Theorem f1cocnv2 6802

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 6732	. 2 ⊢ (𝐹:𝐴–1-1→𝐵 → Fun 𝐹)
2		funcocnv2 6799	. 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 1542   I cid 5518  ^◡ccnv 5623  ran crn 5625   ↾ cres 5626   ∘ ccom 5628  Fun wfun 6486  –1-1→wf1 6489

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709  ax-sep 5231  ax-pr 5370

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-ral 3053  df-rex 3063  df-rab 3391  df-v 3432  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-sn 4569  df-pr 4571  df-op 4575  df-br 5087  df-opab 5149  df-id 5519  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497

This theorem is referenced by: (None)