funcocnv2 - Intuitionistic Logic Explorer

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Theorem funcocnv2 5400

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 5133	. . 3 ⊢ (Fun 𝐹 ↔ (Rel 𝐹 ∧ (𝐹 ∘ ^◡𝐹) ⊆ I ))
2	1	simprbi 273	. 2 ⊢ (Fun 𝐹 → (𝐹 ∘ ^◡𝐹) ⊆ I )
3		iss 4873	. . 3 ⊢ ((𝐹 ∘ ^◡𝐹) ⊆ I ↔ (𝐹 ∘ ^◡𝐹) = ( I ↾ dom (𝐹 ∘ ^◡𝐹)))
4		dfdm4 4739	. . . . . . . 8 ⊢ dom 𝐹 = ran ^◡𝐹
5		dmcoeq 4819	. . . . . . . 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 4558	. . . . . . 7 ⊢ ran 𝐹 = dom ^◡𝐹
8	6, 7	eqtr4i 2164	. . . . . 6 ⊢ dom (𝐹 ∘ ^◡𝐹) = ran 𝐹
9	8	a1i 9	. . . . 5 ⊢ (Fun 𝐹 → dom (𝐹 ∘ ^◡𝐹) = ran 𝐹)
10	9	reseq2d 4827	. . . 4 ⊢ (Fun 𝐹 → ( I ↾ dom (𝐹 ∘ ^◡𝐹)) = ( I ↾ ran 𝐹))
11	10	eqeq2d 2152	. . 3 ⊢ (Fun 𝐹 → ((𝐹 ∘ ^◡𝐹) = ( I ↾ dom (𝐹 ∘ ^◡𝐹)) ↔ (𝐹 ∘ ^◡𝐹) = ( I ↾ ran 𝐹)))
12	3, 11	syl5bb 191	. 2 ⊢ (Fun 𝐹 → ((𝐹 ∘ ^◡𝐹) ⊆ I ↔ (𝐹 ∘ ^◡𝐹) = ( I ↾ ran 𝐹)))
13	2, 12	mpbid 146	1 ⊢ (Fun 𝐹 → (𝐹 ∘ ^◡𝐹) = ( I ↾ ran 𝐹))

Colors of variables: wff set class

Syntax hints: → wi 4 = wceq 1332 ⊆ wss 3076 I cid 4218 ^◡ccnv 4546 dom cdm 4547 ran crn 4548 ↾ cres 4549 ∘ ccom 4551 Rel wrel 4552 Fun wfun 5125

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 699 ax-5 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-14 1493 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 ax-sep 4054 ax-pow 4106 ax-pr 4139

This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1335 df-nf 1438 df-sb 1737 df-eu 2003 df-mo 2004 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-ral 2422 df-rex 2423 df-v 2691 df-un 3080 df-in 3082 df-ss 3089 df-pw 3517 df-sn 3538 df-pr 3539 df-op 3541 df-br 3938 df-opab 3998 df-id 4223 df-xp 4553 df-rel 4554 df-cnv 4555 df-co 4556 df-dm 4557 df-rn 4558 df-res 4559 df-fun 5133

This theorem is referenced by: fococnv2 5401 f1cocnv2 5403 funcoeqres 5406