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Theorem funcoressn 40970
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 5426 . . . . . . . 8 ((𝐺𝑋) ∈ dom 𝐹 → dom (𝐹 ↾ {(𝐺𝑋)}) = {(𝐺𝑋)})
2 df-fn 5879 . . . . . . . . 9 ((𝐹 ↾ {(𝐺𝑋)}) Fn {(𝐺𝑋)} ↔ (Fun (𝐹 ↾ {(𝐺𝑋)}) ∧ dom (𝐹 ↾ {(𝐺𝑋)}) = {(𝐺𝑋)}))
32simplbi2com 656 . . . . . . . 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 445 . . . . . 6 (((𝐺𝑋) ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {(𝐺𝑋)})) → (𝐹 ↾ {(𝐺𝑋)}) Fn {(𝐺𝑋)})
65adantr 481 . . . . 5 ((((𝐺𝑋) ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {(𝐺𝑋)})) ∧ (𝐺 Fn 𝐴𝑋𝐴)) → (𝐹 ↾ {(𝐺𝑋)}) Fn {(𝐺𝑋)})
7 fnsnfv 6245 . . . . . . . . 9 ((𝐺 Fn 𝐴𝑋𝐴) → {(𝐺𝑋)} = (𝐺 “ {𝑋}))
87adantl 482 . . . . . . . 8 ((((𝐺𝑋) ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {(𝐺𝑋)})) ∧ (𝐺 Fn 𝐴𝑋𝐴)) → {(𝐺𝑋)} = (𝐺 “ {𝑋}))
9 df-ima 5117 . . . . . . . 8 (𝐺 “ {𝑋}) = ran (𝐺 ↾ {𝑋})
108, 9syl6eq 2670 . . . . . . 7 ((((𝐺𝑋) ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {(𝐺𝑋)})) ∧ (𝐺 Fn 𝐴𝑋𝐴)) → {(𝐺𝑋)} = ran (𝐺 ↾ {𝑋}))
1110reseq2d 5385 . . . . . 6 ((((𝐺𝑋) ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {(𝐺𝑋)})) ∧ (𝐺 Fn 𝐴𝑋𝐴)) → (𝐹 ↾ {(𝐺𝑋)}) = (𝐹 ↾ ran (𝐺 ↾ {𝑋})))
1211, 10fneq12d 5971 . . . . 5 ((((𝐺𝑋) ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {(𝐺𝑋)})) ∧ (𝐺 Fn 𝐴𝑋𝐴)) → ((𝐹 ↾ {(𝐺𝑋)}) Fn {(𝐺𝑋)} ↔ (𝐹 ↾ ran (𝐺 ↾ {𝑋})) Fn ran (𝐺 ↾ {𝑋})))
136, 12mpbid 222 . . . 4 ((((𝐺𝑋) ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {(𝐺𝑋)})) ∧ (𝐺 Fn 𝐴𝑋𝐴)) → (𝐹 ↾ ran (𝐺 ↾ {𝑋})) Fn ran (𝐺 ↾ {𝑋}))
14 fnfun 5976 . . . . . . 7 (𝐺 Fn 𝐴 → Fun 𝐺)
15 funres 5917 . . . . . . . 8 (Fun 𝐺 → Fun (𝐺 ↾ {𝑋}))
16 funfn 5906 . . . . . . . 8 (Fun (𝐺 ↾ {𝑋}) ↔ (𝐺 ↾ {𝑋}) Fn dom (𝐺 ↾ {𝑋}))
1715, 16sylib 208 . . . . . . 7 (Fun 𝐺 → (𝐺 ↾ {𝑋}) Fn dom (𝐺 ↾ {𝑋}))
1814, 17syl 17 . . . . . 6 (𝐺 Fn 𝐴 → (𝐺 ↾ {𝑋}) Fn dom (𝐺 ↾ {𝑋}))
1918adantr 481 . . . . 5 ((𝐺 Fn 𝐴𝑋𝐴) → (𝐺 ↾ {𝑋}) Fn dom (𝐺 ↾ {𝑋}))
2019adantl 482 . . . 4 ((((𝐺𝑋) ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {(𝐺𝑋)})) ∧ (𝐺 Fn 𝐴𝑋𝐴)) → (𝐺 ↾ {𝑋}) Fn dom (𝐺 ↾ {𝑋}))
21 fnresfnco 40969 . . . 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 692 . . 3 ((((𝐺𝑋) ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {(𝐺𝑋)})) ∧ (𝐺 Fn 𝐴𝑋𝐴)) → (𝐹 ∘ (𝐺 ↾ {𝑋})) Fn dom (𝐺 ↾ {𝑋}))
23 fnfun 5976 . . 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 5627 . . 3 ((𝐹𝐺) ↾ {𝑋}) = (𝐹 ∘ (𝐺 ↾ {𝑋}))
2625funeqi 5897 . 2 (Fun ((𝐹𝐺) ↾ {𝑋}) ↔ Fun (𝐹 ∘ (𝐺 ↾ {𝑋})))
2724, 26sylibr 224 1 ((((𝐺𝑋) ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {(𝐺𝑋)})) ∧ (𝐺 Fn 𝐴𝑋𝐴)) → Fun ((𝐹𝐺) ↾ {𝑋}))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1481  wcel 1988  {csn 4168  dom cdm 5104  ran crn 5105  cres 5106  cima 5107  ccom 5108  Fun wfun 5870   Fn wfn 5871  cfv 5876
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-sep 4772  ax-nul 4780  ax-pr 4897
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ne 2792  df-ral 2914  df-rex 2915  df-rab 2918  df-v 3197  df-sbc 3430  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-nul 3908  df-if 4078  df-sn 4169  df-pr 4171  df-op 4175  df-uni 4428  df-br 4645  df-opab 4704  df-id 5014  df-xp 5110  df-rel 5111  df-cnv 5112  df-co 5113  df-dm 5114  df-rn 5115  df-res 5116  df-ima 5117  df-iota 5839  df-fun 5878  df-fn 5879  df-fv 5884
This theorem is referenced by:  afvco2  41019
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