Theorem funcocnv2 5488

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 5220	. . 3 ⊢ (Fun 𝐹 ↔ (Rel 𝐹 ∧ (𝐹 ∘ ◡𝐹) ⊆ I ))
2	1	simprbi 275	. 2 ⊢ (Fun 𝐹 → (𝐹 ∘ ◡𝐹) ⊆ I )
3		iss 4955	. . 3 ⊢ ((𝐹 ∘ ◡𝐹) ⊆ I ↔ (𝐹 ∘ ◡𝐹) = ( I ↾ dom (𝐹 ∘ ◡𝐹)))
4		dfdm4 4821	. . . . . . . 8 ⊢ dom 𝐹 = ran ◡𝐹
5		dmcoeq 4901	. . . . . . . 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 4639	. . . . . . 7 ⊢ ran 𝐹 = dom ◡𝐹
8	6, 7	eqtr4i 2201	. . . . . 6 ⊢ dom (𝐹 ∘ ◡𝐹) = ran 𝐹
9	8	a1i 9	. . . . 5 ⊢ (Fun 𝐹 → dom (𝐹 ∘ ◡𝐹) = ran 𝐹)
10	9	reseq2d 4909	. . . 4 ⊢ (Fun 𝐹 → ( I ↾ dom (𝐹 ∘ ◡𝐹)) = ( I ↾ ran 𝐹))
11	10	eqeq2d 2189	. . 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 1353   ⊆ wss 3131   I cid 4290  ^◡ccnv 4627  dom cdm 4628  ran crn 4629   ↾ cres 4630   ∘ ccom 4632  Rel wrel 4633  Fun wfun 5212

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4123  ax-pow 4176  ax-pr 4211

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2741  df-un 3135  df-in 3137  df-ss 3144  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-br 4006  df-opab 4067  df-id 4295  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-rn 4639  df-res 4640  df-fun 5220

This theorem is referenced by:  fococnv2  5489  f1cocnv2  5491  funcoeqres  5494