2 6875	. . 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 2779	1 ⊢ (𝐹:𝐴–1-1-onto→𝐵 → (◡𝐹 ∘ 𝐹) = ( I ↾ 𝐴))

Syntax hints:   → wi 4   = wceq 1540   I cid 5577  ^◡ccnv 5684   ↾ cres 5687   ∘ ccom 5689  Rel wrel 5690  –1-1-onto→wf1o 6560

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-br 5144  df-opab 5206  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568

This theorem is referenced by:  f1cocnv1  6878  f1ocnvfv1  7296  fcof1oinvd  7313  mapen  9181  mapfien  9448  hashfacen  14493  setcinv  18135  catcisolem  18155  symggrp  19418  f1omvdco2  19466  rngcinv  20637  ringcinv  20671  pf1mpf  22356  ufldom  23970  motgrp  28551  fmptco1f1o  32643  fcobij  32733  symgfcoeu  33102  pmtrcnel2  33110  cycpmconjslem1  33174  cycpmconjslem2  33175  reprpmtf1o  34641  subfacp1lem5  35189  ltrncoidN  40130  trlcoabs2N  40724  trlcoat  40725  trlcone  40730  cdlemg47  40738  tgrpgrplem  40751  tendoipl  40799  cdlemi2  40821  cdlemk2  40834  cdlemk4  40836  cdlemk8  40840  tendocnv  41023  dvhgrp  41109  cdlemn8  41206  dihopelvalcpre  41250  aks6d1c6lem5  42178  dssmap2d  44035  rngcinvALTV  48192  ringcinvALTV  48226