Theorem funcocnv2 6161

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 5890	. . 3 ⊢ (Fun 𝐹 ↔ (Rel 𝐹 ∧ (𝐹 ∘ ◡𝐹) ⊆ I ))
2	1	simprbi 480	. 2 ⊢ (Fun 𝐹 → (𝐹 ∘ ◡𝐹) ⊆ I )
3		iss 5447	. . 3 ⊢ ((𝐹 ∘ ◡𝐹) ⊆ I ↔ (𝐹 ∘ ◡𝐹) = ( I ↾ dom (𝐹 ∘ ◡𝐹)))
4		dfdm4 5316	. . . . . . 7 ⊢ dom 𝐹 = ran ◡𝐹
5		dmcoeq 5388	. . . . . . 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 5125	. . . . . 6 ⊢ ran 𝐹 = dom ◡𝐹
8	6, 7	eqtr4i 2647	. . . . 5 ⊢ dom (𝐹 ∘ ◡𝐹) = ran 𝐹
9	8	reseq2i 5393	. . . 4 ⊢ ( I ↾ dom (𝐹 ∘ ◡𝐹)) = ( I ↾ ran 𝐹)
10	9	eqeq2i 2634	. . 3 ⊢ ((𝐹 ∘ ◡𝐹) = ( I ↾ dom (𝐹 ∘ ◡𝐹)) ↔ (𝐹 ∘ ◡𝐹) = ( I ↾ ran 𝐹))
11	3, 10	bitri 264	. 2 ⊢ ((𝐹 ∘ ◡𝐹) ⊆ I ↔ (𝐹 ∘ ◡𝐹) = ( I ↾ ran 𝐹))
12	2, 11	sylib 208	1 ⊢ (Fun 𝐹 → (𝐹 ∘ ◡𝐹) = ( I ↾ ran 𝐹))

Syntax hints:   → wi 4   = wceq 1483   ⊆ wss 3574   I cid 5023  ^◡ccnv 5113  dom cdm 5114  ran crn 5115   ↾ cres 5116   ∘ ccom 5118  Rel wrel 5119  Fun wfun 5882

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-br 4654  df-opab 4713  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-fun 5890

This theorem is referenced by:  fococnv2  6162  f1cocnv2  6164  funcoeqres  6167  fcoinver  29418  cocnv  33520  frege131d  38056