funcocnv2 - Intuitionistic Logic Explorer

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Theorem funcocnv2 5151

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 4904	. . 3 ⊢ (Fun 𝐹 ↔ (Rel 𝐹 ∧ (𝐹 ∘ ^◡𝐹) ⊆ I ))
2	1	simprbi 260	. 2 ⊢ (Fun 𝐹 → (𝐹 ∘ ^◡𝐹) ⊆ I )
3		iss 4654	. . 3 ⊢ ((𝐹 ∘ ^◡𝐹) ⊆ I ↔ (𝐹 ∘ ^◡𝐹) = ( I ↾ dom (𝐹 ∘ ^◡𝐹)))
4		dfdm4 4527	. . . . . . . 8 ⊢ dom 𝐹 = ran ^◡𝐹
5		dmcoeq 4604	. . . . . . . 8 ⊢ (dom 𝐹 = ran ^◡𝐹 → dom (𝐹 ∘ ^◡𝐹) = dom ^◡𝐹)
6	4, 5	ax-mp 7	. . . . . . 7 ⊢ dom (𝐹 ∘ ^◡𝐹) = dom ^◡𝐹
7		df-rn 4356	. . . . . . 7 ⊢ ran 𝐹 = dom ^◡𝐹
8	6, 7	eqtr4i 2063	. . . . . 6 ⊢ dom (𝐹 ∘ ^◡𝐹) = ran 𝐹
9	8	a1i 9	. . . . 5 ⊢ (Fun 𝐹 → dom (𝐹 ∘ ^◡𝐹) = ran 𝐹)
10	9	reseq2d 4612	. . . 4 ⊢ (Fun 𝐹 → ( I ↾ dom (𝐹 ∘ ^◡𝐹)) = ( I ↾ ran 𝐹))
11	10	eqeq2d 2051	. . 3 ⊢ (Fun 𝐹 → ((𝐹 ∘ ^◡𝐹) = ( I ↾ dom (𝐹 ∘ ^◡𝐹)) ↔ (𝐹 ∘ ^◡𝐹) = ( I ↾ ran 𝐹)))
12	3, 11	syl5bb 181	. 2 ⊢ (Fun 𝐹 → ((𝐹 ∘ ^◡𝐹) ⊆ I ↔ (𝐹 ∘ ^◡𝐹) = ( I ↾ ran 𝐹)))
13	2, 12	mpbid 135	1 ⊢ (Fun 𝐹 → (𝐹 ∘ ^◡𝐹) = ( I ↾ ran 𝐹))

Colors of variables: wff set class

Syntax hints: → wi 4 = wceq 1243 ⊆ wss 2917 I cid 4025 ^◡ccnv 4344 dom cdm 4345 ran crn 4346 ↾ cres 4347 ∘ ccom 4349 Rel wrel 4350 Fun wfun 4896

This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-14 1405 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 ax-sep 3875 ax-pow 3927 ax-pr 3944

This theorem depends on definitions: df-bi 110 df-3an 887 df-tru 1246 df-nf 1350 df-sb 1646 df-eu 1903 df-mo 1904 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-ral 2311 df-rex 2312 df-v 2559 df-un 2922 df-in 2924 df-ss 2931 df-pw 3361 df-sn 3381 df-pr 3382 df-op 3384 df-br 3765 df-opab 3819 df-id 4030 df-xp 4351 df-rel 4352 df-cnv 4353 df-co 4354 df-dm 4355 df-rn 4356 df-res 4357 df-fun 4904

This theorem is referenced by: fococnv2 5152 f1cocnv2 5154 funcoeqres 5157