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Theorem funcoressn 43632
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 5860 . . . . . . . 8 ((𝐺𝑋) ∈ dom 𝐹 → dom (𝐹 ↾ {(𝐺𝑋)}) = {(𝐺𝑋)})
2 df-fn 6327 . . . . . . . . 9 ((𝐹 ↾ {(𝐺𝑋)}) Fn {(𝐺𝑋)} ↔ (Fun (𝐹 ↾ {(𝐺𝑋)}) ∧ dom (𝐹 ↾ {(𝐺𝑋)}) = {(𝐺𝑋)}))
32simplbi2com 506 . . . . . . . 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 410 . . . . . 6 (((𝐺𝑋) ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {(𝐺𝑋)})) → (𝐹 ↾ {(𝐺𝑋)}) Fn {(𝐺𝑋)})
65adantr 484 . . . . 5 ((((𝐺𝑋) ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {(𝐺𝑋)})) ∧ (𝐺 Fn 𝐴𝑋𝐴)) → (𝐹 ↾ {(𝐺𝑋)}) Fn {(𝐺𝑋)})
7 fnsnfv 6718 . . . . . . . . 9 ((𝐺 Fn 𝐴𝑋𝐴) → {(𝐺𝑋)} = (𝐺 “ {𝑋}))
87adantl 485 . . . . . . . 8 ((((𝐺𝑋) ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {(𝐺𝑋)})) ∧ (𝐺 Fn 𝐴𝑋𝐴)) → {(𝐺𝑋)} = (𝐺 “ {𝑋}))
9 df-ima 5532 . . . . . . . 8 (𝐺 “ {𝑋}) = ran (𝐺 ↾ {𝑋})
108, 9eqtrdi 2849 . . . . . . 7 ((((𝐺𝑋) ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {(𝐺𝑋)})) ∧ (𝐺 Fn 𝐴𝑋𝐴)) → {(𝐺𝑋)} = ran (𝐺 ↾ {𝑋}))
1110reseq2d 5818 . . . . . 6 ((((𝐺𝑋) ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {(𝐺𝑋)})) ∧ (𝐺 Fn 𝐴𝑋𝐴)) → (𝐹 ↾ {(𝐺𝑋)}) = (𝐹 ↾ ran (𝐺 ↾ {𝑋})))
1211, 10fneq12d 6418 . . . . 5 ((((𝐺𝑋) ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {(𝐺𝑋)})) ∧ (𝐺 Fn 𝐴𝑋𝐴)) → ((𝐹 ↾ {(𝐺𝑋)}) Fn {(𝐺𝑋)} ↔ (𝐹 ↾ ran (𝐺 ↾ {𝑋})) Fn ran (𝐺 ↾ {𝑋})))
136, 12mpbid 235 . . . 4 ((((𝐺𝑋) ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {(𝐺𝑋)})) ∧ (𝐺 Fn 𝐴𝑋𝐴)) → (𝐹 ↾ ran (𝐺 ↾ {𝑋})) Fn ran (𝐺 ↾ {𝑋}))
14 fnfun 6423 . . . . . . 7 (𝐺 Fn 𝐴 → Fun 𝐺)
15 funres 6366 . . . . . . . 8 (Fun 𝐺 → Fun (𝐺 ↾ {𝑋}))
1615funfnd 6355 . . . . . . 7 (Fun 𝐺 → (𝐺 ↾ {𝑋}) Fn dom (𝐺 ↾ {𝑋}))
1714, 16syl 17 . . . . . 6 (𝐺 Fn 𝐴 → (𝐺 ↾ {𝑋}) Fn dom (𝐺 ↾ {𝑋}))
1817adantr 484 . . . . 5 ((𝐺 Fn 𝐴𝑋𝐴) → (𝐺 ↾ {𝑋}) Fn dom (𝐺 ↾ {𝑋}))
1918adantl 485 . . . 4 ((((𝐺𝑋) ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {(𝐺𝑋)})) ∧ (𝐺 Fn 𝐴𝑋𝐴)) → (𝐺 ↾ {𝑋}) Fn dom (𝐺 ↾ {𝑋}))
20 fnresfnco 43631 . . . 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 587 . . 3 ((((𝐺𝑋) ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {(𝐺𝑋)})) ∧ (𝐺 Fn 𝐴𝑋𝐴)) → (𝐹 ∘ (𝐺 ↾ {𝑋})) Fn dom (𝐺 ↾ {𝑋}))
22 fnfun 6423 . . 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 6070 . . 3 ((𝐹𝐺) ↾ {𝑋}) = (𝐹 ∘ (𝐺 ↾ {𝑋}))
2524funeqi 6345 . 2 (Fun ((𝐹𝐺) ↾ {𝑋}) ↔ Fun (𝐹 ∘ (𝐺 ↾ {𝑋})))
2623, 25sylibr 237 1 ((((𝐺𝑋) ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {(𝐺𝑋)})) ∧ (𝐺 Fn 𝐴𝑋𝐴)) → Fun ((𝐹𝐺) ↾ {𝑋}))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399   = wceq 1538  wcel 2111  {csn 4525  dom cdm 5519  ran crn 5520  cres 5521  cima 5522  ccom 5523  Fun wfun 6318   Fn wfn 6319  cfv 6324
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pr 5295
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3443  df-sbc 3721  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-br 5031  df-opab 5093  df-id 5425  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-iota 6283  df-fun 6326  df-fn 6327  df-fv 6332
This theorem is referenced by:  afvco2  43730
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