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Theorem cocnv 35883
Description: Composition with a function and then with the converse. (Contributed by Jeff Madsen, 2-Sep-2009.)
Assertion
Ref Expression
cocnv ((Fun 𝐹 ∧ Fun 𝐺) → ((𝐹𝐺) ∘ 𝐺) = (𝐹 ↾ ran 𝐺))

Proof of Theorem cocnv
StepHypRef Expression
1 coass 6169 . 2 ((𝐹𝐺) ∘ 𝐺) = (𝐹 ∘ (𝐺𝐺))
2 funcocnv2 6741 . . . . 5 (Fun 𝐺 → (𝐺𝐺) = ( I ↾ ran 𝐺))
32adantl 482 . . . 4 ((Fun 𝐹 ∧ Fun 𝐺) → (𝐺𝐺) = ( I ↾ ran 𝐺))
43coeq2d 5771 . . 3 ((Fun 𝐹 ∧ Fun 𝐺) → (𝐹 ∘ (𝐺𝐺)) = (𝐹 ∘ ( I ↾ ran 𝐺)))
5 resco 6154 . . . 4 ((𝐹 ∘ I ) ↾ ran 𝐺) = (𝐹 ∘ ( I ↾ ran 𝐺))
6 funrel 6451 . . . . . . 7 (Fun 𝐹 → Rel 𝐹)
7 coi1 6166 . . . . . . 7 (Rel 𝐹 → (𝐹 ∘ I ) = 𝐹)
86, 7syl 17 . . . . . 6 (Fun 𝐹 → (𝐹 ∘ I ) = 𝐹)
98reseq1d 5890 . . . . 5 (Fun 𝐹 → ((𝐹 ∘ I ) ↾ ran 𝐺) = (𝐹 ↾ ran 𝐺))
109adantr 481 . . . 4 ((Fun 𝐹 ∧ Fun 𝐺) → ((𝐹 ∘ I ) ↾ ran 𝐺) = (𝐹 ↾ ran 𝐺))
115, 10eqtr3id 2792 . . 3 ((Fun 𝐹 ∧ Fun 𝐺) → (𝐹 ∘ ( I ↾ ran 𝐺)) = (𝐹 ↾ ran 𝐺))
124, 11eqtrd 2778 . 2 ((Fun 𝐹 ∧ Fun 𝐺) → (𝐹 ∘ (𝐺𝐺)) = (𝐹 ↾ ran 𝐺))
131, 12eqtrid 2790 1 ((Fun 𝐹 ∧ Fun 𝐺) → ((𝐹𝐺) ∘ 𝐺) = (𝐹 ↾ ran 𝐺))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1539   I cid 5488  ccnv 5588  ran crn 5590  cres 5591  ccom 5593  Rel wrel 5594  Fun wfun 6427
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-br 5075  df-opab 5137  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-fun 6435
This theorem is referenced by: (None)
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