Theorem f1cocnv2 6877

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 6807	. 2 ⊢ (𝐹:𝐴–1-1→𝐵 → Fun 𝐹)
2		funcocnv2 6874	. 2 ⊢ (Fun 𝐹 → (𝐹 ∘ ◡𝐹) = ( I ↾ ran 𝐹))
3	1, 2	syl 17	1 ⊢ (𝐹:𝐴–1-1→𝐵 → (𝐹 ∘ ◡𝐹) = ( I ↾ ran 𝐹))

Syntax hints:   → wi 4   = wceq 1537   I cid 5582  ^◡ccnv 5688  ran crn 5690   ↾ cres 5691   ∘ ccom 5693  Fun wfun 6557  –1-1→wf1 6560

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-br 5149  df-opab 5211  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568

This theorem is referenced by: (None)