Theorem f1cocnv1 6801

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

Assertion

Ref	Expression
f1cocnv1	⊢ (𝐹:𝐴–1-1→𝐵 → (◡𝐹 ∘ 𝐹) = ( I ↾ 𝐴))

Proof of Theorem f1cocnv1

Step	Hyp	Ref	Expression
1		f1f1orn 6782	. 2 ⊢ (𝐹:𝐴–1-1→𝐵 → 𝐹:𝐴–1-1-onto→ran 𝐹)
2		f1ococnv1 6800	. 2 ⊢ (𝐹:𝐴–1-1-onto→ran 𝐹 → (◡𝐹 ∘ 𝐹) = ( I ↾ 𝐴))
3	1, 2	syl 17	1 ⊢ (𝐹:𝐴–1-1→𝐵 → (◡𝐹 ∘ 𝐹) = ( I ↾ 𝐴))

Syntax hints:   → wi 4   = wceq 1548   I cid 5515  ^◡ccnv 5620  ran crn 5622   ↾ cres 5623   ∘ ccom 5625  –1-1→wf1 6486  –1-1-onto→wf1o 6488

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1975  ax-7 2016  ax-8 2123  ax-9 2131  ax-ext 2713  ax-sep 5221  ax-pr 5365

This theorem depends on definitions:  df-bi 209  df-an 398  df-or 855  df-3an 1095  df-tru 1551  df-fal 1561  df-ex 1788  df-sb 2075  df-clab 2720  df-cleq 2733  df-clel 2816  df-ral 3056  df-rex 3066  df-rab 3394  df-v 3435  df-dif 3888  df-un 3890  df-in 3892  df-ss 3902  df-nul 4265  df-if 4458  df-sn 4559  df-pr 4561  df-op 4565  df-br 5076  df-opab 5138  df-id 5516  df-xp 5627  df-rel 5628  df-cnv 5629  df-co 5630  df-dm 5631  df-rn 5632  df-res 5633  df-fun 6491  df-fn 6492  df-f 6493  df-f1 6494  df-fo 6495  df-f1o 6496

This theorem is referenced by:  f1eqcocnv  7249  domss2  9068  1arithidomlem2  33631  diophrw  43223