funcocnv2 - Intuitionistic Logic Explorer

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Theorem funcocnv2 5667

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 5382	. . 3 ⊢ (Fun 𝐹 ↔ (Rel 𝐹 ∧ (𝐹 ∘ ^◡𝐹) ⊆ I ))
2	1	simprbi 275	. 2 ⊢ (Fun 𝐹 → (𝐹 ∘ ^◡𝐹) ⊆ I )
3		iss 5112	. . 3 ⊢ ((𝐹 ∘ ^◡𝐹) ⊆ I ↔ (𝐹 ∘ ^◡𝐹) = ( I ↾ dom (𝐹 ∘ ^◡𝐹)))
4		dfdm4 4975	. . . . . . . 8 ⊢ dom 𝐹 = ran ^◡𝐹
5		dmcoeq 5057	. . . . . . . 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 4787	. . . . . . 7 ⊢ ran 𝐹 = dom ^◡𝐹
8	6, 7	eqtr4i 2262	. . . . . 6 ⊢ dom (𝐹 ∘ ^◡𝐹) = ran 𝐹
9	8	a1i 9	. . . . 5 ⊢ (Fun 𝐹 → dom (𝐹 ∘ ^◡𝐹) = ran 𝐹)
10	9	reseq2d 5065	. . . 4 ⊢ (Fun 𝐹 → ( I ↾ dom (𝐹 ∘ ^◡𝐹)) = ( I ↾ ran 𝐹))
11	10	eqeq2d 2250	. . 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))

Colors of variables: wff set class

This proof depends on syntax axioms: → wi 4 = wceq 1402 ⊆ wss 3220 I cid 4435 ^◡ccnv 4775 dom cdm 4776 ran crn 4777 ↾ cres 4778 ∘ ccom 4780 Rel wrel 4781 Fun wfun 5374

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 721 ax-5 1500 ax-7 1501 ax-gen 1502 ax-ie1 1546 ax-ie2 1547 ax-8 1557 ax-10 1558 ax-11 1559 ax-i12 1560 ax-bndl 1562 ax-4 1563 ax-17 1579 ax-i9 1583 ax-ial 1587 ax-i5r 1588 ax-14 2212 ax-ext 2220 ax-sep 4251 ax-pow 4313 ax-pr 4348

This proof depends on definitions: df-bi 117 df-3an 1011 df-tru 1405 df-nf 1514 df-sb 1816 df-eu 2089 df-mo 2090 df-clab 2225 df-cleq 2231 df-clel 2234 df-nfc 2381 df-ral 2533 df-rex 2534 df-v 2823 df-un 3224 df-in 3226 df-ss 3233 df-pw 3690 df-sn 3715 df-pr 3716 df-op 3718 df-br 4132 df-opab 4194 df-id 4440 df-xp 4782 df-rel 4783 df-cnv 4784 df-co 4785 df-dm 4786 df-rn 4787 df-res 4788 df-fun 5382

This theorem is used by: fococnv2 5668 f1cocnv2 5670 funcoeqres 5673

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