Theorem funcocnv2 5526

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 5257	. . 3 ⊢ (Fun 𝐹 ↔ (Rel 𝐹 ∧ (𝐹 ∘ ◡𝐹) ⊆ I ))
2	1	simprbi 275	. 2 ⊢ (Fun 𝐹 → (𝐹 ∘ ◡𝐹) ⊆ I )
3		iss 4989	. . 3 ⊢ ((𝐹 ∘ ◡𝐹) ⊆ I ↔ (𝐹 ∘ ◡𝐹) = ( I ↾ dom (𝐹 ∘ ◡𝐹)))
4		dfdm4 4855	. . . . . . . 8 ⊢ dom 𝐹 = ran ◡𝐹
5		dmcoeq 4935	. . . . . . . 8 ⊢ (dom 𝐹 = ran ◡𝐹 → dom (𝐹 ∘ ◡𝐹) = dom ◡𝐹)
6	4, 5	ax-mp 5	. . . . . . 7 ⊢ dom (𝐹 ∘ ◡𝐹) = dom ◡𝐹
7		df-rn 4671	. . . . . . 7 ⊢ ran 𝐹 = dom ◡𝐹
8	6, 7	eqtr4i 2217	. . . . . 6 ⊢ dom (𝐹 ∘ ◡𝐹) = ran 𝐹
9	8	a1i 9	. . . . 5 ⊢ (Fun 𝐹 → dom (𝐹 ∘ ◡𝐹) = ran 𝐹)
10	9	reseq2d 4943	. . . 4 ⊢ (Fun 𝐹 → ( I ↾ dom (𝐹 ∘ ◡𝐹)) = ( I ↾ ran 𝐹))
11	10	eqeq2d 2205	. . 3 ⊢ (Fun 𝐹 → ((𝐹 ∘ ◡𝐹) = ( I ↾ dom (𝐹 ∘ ◡𝐹)) ↔ (𝐹 ∘ ◡𝐹) = ( I ↾ ran 𝐹)))
12	3, 11	bitrid 192	. 2 ⊢ (Fun 𝐹 → ((𝐹 ∘ ◡𝐹) ⊆ I ↔ (𝐹 ∘ ◡𝐹) = ( I ↾ ran 𝐹)))
13	2, 12	mpbid 147	1 ⊢ (Fun 𝐹 → (𝐹 ∘ ◡𝐹) = ( I ↾ ran 𝐹))

Syntax hints:   → wi 4   = wceq 1364   ⊆ wss 3154   I cid 4320  ^◡ccnv 4659  dom cdm 4660  ran crn 4661   ↾ cres 4662   ∘ ccom 4664  Rel wrel 4665  Fun wfun 5249

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2167  ax-ext 2175  ax-sep 4148  ax-pow 4204  ax-pr 4239

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-v 2762  df-un 3158  df-in 3160  df-ss 3167  df-pw 3604  df-sn 3625  df-pr 3626  df-op 3628  df-br 4031  df-opab 4092  df-id 4325  df-xp 4666  df-rel 4667  df-cnv 4668  df-co 4669  df-dm 4670  df-rn 4671  df-res 4672  df-fun 5257

This theorem is referenced by:  fococnv2  5527  f1cocnv2  5529  funcoeqres  5532