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Theorem funcoressn 45283
Description: A composition restricted to a singleton is a function under certain conditions. (Contributed by Alexander van der Vekens, 25-Jul-2017.)
Assertion
Ref Expression
funcoressn ((((𝐺𝑋) ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {(𝐺𝑋)})) ∧ (𝐺 Fn 𝐴𝑋𝐴)) → Fun ((𝐹𝐺) ↾ {𝑋}))

Proof of Theorem funcoressn
StepHypRef Expression
1 dmressnsn 5980 . . . . . . . 8 ((𝐺𝑋) ∈ dom 𝐹 → dom (𝐹 ↾ {(𝐺𝑋)}) = {(𝐺𝑋)})
2 df-fn 6500 . . . . . . . . 9 ((𝐹 ↾ {(𝐺𝑋)}) Fn {(𝐺𝑋)} ↔ (Fun (𝐹 ↾ {(𝐺𝑋)}) ∧ dom (𝐹 ↾ {(𝐺𝑋)}) = {(𝐺𝑋)}))
32simplbi2com 504 . . . . . . . 8 (dom (𝐹 ↾ {(𝐺𝑋)}) = {(𝐺𝑋)} → (Fun (𝐹 ↾ {(𝐺𝑋)}) → (𝐹 ↾ {(𝐺𝑋)}) Fn {(𝐺𝑋)}))
41, 3syl 17 . . . . . . 7 ((𝐺𝑋) ∈ dom 𝐹 → (Fun (𝐹 ↾ {(𝐺𝑋)}) → (𝐹 ↾ {(𝐺𝑋)}) Fn {(𝐺𝑋)}))
54imp 408 . . . . . 6 (((𝐺𝑋) ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {(𝐺𝑋)})) → (𝐹 ↾ {(𝐺𝑋)}) Fn {(𝐺𝑋)})
65adantr 482 . . . . 5 ((((𝐺𝑋) ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {(𝐺𝑋)})) ∧ (𝐺 Fn 𝐴𝑋𝐴)) → (𝐹 ↾ {(𝐺𝑋)}) Fn {(𝐺𝑋)})
7 fnsnfv 6921 . . . . . . . . 9 ((𝐺 Fn 𝐴𝑋𝐴) → {(𝐺𝑋)} = (𝐺 “ {𝑋}))
87adantl 483 . . . . . . . 8 ((((𝐺𝑋) ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {(𝐺𝑋)})) ∧ (𝐺 Fn 𝐴𝑋𝐴)) → {(𝐺𝑋)} = (𝐺 “ {𝑋}))
9 df-ima 5647 . . . . . . . 8 (𝐺 “ {𝑋}) = ran (𝐺 ↾ {𝑋})
108, 9eqtrdi 2793 . . . . . . 7 ((((𝐺𝑋) ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {(𝐺𝑋)})) ∧ (𝐺 Fn 𝐴𝑋𝐴)) → {(𝐺𝑋)} = ran (𝐺 ↾ {𝑋}))
1110reseq2d 5938 . . . . . 6 ((((𝐺𝑋) ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {(𝐺𝑋)})) ∧ (𝐺 Fn 𝐴𝑋𝐴)) → (𝐹 ↾ {(𝐺𝑋)}) = (𝐹 ↾ ran (𝐺 ↾ {𝑋})))
1211, 10fneq12d 6598 . . . . 5 ((((𝐺𝑋) ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {(𝐺𝑋)})) ∧ (𝐺 Fn 𝐴𝑋𝐴)) → ((𝐹 ↾ {(𝐺𝑋)}) Fn {(𝐺𝑋)} ↔ (𝐹 ↾ ran (𝐺 ↾ {𝑋})) Fn ran (𝐺 ↾ {𝑋})))
136, 12mpbid 231 . . . 4 ((((𝐺𝑋) ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {(𝐺𝑋)})) ∧ (𝐺 Fn 𝐴𝑋𝐴)) → (𝐹 ↾ ran (𝐺 ↾ {𝑋})) Fn ran (𝐺 ↾ {𝑋}))
14 fnfun 6603 . . . . . . 7 (𝐺 Fn 𝐴 → Fun 𝐺)
15 funres 6544 . . . . . . . 8 (Fun 𝐺 → Fun (𝐺 ↾ {𝑋}))
1615funfnd 6533 . . . . . . 7 (Fun 𝐺 → (𝐺 ↾ {𝑋}) Fn dom (𝐺 ↾ {𝑋}))
1714, 16syl 17 . . . . . 6 (𝐺 Fn 𝐴 → (𝐺 ↾ {𝑋}) Fn dom (𝐺 ↾ {𝑋}))
1817adantr 482 . . . . 5 ((𝐺 Fn 𝐴𝑋𝐴) → (𝐺 ↾ {𝑋}) Fn dom (𝐺 ↾ {𝑋}))
1918adantl 483 . . . 4 ((((𝐺𝑋) ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {(𝐺𝑋)})) ∧ (𝐺 Fn 𝐴𝑋𝐴)) → (𝐺 ↾ {𝑋}) Fn dom (𝐺 ↾ {𝑋}))
20 fnresfnco 45282 . . . 4 (((𝐹 ↾ ran (𝐺 ↾ {𝑋})) Fn ran (𝐺 ↾ {𝑋}) ∧ (𝐺 ↾ {𝑋}) Fn dom (𝐺 ↾ {𝑋})) → (𝐹 ∘ (𝐺 ↾ {𝑋})) Fn dom (𝐺 ↾ {𝑋}))
2113, 19, 20syl2anc 585 . . 3 ((((𝐺𝑋) ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {(𝐺𝑋)})) ∧ (𝐺 Fn 𝐴𝑋𝐴)) → (𝐹 ∘ (𝐺 ↾ {𝑋})) Fn dom (𝐺 ↾ {𝑋}))
22 fnfun 6603 . . 3 ((𝐹 ∘ (𝐺 ↾ {𝑋})) Fn dom (𝐺 ↾ {𝑋}) → Fun (𝐹 ∘ (𝐺 ↾ {𝑋})))
2321, 22syl 17 . 2 ((((𝐺𝑋) ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {(𝐺𝑋)})) ∧ (𝐺 Fn 𝐴𝑋𝐴)) → Fun (𝐹 ∘ (𝐺 ↾ {𝑋})))
24 resco 6203 . . 3 ((𝐹𝐺) ↾ {𝑋}) = (𝐹 ∘ (𝐺 ↾ {𝑋}))
2524funeqi 6523 . 2 (Fun ((𝐹𝐺) ↾ {𝑋}) ↔ Fun (𝐹 ∘ (𝐺 ↾ {𝑋})))
2623, 25sylibr 233 1 ((((𝐺𝑋) ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {(𝐺𝑋)})) ∧ (𝐺 Fn 𝐴𝑋𝐴)) → Fun ((𝐹𝐺) ↾ {𝑋}))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397   = wceq 1542  wcel 2107  {csn 4587  dom cdm 5634  ran crn 5635  cres 5636  cima 5637  ccom 5638  Fun wfun 6491   Fn wfn 6492  cfv 6497
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 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708  ax-sep 5257  ax-nul 5264  ax-pr 5385
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ne 2945  df-ral 3066  df-rex 3075  df-rab 3409  df-v 3448  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-br 5107  df-opab 5169  df-id 5532  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-iota 6449  df-fun 6499  df-fn 6500  df-fv 6505
This theorem is referenced by:  afvco2  45415
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