Theorem f1cocnv2 6851

Description: Composition of an injective function with its converse. (Contributed by FL, 11-Nov-2011.)

Assertion

Ref	Expression
f1cocnv2	⊢ (𝐹:𝐴–1-1→𝐵 → (𝐹 ∘ ◡𝐹) = ( I ↾ ran 𝐹))

Proof of Theorem f1cocnv2

Step	Hyp	Ref	Expression
1		f1fun 6778	. 2 ⊢ (𝐹:𝐴–1-1→𝐵 → Fun 𝐹)
2		funcocnv2 6848	. 2 ⊢ (Fun 𝐹 → (𝐹 ∘ ◡𝐹) = ( I ↾ ran 𝐹))
3	1, 2	syl 18	1 ⊢ (𝐹:𝐴–1-1→𝐵 → (𝐹 ∘ ◡𝐹) = ( I ↾ ran 𝐹))

Syntax hints:   → wi 4   = wceq 1570   I cid 5557  ^◡ccnv 5662  ran crn 5664   ↾ cres 5665   ∘ ccom 5667  Fun wfun 6532  –1-1→wf1 6535

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-ext 2735  ax-sep 5258  ax-pr 5406

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-sb 2097  df-clab 2742  df-cleq 2755  df-clel 2838  df-ral 3080  df-rex 3090  df-rab 3417  df-v 3457  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4288  df-if 4489  df-sn 4591  df-pr 4593  df-op 4597  df-br 5111  df-opab 5175  df-id 5558  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543

This theorem is referenced by: (None)