funcocnv2 - Intuitionistic Logic Explorer

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Theorem funcocnv2 5392

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 5125	. . 3 ⊢ (Fun 𝐹 ↔ (Rel 𝐹 ∧ (𝐹 ∘ ^◡𝐹) ⊆ I ))
2	1	simprbi 273	. 2 ⊢ (Fun 𝐹 → (𝐹 ∘ ^◡𝐹) ⊆ I )
3		iss 4865	. . 3 ⊢ ((𝐹 ∘ ^◡𝐹) ⊆ I ↔ (𝐹 ∘ ^◡𝐹) = ( I ↾ dom (𝐹 ∘ ^◡𝐹)))
4		dfdm4 4731	. . . . . . . 8 ⊢ dom 𝐹 = ran ^◡𝐹
5		dmcoeq 4811	. . . . . . . 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 4550	. . . . . . 7 ⊢ ran 𝐹 = dom ^◡𝐹
8	6, 7	eqtr4i 2163	. . . . . 6 ⊢ dom (𝐹 ∘ ^◡𝐹) = ran 𝐹
9	8	a1i 9	. . . . 5 ⊢ (Fun 𝐹 → dom (𝐹 ∘ ^◡𝐹) = ran 𝐹)
10	9	reseq2d 4819	. . . 4 ⊢ (Fun 𝐹 → ( I ↾ dom (𝐹 ∘ ^◡𝐹)) = ( I ↾ ran 𝐹))
11	10	eqeq2d 2151	. . 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 1331 ⊆ wss 3071 I cid 4210 ^◡ccnv 4538 dom cdm 4539 ran crn 4540 ↾ cres 4541 ∘ ccom 4543 Rel wrel 4544 Fun wfun 5117

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 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-sep 4046 ax-pow 4098 ax-pr 4131

This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ral 2421 df-rex 2422 df-v 2688 df-un 3075 df-in 3077 df-ss 3084 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-br 3930 df-opab 3990 df-id 4215 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-rn 4550 df-res 4551 df-fun 5125

This theorem is referenced by: fococnv2 5393 f1cocnv2 5395 funcoeqres 5398