Theorem funcocnv2 6799

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 6494	. . 3 ⊢ (Fun 𝐹 ↔ (Rel 𝐹 ∧ (𝐹 ∘ ◡𝐹) ⊆ I ))
2	1	simprbi 497	. 2 ⊢ (Fun 𝐹 → (𝐹 ∘ ◡𝐹) ⊆ I )
3		iss 5994	. . 3 ⊢ ((𝐹 ∘ ◡𝐹) ⊆ I ↔ (𝐹 ∘ ◡𝐹) = ( I ↾ dom (𝐹 ∘ ◡𝐹)))
4		dfdm4 5844	. . . . . . 7 ⊢ dom 𝐹 = ran ◡𝐹
5		dmcoeq 5930	. . . . . . 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 5635	. . . . . 6 ⊢ ran 𝐹 = dom ◡𝐹
8	6, 7	eqtr4i 2763	. . . . 5 ⊢ dom (𝐹 ∘ ◡𝐹) = ran 𝐹
9	8	reseq2i 5935	. . . 4 ⊢ ( I ↾ dom (𝐹 ∘ ◡𝐹)) = ( I ↾ ran 𝐹)
10	9	eqeq2i 2750	. . 3 ⊢ ((𝐹 ∘ ◡𝐹) = ( I ↾ dom (𝐹 ∘ ◡𝐹)) ↔ (𝐹 ∘ ◡𝐹) = ( I ↾ ran 𝐹))
11	3, 10	bitri 275	. 2 ⊢ ((𝐹 ∘ ◡𝐹) ⊆ I ↔ (𝐹 ∘ ◡𝐹) = ( I ↾ ran 𝐹))
12	2, 11	sylib 218	1 ⊢ (Fun 𝐹 → (𝐹 ∘ ◡𝐹) = ( I ↾ ran 𝐹))

Syntax hints:   → wi 4   = wceq 1542   ⊆ wss 3890   I cid 5518  ^◡ccnv 5623  dom cdm 5624  ran crn 5625   ↾ cres 5626   ∘ ccom 5628  Rel wrel 5629  Fun wfun 6486

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

This theorem is referenced by:  fococnv2  6800  f1cocnv2  6802  funcoeqres  6805  fcoinver  32689  tocyc01  33194  cocnv  38060  frege131d  44209  gricushgr  48405