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Theorem cocnv 34881
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 6111 . 2 ((𝐹𝐺) ∘ 𝐺) = (𝐹 ∘ (𝐺𝐺))
2 funcocnv2 6632 . . . . 5 (Fun 𝐺 → (𝐺𝐺) = ( I ↾ ran 𝐺))
32adantl 482 . . . 4 ((Fun 𝐹 ∧ Fun 𝐺) → (𝐺𝐺) = ( I ↾ ran 𝐺))
43coeq2d 5726 . . 3 ((Fun 𝐹 ∧ Fun 𝐺) → (𝐹 ∘ (𝐺𝐺)) = (𝐹 ∘ ( I ↾ ran 𝐺)))
5 resco 6096 . . . 4 ((𝐹 ∘ I ) ↾ ran 𝐺) = (𝐹 ∘ ( I ↾ ran 𝐺))
6 funrel 6365 . . . . . . 7 (Fun 𝐹 → Rel 𝐹)
7 coi1 6108 . . . . . . 7 (Rel 𝐹 → (𝐹 ∘ I ) = 𝐹)
86, 7syl 17 . . . . . 6 (Fun 𝐹 → (𝐹 ∘ I ) = 𝐹)
98reseq1d 5845 . . . . 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, 10syl5eqr 2867 . . 3 ((Fun 𝐹 ∧ Fun 𝐺) → (𝐹 ∘ ( I ↾ ran 𝐺)) = (𝐹 ↾ ran 𝐺))
124, 11eqtrd 2853 . 2 ((Fun 𝐹 ∧ Fun 𝐺) → (𝐹 ∘ (𝐺𝐺)) = (𝐹 ↾ ran 𝐺))
131, 12syl5eq 2865 1 ((Fun 𝐹 ∧ Fun 𝐺) → ((𝐹𝐺) ∘ 𝐺) = (𝐹 ↾ ran 𝐺))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1528   I cid 5452  ccnv 5547  ran crn 5549  cres 5550  ccom 5552  Rel wrel 5553  Fun wfun 6342
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pr 5320
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-br 5058  df-opab 5120  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-fun 6350
This theorem is referenced by: (None)
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