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Theorem cocnv 33831
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 5813 . 2 ((𝐹𝐺) ∘ 𝐺) = (𝐹 ∘ (𝐺𝐺))
2 funcocnv2 6320 . . . . 5 (Fun 𝐺 → (𝐺𝐺) = ( I ↾ ran 𝐺))
32adantl 473 . . . 4 ((Fun 𝐹 ∧ Fun 𝐺) → (𝐺𝐺) = ( I ↾ ran 𝐺))
43coeq2d 5438 . . 3 ((Fun 𝐹 ∧ Fun 𝐺) → (𝐹 ∘ (𝐺𝐺)) = (𝐹 ∘ ( I ↾ ran 𝐺)))
5 resco 5798 . . . 4 ((𝐹 ∘ I ) ↾ ran 𝐺) = (𝐹 ∘ ( I ↾ ran 𝐺))
6 funrel 6064 . . . . . . 7 (Fun 𝐹 → Rel 𝐹)
7 coi1 5810 . . . . . . 7 (Rel 𝐹 → (𝐹 ∘ I ) = 𝐹)
86, 7syl 17 . . . . . 6 (Fun 𝐹 → (𝐹 ∘ I ) = 𝐹)
98reseq1d 5548 . . . . 5 (Fun 𝐹 → ((𝐹 ∘ I ) ↾ ran 𝐺) = (𝐹 ↾ ran 𝐺))
109adantr 472 . . . 4 ((Fun 𝐹 ∧ Fun 𝐺) → ((𝐹 ∘ I ) ↾ ran 𝐺) = (𝐹 ↾ ran 𝐺))
115, 10syl5eqr 2806 . . 3 ((Fun 𝐹 ∧ Fun 𝐺) → (𝐹 ∘ ( I ↾ ran 𝐺)) = (𝐹 ↾ ran 𝐺))
124, 11eqtrd 2792 . 2 ((Fun 𝐹 ∧ Fun 𝐺) → (𝐹 ∘ (𝐺𝐺)) = (𝐹 ↾ ran 𝐺))
131, 12syl5eq 2804 1 ((Fun 𝐹 ∧ Fun 𝐺) → ((𝐹𝐺) ∘ 𝐺) = (𝐹 ↾ ran 𝐺))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1630   I cid 5171  ccnv 5263  ran crn 5265  cres 5266  ccom 5268  Rel wrel 5269  Fun wfun 6041
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1986  ax-6 2052  ax-7 2088  ax-9 2146  ax-10 2166  ax-11 2181  ax-12 2194  ax-13 2389  ax-ext 2738  ax-sep 4931  ax-nul 4939  ax-pr 5053
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2045  df-eu 2609  df-mo 2610  df-clab 2745  df-cleq 2751  df-clel 2754  df-nfc 2889  df-ral 3053  df-rex 3054  df-rab 3057  df-v 3340  df-dif 3716  df-un 3718  df-in 3720  df-ss 3727  df-nul 4057  df-if 4229  df-sn 4320  df-pr 4322  df-op 4326  df-br 4803  df-opab 4863  df-id 5172  df-xp 5270  df-rel 5271  df-cnv 5272  df-co 5273  df-dm 5274  df-rn 5275  df-res 5276  df-fun 6049
This theorem is referenced by: (None)
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