funcocnv2 - Intuitionistic Logic Explorer

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Theorem funcocnv2 5467

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 5200	. . 3 ⊢ (Fun 𝐹 ↔ (Rel 𝐹 ∧ (𝐹 ∘ ^◡𝐹) ⊆ I ))
2	1	simprbi 273	. 2 ⊢ (Fun 𝐹 → (𝐹 ∘ ^◡𝐹) ⊆ I )
3		iss 4937	. . 3 ⊢ ((𝐹 ∘ ^◡𝐹) ⊆ I ↔ (𝐹 ∘ ^◡𝐹) = ( I ↾ dom (𝐹 ∘ ^◡𝐹)))
4		dfdm4 4803	. . . . . . . 8 ⊢ dom 𝐹 = ran ^◡𝐹
5		dmcoeq 4883	. . . . . . . 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 4622	. . . . . . 7 ⊢ ran 𝐹 = dom ^◡𝐹
8	6, 7	eqtr4i 2194	. . . . . 6 ⊢ dom (𝐹 ∘ ^◡𝐹) = ran 𝐹
9	8	a1i 9	. . . . 5 ⊢ (Fun 𝐹 → dom (𝐹 ∘ ^◡𝐹) = ran 𝐹)
10	9	reseq2d 4891	. . . 4 ⊢ (Fun 𝐹 → ( I ↾ dom (𝐹 ∘ ^◡𝐹)) = ( I ↾ ran 𝐹))
11	10	eqeq2d 2182	. . 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 1348 ⊆ wss 3121 I cid 4273 ^◡ccnv 4610 dom cdm 4611 ran crn 4612 ↾ cres 4613 ∘ ccom 4615 Rel wrel 4616 Fun wfun 5192

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 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-14 2144 ax-ext 2152 ax-sep 4107 ax-pow 4160 ax-pr 4194

This theorem depends on definitions: df-bi 116 df-3an 975 df-tru 1351 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ral 2453 df-rex 2454 df-v 2732 df-un 3125 df-in 3127 df-ss 3134 df-pw 3568 df-sn 3589 df-pr 3590 df-op 3592 df-br 3990 df-opab 4051 df-id 4278 df-xp 4617 df-rel 4618 df-cnv 4619 df-co 4620 df-dm 4621 df-rn 4622 df-res 4623 df-fun 5200

This theorem is referenced by: fococnv2 5468 f1cocnv2 5470 funcoeqres 5473