Theorem f1ococnv1 6830

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 6803	. . . 4 ⊢ (𝐹:𝐴–1-1-onto→𝐵 → Rel 𝐹)
2		dfrel2 6169	. . . 4 ⊢ (Rel 𝐹 ↔ ◡◡𝐹 = 𝐹)
3	1, 2	sylib 220	. . 3 ⊢ (𝐹:𝐴–1-1-onto→𝐵 → ◡◡𝐹 = 𝐹)
4	3	coeq2d 5830	. 2 ⊢ (𝐹:𝐴–1-1-onto→𝐵 → (◡𝐹 ∘ ◡◡𝐹) = (◡𝐹 ∘ 𝐹))
5		f1ocnv 6813	. . 3 ⊢ (𝐹:𝐴–1-1-onto→𝐵 → ◡𝐹:𝐵–1-1-onto→𝐴)
6		f1ococnv2 6828	. . 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 2798	1 ⊢ (𝐹:𝐴–1-1-onto→𝐵 → (◡𝐹 ∘ 𝐹) = ( I ↾ 𝐴))

Syntax hints:   → wi 4   = wceq 1559   I cid 5537  ^◡ccnv 5642   ↾ cres 5645   ∘ ccom 5647  Rel wrel 5648  –1-1-onto→wf1o 6514

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-ext 2733  ax-sep 5243  ax-pr 5387

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-sb 2090  df-clab 2740  df-cleq 2753  df-clel 2836  df-ral 3076  df-rex 3086  df-rab 3414  df-v 3455  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4284  df-if 4478  df-sn 4580  df-pr 4582  df-op 4586  df-br 5098  df-opab 5160  df-id 5538  df-xp 5649  df-rel 5650  df-cnv 5651  df-co 5652  df-dm 5653  df-rn 5654  df-res 5655  df-fun 6517  df-fn 6518  df-f 6519  df-f1 6520  df-fo 6521  df-f1o 6522

This theorem is referenced by:  f1cocnv1  6831  f1ocnvfv1  7254  fcof1oinvd  7271  mapen  9106  mapfien  9347  hashfacen  14460  setcinv  18113  catcisolem  18133  symggrp  19430  f1omvdco2  19478  rngcinv  20673  ringcinv  20707  pf1mpf  22402  ufldom  24009  motgrp  28699  fmptco1f1o  32795  fcobij  32882  cocnvf1o  32891  symgfcoeu  33222  pmtrcnel2  33230  cycpmconjslem1  33294  cycpmconjslem2  33295  reprpmtf1o  34880  subfacp1lem5  35494  ltrncoidN  40712  trlcoabs2N  41306  trlcoat  41307  trlcone  41312  cdlemg47  41320  tgrpgrplem  41333  tendoipl  41381  cdlemi2  41403  cdlemk2  41416  cdlemk4  41418  cdlemk8  41422  tendocnv  41605  dvhgrp  41691  cdlemn8  41788  dihopelvalcpre  41832  aks6d1c6lem5  42754  dssmap2d  44558  rngcinvALTV  48858  ringcinvALTV  48892