Theorem f1ococnv1 5668

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 5642	. . . 4 ⊢ (𝐹:𝐴–1-1-onto→𝐵 → Rel 𝐹)
2		dfrel2 5238	. . . 4 ⊢ (Rel 𝐹 ↔ ◡◡𝐹 = 𝐹)
3	1, 2	sylib 122	. . 3 ⊢ (𝐹:𝐴–1-1-onto→𝐵 → ◡◡𝐹 = 𝐹)
4	3	coeq2d 4942	. 2 ⊢ (𝐹:𝐴–1-1-onto→𝐵 → (◡𝐹 ∘ ◡◡𝐹) = (◡𝐹 ∘ 𝐹))
5		f1ocnv 5652	. . 3 ⊢ (𝐹:𝐴–1-1-onto→𝐵 → ◡𝐹:𝐵–1-1-onto→𝐴)
6		f1ococnv2 5666	. . 3 ⊢ (◡𝐹:𝐵–1-1-onto→𝐴 → (◡𝐹 ∘ ◡◡𝐹) = ( I ↾ 𝐴))
7	5, 6	syl 14	. 2 ⊢ (𝐹:𝐴–1-1-onto→𝐵 → (◡𝐹 ∘ ◡◡𝐹) = ( I ↾ 𝐴))
8	4, 7	eqtr3d 2273	1 ⊢ (𝐹:𝐴–1-1-onto→𝐵 → (◡𝐹 ∘ 𝐹) = ( I ↾ 𝐴))

This proof depends on syntax axioms:   → wi 4   = wceq 1402   I cid 4433  ^◡ccnv 4773   ↾ cres 4776   ∘ ccom 4778  Rel wrel 4779  –1-1-onto→wf1o 5376

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4249  ax-pow 4311  ax-pr 4346

This proof depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-v 2823  df-un 3224  df-in 3226  df-ss 3233  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-br 4131  df-opab 4193  df-id 4438  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-fun 5379  df-fn 5380  df-f 5381  df-f1 5382  df-fo 5383  df-f1o 5384

This theorem is used by:  f1cocnv1  5669  f1ocnvfv1  5983  fcof1o  5995  mapen  7146  hashfacen  11286