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Theorem funcoressn 46329
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 6017 . . . . . . . 8 ((𝐺𝑋) ∈ dom 𝐹 → dom (𝐹 ↾ {(𝐺𝑋)}) = {(𝐺𝑋)})
2 df-fn 6540 . . . . . . . . 9 ((𝐹 ↾ {(𝐺𝑋)}) Fn {(𝐺𝑋)} ↔ (Fun (𝐹 ↾ {(𝐺𝑋)}) ∧ dom (𝐹 ↾ {(𝐺𝑋)}) = {(𝐺𝑋)}))
32simplbi2com 502 . . . . . . . 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 406 . . . . . 6 (((𝐺𝑋) ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {(𝐺𝑋)})) → (𝐹 ↾ {(𝐺𝑋)}) Fn {(𝐺𝑋)})
65adantr 480 . . . . 5 ((((𝐺𝑋) ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {(𝐺𝑋)})) ∧ (𝐺 Fn 𝐴𝑋𝐴)) → (𝐹 ↾ {(𝐺𝑋)}) Fn {(𝐺𝑋)})
7 fnsnfv 6964 . . . . . . . . 9 ((𝐺 Fn 𝐴𝑋𝐴) → {(𝐺𝑋)} = (𝐺 “ {𝑋}))
87adantl 481 . . . . . . . 8 ((((𝐺𝑋) ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {(𝐺𝑋)})) ∧ (𝐺 Fn 𝐴𝑋𝐴)) → {(𝐺𝑋)} = (𝐺 “ {𝑋}))
9 df-ima 5682 . . . . . . . 8 (𝐺 “ {𝑋}) = ran (𝐺 ↾ {𝑋})
108, 9eqtrdi 2782 . . . . . . 7 ((((𝐺𝑋) ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {(𝐺𝑋)})) ∧ (𝐺 Fn 𝐴𝑋𝐴)) → {(𝐺𝑋)} = ran (𝐺 ↾ {𝑋}))
1110reseq2d 5975 . . . . . 6 ((((𝐺𝑋) ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {(𝐺𝑋)})) ∧ (𝐺 Fn 𝐴𝑋𝐴)) → (𝐹 ↾ {(𝐺𝑋)}) = (𝐹 ↾ ran (𝐺 ↾ {𝑋})))
1211, 10fneq12d 6638 . . . . 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 6643 . . . . . . 7 (𝐺 Fn 𝐴 → Fun 𝐺)
15 funres 6584 . . . . . . . 8 (Fun 𝐺 → Fun (𝐺 ↾ {𝑋}))
1615funfnd 6573 . . . . . . 7 (Fun 𝐺 → (𝐺 ↾ {𝑋}) Fn dom (𝐺 ↾ {𝑋}))
1714, 16syl 17 . . . . . 6 (𝐺 Fn 𝐴 → (𝐺 ↾ {𝑋}) Fn dom (𝐺 ↾ {𝑋}))
1817adantr 480 . . . . 5 ((𝐺 Fn 𝐴𝑋𝐴) → (𝐺 ↾ {𝑋}) Fn dom (𝐺 ↾ {𝑋}))
1918adantl 481 . . . 4 ((((𝐺𝑋) ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {(𝐺𝑋)})) ∧ (𝐺 Fn 𝐴𝑋𝐴)) → (𝐺 ↾ {𝑋}) Fn dom (𝐺 ↾ {𝑋}))
20 fnresfnco 46328 . . . 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 583 . . 3 ((((𝐺𝑋) ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {(𝐺𝑋)})) ∧ (𝐺 Fn 𝐴𝑋𝐴)) → (𝐹 ∘ (𝐺 ↾ {𝑋})) Fn dom (𝐺 ↾ {𝑋}))
22 fnfun 6643 . . 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 6243 . . 3 ((𝐹𝐺) ↾ {𝑋}) = (𝐹 ∘ (𝐺 ↾ {𝑋}))
2524funeqi 6563 . 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 395   = wceq 1533  wcel 2098  {csn 4623  dom cdm 5669  ran crn 5670  cres 5671  cima 5672  ccom 5673  Fun wfun 6531   Fn wfn 6532  cfv 6537
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pr 5420
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-br 5142  df-opab 5204  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-iota 6489  df-fun 6539  df-fn 6540  df-fv 6545
This theorem is referenced by:  afvco2  46461
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