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Theorem funcoressn 42106
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 5688 . . . . . . . 8 ((𝐺𝑋) ∈ dom 𝐹 → dom (𝐹 ↾ {(𝐺𝑋)}) = {(𝐺𝑋)})
2 df-fn 6138 . . . . . . . . 9 ((𝐹 ↾ {(𝐺𝑋)}) Fn {(𝐺𝑋)} ↔ (Fun (𝐹 ↾ {(𝐺𝑋)}) ∧ dom (𝐹 ↾ {(𝐺𝑋)}) = {(𝐺𝑋)}))
32simplbi2com 498 . . . . . . . 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 397 . . . . . 6 (((𝐺𝑋) ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {(𝐺𝑋)})) → (𝐹 ↾ {(𝐺𝑋)}) Fn {(𝐺𝑋)})
65adantr 474 . . . . 5 ((((𝐺𝑋) ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {(𝐺𝑋)})) ∧ (𝐺 Fn 𝐴𝑋𝐴)) → (𝐹 ↾ {(𝐺𝑋)}) Fn {(𝐺𝑋)})
7 fnsnfv 6518 . . . . . . . . 9 ((𝐺 Fn 𝐴𝑋𝐴) → {(𝐺𝑋)} = (𝐺 “ {𝑋}))
87adantl 475 . . . . . . . 8 ((((𝐺𝑋) ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {(𝐺𝑋)})) ∧ (𝐺 Fn 𝐴𝑋𝐴)) → {(𝐺𝑋)} = (𝐺 “ {𝑋}))
9 df-ima 5368 . . . . . . . 8 (𝐺 “ {𝑋}) = ran (𝐺 ↾ {𝑋})
108, 9syl6eq 2830 . . . . . . 7 ((((𝐺𝑋) ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {(𝐺𝑋)})) ∧ (𝐺 Fn 𝐴𝑋𝐴)) → {(𝐺𝑋)} = ran (𝐺 ↾ {𝑋}))
1110reseq2d 5642 . . . . . 6 ((((𝐺𝑋) ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {(𝐺𝑋)})) ∧ (𝐺 Fn 𝐴𝑋𝐴)) → (𝐹 ↾ {(𝐺𝑋)}) = (𝐹 ↾ ran (𝐺 ↾ {𝑋})))
1211, 10fneq12d 6228 . . . . 5 ((((𝐺𝑋) ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {(𝐺𝑋)})) ∧ (𝐺 Fn 𝐴𝑋𝐴)) → ((𝐹 ↾ {(𝐺𝑋)}) Fn {(𝐺𝑋)} ↔ (𝐹 ↾ ran (𝐺 ↾ {𝑋})) Fn ran (𝐺 ↾ {𝑋})))
136, 12mpbid 224 . . . 4 ((((𝐺𝑋) ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {(𝐺𝑋)})) ∧ (𝐺 Fn 𝐴𝑋𝐴)) → (𝐹 ↾ ran (𝐺 ↾ {𝑋})) Fn ran (𝐺 ↾ {𝑋}))
14 fnfun 6233 . . . . . . 7 (𝐺 Fn 𝐴 → Fun 𝐺)
15 funres 6177 . . . . . . . 8 (Fun 𝐺 → Fun (𝐺 ↾ {𝑋}))
16 funfn 6165 . . . . . . . 8 (Fun (𝐺 ↾ {𝑋}) ↔ (𝐺 ↾ {𝑋}) Fn dom (𝐺 ↾ {𝑋}))
1715, 16sylib 210 . . . . . . 7 (Fun 𝐺 → (𝐺 ↾ {𝑋}) Fn dom (𝐺 ↾ {𝑋}))
1814, 17syl 17 . . . . . 6 (𝐺 Fn 𝐴 → (𝐺 ↾ {𝑋}) Fn dom (𝐺 ↾ {𝑋}))
1918adantr 474 . . . . 5 ((𝐺 Fn 𝐴𝑋𝐴) → (𝐺 ↾ {𝑋}) Fn dom (𝐺 ↾ {𝑋}))
2019adantl 475 . . . 4 ((((𝐺𝑋) ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {(𝐺𝑋)})) ∧ (𝐺 Fn 𝐴𝑋𝐴)) → (𝐺 ↾ {𝑋}) Fn dom (𝐺 ↾ {𝑋}))
21 fnresfnco 42105 . . . 4 (((𝐹 ↾ ran (𝐺 ↾ {𝑋})) Fn ran (𝐺 ↾ {𝑋}) ∧ (𝐺 ↾ {𝑋}) Fn dom (𝐺 ↾ {𝑋})) → (𝐹 ∘ (𝐺 ↾ {𝑋})) Fn dom (𝐺 ↾ {𝑋}))
2213, 20, 21syl2anc 579 . . 3 ((((𝐺𝑋) ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {(𝐺𝑋)})) ∧ (𝐺 Fn 𝐴𝑋𝐴)) → (𝐹 ∘ (𝐺 ↾ {𝑋})) Fn dom (𝐺 ↾ {𝑋}))
23 fnfun 6233 . . 3 ((𝐹 ∘ (𝐺 ↾ {𝑋})) Fn dom (𝐺 ↾ {𝑋}) → Fun (𝐹 ∘ (𝐺 ↾ {𝑋})))
2422, 23syl 17 . 2 ((((𝐺𝑋) ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {(𝐺𝑋)})) ∧ (𝐺 Fn 𝐴𝑋𝐴)) → Fun (𝐹 ∘ (𝐺 ↾ {𝑋})))
25 resco 5893 . . 3 ((𝐹𝐺) ↾ {𝑋}) = (𝐹 ∘ (𝐺 ↾ {𝑋}))
2625funeqi 6156 . 2 (Fun ((𝐹𝐺) ↾ {𝑋}) ↔ Fun (𝐹 ∘ (𝐺 ↾ {𝑋})))
2724, 26sylibr 226 1 ((((𝐺𝑋) ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {(𝐺𝑋)})) ∧ (𝐺 Fn 𝐴𝑋𝐴)) → Fun ((𝐹𝐺) ↾ {𝑋}))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 386   = wceq 1601  wcel 2107  {csn 4398  dom cdm 5355  ran crn 5356  cres 5357  cima 5358  ccom 5359  Fun wfun 6129   Fn wfn 6130  cfv 6135
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  ax-sep 5017  ax-nul 5025  ax-pr 5138
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2551  df-eu 2587  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ne 2970  df-ral 3095  df-rex 3096  df-rab 3099  df-v 3400  df-sbc 3653  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-nul 4142  df-if 4308  df-sn 4399  df-pr 4401  df-op 4405  df-uni 4672  df-br 4887  df-opab 4949  df-id 5261  df-xp 5361  df-rel 5362  df-cnv 5363  df-co 5364  df-dm 5365  df-rn 5366  df-res 5367  df-ima 5368  df-iota 6099  df-fun 6137  df-fn 6138  df-fv 6143
This theorem is referenced by:  afvco2  42217
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