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Theorem cocnv 34063
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 5895 . 2 ((𝐹𝐺) ∘ 𝐺) = (𝐹 ∘ (𝐺𝐺))
2 funcocnv2 6402 . . . . 5 (Fun 𝐺 → (𝐺𝐺) = ( I ↾ ran 𝐺))
32adantl 475 . . . 4 ((Fun 𝐹 ∧ Fun 𝐺) → (𝐺𝐺) = ( I ↾ ran 𝐺))
43coeq2d 5517 . . 3 ((Fun 𝐹 ∧ Fun 𝐺) → (𝐹 ∘ (𝐺𝐺)) = (𝐹 ∘ ( I ↾ ran 𝐺)))
5 resco 5880 . . . 4 ((𝐹 ∘ I ) ↾ ran 𝐺) = (𝐹 ∘ ( I ↾ ran 𝐺))
6 funrel 6140 . . . . . . 7 (Fun 𝐹 → Rel 𝐹)
7 coi1 5892 . . . . . . 7 (Rel 𝐹 → (𝐹 ∘ I ) = 𝐹)
86, 7syl 17 . . . . . 6 (Fun 𝐹 → (𝐹 ∘ I ) = 𝐹)
98reseq1d 5628 . . . . 5 (Fun 𝐹 → ((𝐹 ∘ I ) ↾ ran 𝐺) = (𝐹 ↾ ran 𝐺))
109adantr 474 . . . 4 ((Fun 𝐹 ∧ Fun 𝐺) → ((𝐹 ∘ I ) ↾ ran 𝐺) = (𝐹 ↾ ran 𝐺))
115, 10syl5eqr 2875 . . 3 ((Fun 𝐹 ∧ Fun 𝐺) → (𝐹 ∘ ( I ↾ ran 𝐺)) = (𝐹 ↾ ran 𝐺))
124, 11eqtrd 2861 . 2 ((Fun 𝐹 ∧ Fun 𝐺) → (𝐹 ∘ (𝐺𝐺)) = (𝐹 ↾ ran 𝐺))
131, 12syl5eq 2873 1 ((Fun 𝐹 ∧ Fun 𝐺) → ((𝐹𝐺) ∘ 𝐺) = (𝐹 ↾ ran 𝐺))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 386   = wceq 1658   I cid 5249  ccnv 5341  ran crn 5343  cres 5344  ccom 5346  Rel wrel 5347  Fun wfun 6117
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-13 2391  ax-ext 2803  ax-sep 5005  ax-nul 5013  ax-pr 5127
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-3an 1115  df-tru 1662  df-ex 1881  df-nf 1885  df-sb 2070  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ral 3122  df-rex 3123  df-rab 3126  df-v 3416  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-nul 4145  df-if 4307  df-sn 4398  df-pr 4400  df-op 4404  df-br 4874  df-opab 4936  df-id 5250  df-xp 5348  df-rel 5349  df-cnv 5350  df-co 5351  df-dm 5352  df-rn 5353  df-res 5354  df-fun 6125
This theorem is referenced by: (None)
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