Theorem funcocnv2 6850

Description: Composition with the converse. (Contributed by Jeff Madsen, 2-Sep-2009.)

Assertion

Ref	Expression
funcocnv2	⊢ (Fun 𝐹 → (𝐹 ∘ ◡𝐹) = ( I ↾ ran 𝐹))

Proof of Theorem funcocnv2

Step	Hyp	Ref	Expression
1		df-fun 6540	. . 3 ⊢ (Fun 𝐹 ↔ (Rel 𝐹 ∧ (𝐹 ∘ ◡𝐹) ⊆ I ))
2	1	simprbi 503	. 2 ⊢ (Fun 𝐹 → (𝐹 ∘ ◡𝐹) ⊆ I )
3		iss 6026	. . 3 ⊢ ((𝐹 ∘ ◡𝐹) ⊆ I ↔ (𝐹 ∘ ◡𝐹) = ( I ↾ dom (𝐹 ∘ ◡𝐹)))
4		dfdm4 5876	. . . . . . 7 ⊢ dom 𝐹 = ran ◡𝐹
5		dmcoeq 5961	. . . . . . 7 ⊢ (dom 𝐹 = ran ◡𝐹 → dom (𝐹 ∘ ◡𝐹) = dom ◡𝐹)
6	4, 5	ax-mp 5	. . . . . 6 ⊢ dom (𝐹 ∘ ◡𝐹) = dom ◡𝐹
7		df-rn 5661	. . . . . 6 ⊢ ran 𝐹 = dom ◡𝐹
8	6, 7	eqtr4i 2786	. . . . 5 ⊢ dom (𝐹 ∘ ◡𝐹) = ran 𝐹
9	8	reseq2i 5966	. . . 4 ⊢ ( I ↾ dom (𝐹 ∘ ◡𝐹)) = ( I ↾ ran 𝐹)
10	9	eqeq2i 2773	. . 3 ⊢ ((𝐹 ∘ ◡𝐹) = ( I ↾ dom (𝐹 ∘ ◡𝐹)) ↔ (𝐹 ∘ ◡𝐹) = ( I ↾ ran 𝐹))
11	3, 10	bitri 278	. 2 ⊢ ((𝐹 ∘ ◡𝐹) ⊆ I ↔ (𝐹 ∘ ◡𝐹) = ( I ↾ ran 𝐹))
12	2, 11	sylib 221	1 ⊢ (Fun 𝐹 → (𝐹 ∘ ◡𝐹) = ( I ↾ ran 𝐹))

This proof depends on syntax axioms:   → wi 4   = wceq 1569   ⊆ wss 3895   I cid 5544  ^◡ccnv 5649  dom cdm 5650  ran crn 5651   ↾ cres 5652   ∘ ccom 5654  Rel wrel 5655  Fun wfun 6532

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-ext 2732  ax-sep 5248  ax-pr 5390

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-ss 3896  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-sn 4585  df-pr 4587  df-op 4835  df-br 5103  df-opab 5167  df-id 5545  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-fun 6540

This theorem is used by:  fococnv2  6851  f1cocnv2  6853  funcoeqres  6856  fcoinver  33245  tocyc01  33725  cocnv  38752  frege131d  44857  gricushgr  49108