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Theorem afvco2 45871
Description: Value of a function composition, analogous to fvco2 6986. (Contributed by Alexander van der Vekens, 23-Jul-2017.)
Assertion
Ref Expression
afvco2 ((𝐺 Fn 𝐴𝑋𝐴) → ((𝐹𝐺)'''𝑋) = (𝐹'''(𝐺'''𝑋)))

Proof of Theorem afvco2
StepHypRef Expression
1 fvco2 6986 . . . . 5 ((𝐺 Fn 𝐴𝑋𝐴) → ((𝐹𝐺)‘𝑋) = (𝐹‘(𝐺𝑋)))
21adantl 483 . . . 4 ((((𝐺𝑋) ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {(𝐺𝑋)})) ∧ (𝐺 Fn 𝐴𝑋𝐴)) → ((𝐹𝐺)‘𝑋) = (𝐹‘(𝐺𝑋)))
3 simpll 766 . . . . . 6 ((((𝐺𝑋) ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {(𝐺𝑋)})) ∧ (𝐺 Fn 𝐴𝑋𝐴)) → (𝐺𝑋) ∈ dom 𝐹)
4 df-fn 6544 . . . . . . . . 9 (𝐺 Fn 𝐴 ↔ (Fun 𝐺 ∧ dom 𝐺 = 𝐴))
5 simpll 766 . . . . . . . . . 10 (((Fun 𝐺 ∧ dom 𝐺 = 𝐴) ∧ 𝑋𝐴) → Fun 𝐺)
6 eleq2 2823 . . . . . . . . . . . . . 14 (𝐴 = dom 𝐺 → (𝑋𝐴𝑋 ∈ dom 𝐺))
76eqcoms 2741 . . . . . . . . . . . . 13 (dom 𝐺 = 𝐴 → (𝑋𝐴𝑋 ∈ dom 𝐺))
87biimpd 228 . . . . . . . . . . . 12 (dom 𝐺 = 𝐴 → (𝑋𝐴𝑋 ∈ dom 𝐺))
98adantl 483 . . . . . . . . . . 11 ((Fun 𝐺 ∧ dom 𝐺 = 𝐴) → (𝑋𝐴𝑋 ∈ dom 𝐺))
109imp 408 . . . . . . . . . 10 (((Fun 𝐺 ∧ dom 𝐺 = 𝐴) ∧ 𝑋𝐴) → 𝑋 ∈ dom 𝐺)
115, 10jca 513 . . . . . . . . 9 (((Fun 𝐺 ∧ dom 𝐺 = 𝐴) ∧ 𝑋𝐴) → (Fun 𝐺𝑋 ∈ dom 𝐺))
124, 11sylanb 582 . . . . . . . 8 ((𝐺 Fn 𝐴𝑋𝐴) → (Fun 𝐺𝑋 ∈ dom 𝐺))
1312adantl 483 . . . . . . 7 ((((𝐺𝑋) ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {(𝐺𝑋)})) ∧ (𝐺 Fn 𝐴𝑋𝐴)) → (Fun 𝐺𝑋 ∈ dom 𝐺))
14 dmfco 6985 . . . . . . 7 ((Fun 𝐺𝑋 ∈ dom 𝐺) → (𝑋 ∈ dom (𝐹𝐺) ↔ (𝐺𝑋) ∈ dom 𝐹))
1513, 14syl 17 . . . . . 6 ((((𝐺𝑋) ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {(𝐺𝑋)})) ∧ (𝐺 Fn 𝐴𝑋𝐴)) → (𝑋 ∈ dom (𝐹𝐺) ↔ (𝐺𝑋) ∈ dom 𝐹))
163, 15mpbird 257 . . . . 5 ((((𝐺𝑋) ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {(𝐺𝑋)})) ∧ (𝐺 Fn 𝐴𝑋𝐴)) → 𝑋 ∈ dom (𝐹𝐺))
17 funcoressn 45739 . . . . 5 ((((𝐺𝑋) ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {(𝐺𝑋)})) ∧ (𝐺 Fn 𝐴𝑋𝐴)) → Fun ((𝐹𝐺) ↾ {𝑋}))
18 df-dfat 45814 . . . . . 6 ((𝐹𝐺) defAt 𝑋 ↔ (𝑋 ∈ dom (𝐹𝐺) ∧ Fun ((𝐹𝐺) ↾ {𝑋})))
19 afvfundmfveq 45833 . . . . . 6 ((𝐹𝐺) defAt 𝑋 → ((𝐹𝐺)'''𝑋) = ((𝐹𝐺)‘𝑋))
2018, 19sylbir 234 . . . . 5 ((𝑋 ∈ dom (𝐹𝐺) ∧ Fun ((𝐹𝐺) ↾ {𝑋})) → ((𝐹𝐺)'''𝑋) = ((𝐹𝐺)‘𝑋))
2116, 17, 20syl2anc 585 . . . 4 ((((𝐺𝑋) ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {(𝐺𝑋)})) ∧ (𝐺 Fn 𝐴𝑋𝐴)) → ((𝐹𝐺)'''𝑋) = ((𝐹𝐺)‘𝑋))
22 df-dfat 45814 . . . . . 6 (𝐹 defAt (𝐺𝑋) ↔ ((𝐺𝑋) ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {(𝐺𝑋)})))
23 afvfundmfveq 45833 . . . . . 6 (𝐹 defAt (𝐺𝑋) → (𝐹'''(𝐺𝑋)) = (𝐹‘(𝐺𝑋)))
2422, 23sylbir 234 . . . . 5 (((𝐺𝑋) ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {(𝐺𝑋)})) → (𝐹'''(𝐺𝑋)) = (𝐹‘(𝐺𝑋)))
2524adantr 482 . . . 4 ((((𝐺𝑋) ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {(𝐺𝑋)})) ∧ (𝐺 Fn 𝐴𝑋𝐴)) → (𝐹'''(𝐺𝑋)) = (𝐹‘(𝐺𝑋)))
262, 21, 253eqtr4d 2783 . . 3 ((((𝐺𝑋) ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {(𝐺𝑋)})) ∧ (𝐺 Fn 𝐴𝑋𝐴)) → ((𝐹𝐺)'''𝑋) = (𝐹'''(𝐺𝑋)))
27 ianor 981 . . . . . 6 (¬ ((𝐺𝑋) ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {(𝐺𝑋)})) ↔ (¬ (𝐺𝑋) ∈ dom 𝐹 ∨ ¬ Fun (𝐹 ↾ {(𝐺𝑋)})))
2814funfni 6653 . . . . . . . . . . 11 ((𝐺 Fn 𝐴𝑋𝐴) → (𝑋 ∈ dom (𝐹𝐺) ↔ (𝐺𝑋) ∈ dom 𝐹))
2928bicomd 222 . . . . . . . . . 10 ((𝐺 Fn 𝐴𝑋𝐴) → ((𝐺𝑋) ∈ dom 𝐹𝑋 ∈ dom (𝐹𝐺)))
3029notbid 318 . . . . . . . . 9 ((𝐺 Fn 𝐴𝑋𝐴) → (¬ (𝐺𝑋) ∈ dom 𝐹 ↔ ¬ 𝑋 ∈ dom (𝐹𝐺)))
3130biimpd 228 . . . . . . . 8 ((𝐺 Fn 𝐴𝑋𝐴) → (¬ (𝐺𝑋) ∈ dom 𝐹 → ¬ 𝑋 ∈ dom (𝐹𝐺)))
32 ndmafv 45835 . . . . . . . 8 𝑋 ∈ dom (𝐹𝐺) → ((𝐹𝐺)'''𝑋) = V)
3331, 32syl6com 37 . . . . . . 7 (¬ (𝐺𝑋) ∈ dom 𝐹 → ((𝐺 Fn 𝐴𝑋𝐴) → ((𝐹𝐺)'''𝑋) = V))
34 funressnfv 45740 . . . . . . . . . . . 12 (((𝑋 ∈ dom (𝐹𝐺) ∧ Fun ((𝐹𝐺) ↾ {𝑋})) ∧ (𝐺 Fn 𝐴𝑋𝐴)) → Fun (𝐹 ↾ {(𝐺𝑋)}))
3534ex 414 . . . . . . . . . . 11 ((𝑋 ∈ dom (𝐹𝐺) ∧ Fun ((𝐹𝐺) ↾ {𝑋})) → ((𝐺 Fn 𝐴𝑋𝐴) → Fun (𝐹 ↾ {(𝐺𝑋)})))
36 afvnfundmuv 45834 . . . . . . . . . . . 12 (¬ (𝐹𝐺) defAt 𝑋 → ((𝐹𝐺)'''𝑋) = V)
3718, 36sylnbir 331 . . . . . . . . . . 11 (¬ (𝑋 ∈ dom (𝐹𝐺) ∧ Fun ((𝐹𝐺) ↾ {𝑋})) → ((𝐹𝐺)'''𝑋) = V)
3835, 37nsyl4 158 . . . . . . . . . 10 (¬ ((𝐹𝐺)'''𝑋) = V → ((𝐺 Fn 𝐴𝑋𝐴) → Fun (𝐹 ↾ {(𝐺𝑋)})))
3938com12 32 . . . . . . . . 9 ((𝐺 Fn 𝐴𝑋𝐴) → (¬ ((𝐹𝐺)'''𝑋) = V → Fun (𝐹 ↾ {(𝐺𝑋)})))
4039con1d 145 . . . . . . . 8 ((𝐺 Fn 𝐴𝑋𝐴) → (¬ Fun (𝐹 ↾ {(𝐺𝑋)}) → ((𝐹𝐺)'''𝑋) = V))
4140com12 32 . . . . . . 7 (¬ Fun (𝐹 ↾ {(𝐺𝑋)}) → ((𝐺 Fn 𝐴𝑋𝐴) → ((𝐹𝐺)'''𝑋) = V))
4233, 41jaoi 856 . . . . . 6 ((¬ (𝐺𝑋) ∈ dom 𝐹 ∨ ¬ Fun (𝐹 ↾ {(𝐺𝑋)})) → ((𝐺 Fn 𝐴𝑋𝐴) → ((𝐹𝐺)'''𝑋) = V))
4327, 42sylbi 216 . . . . 5 (¬ ((𝐺𝑋) ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {(𝐺𝑋)})) → ((𝐺 Fn 𝐴𝑋𝐴) → ((𝐹𝐺)'''𝑋) = V))
4443imp 408 . . . 4 ((¬ ((𝐺𝑋) ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {(𝐺𝑋)})) ∧ (𝐺 Fn 𝐴𝑋𝐴)) → ((𝐹𝐺)'''𝑋) = V)
45 afvnfundmuv 45834 . . . . . . 7 𝐹 defAt (𝐺𝑋) → (𝐹'''(𝐺𝑋)) = V)
4622, 45sylnbir 331 . . . . . 6 (¬ ((𝐺𝑋) ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {(𝐺𝑋)})) → (𝐹'''(𝐺𝑋)) = V)
4746eqcomd 2739 . . . . 5 (¬ ((𝐺𝑋) ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {(𝐺𝑋)})) → V = (𝐹'''(𝐺𝑋)))
4847adantr 482 . . . 4 ((¬ ((𝐺𝑋) ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {(𝐺𝑋)})) ∧ (𝐺 Fn 𝐴𝑋𝐴)) → V = (𝐹'''(𝐺𝑋)))
4944, 48eqtrd 2773 . . 3 ((¬ ((𝐺𝑋) ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {(𝐺𝑋)})) ∧ (𝐺 Fn 𝐴𝑋𝐴)) → ((𝐹𝐺)'''𝑋) = (𝐹'''(𝐺𝑋)))
5026, 49pm2.61ian 811 . 2 ((𝐺 Fn 𝐴𝑋𝐴) → ((𝐹𝐺)'''𝑋) = (𝐹'''(𝐺𝑋)))
51 eqidd 2734 . . 3 ((𝐺 Fn 𝐴𝑋𝐴) → 𝐹 = 𝐹)
524, 9sylbi 216 . . . . . 6 (𝐺 Fn 𝐴 → (𝑋𝐴𝑋 ∈ dom 𝐺))
5352imp 408 . . . . 5 ((𝐺 Fn 𝐴𝑋𝐴) → 𝑋 ∈ dom 𝐺)
54 fnfun 6647 . . . . . . 7 (𝐺 Fn 𝐴 → Fun 𝐺)
5554funresd 6589 . . . . . 6 (𝐺 Fn 𝐴 → Fun (𝐺 ↾ {𝑋}))
5655adantr 482 . . . . 5 ((𝐺 Fn 𝐴𝑋𝐴) → Fun (𝐺 ↾ {𝑋}))
57 df-dfat 45814 . . . . . 6 (𝐺 defAt 𝑋 ↔ (𝑋 ∈ dom 𝐺 ∧ Fun (𝐺 ↾ {𝑋})))
58 afvfundmfveq 45833 . . . . . 6 (𝐺 defAt 𝑋 → (𝐺'''𝑋) = (𝐺𝑋))
5957, 58sylbir 234 . . . . 5 ((𝑋 ∈ dom 𝐺 ∧ Fun (𝐺 ↾ {𝑋})) → (𝐺'''𝑋) = (𝐺𝑋))
6053, 56, 59syl2anc 585 . . . 4 ((𝐺 Fn 𝐴𝑋𝐴) → (𝐺'''𝑋) = (𝐺𝑋))
6160eqcomd 2739 . . 3 ((𝐺 Fn 𝐴𝑋𝐴) → (𝐺𝑋) = (𝐺'''𝑋))
6251, 61afveq12d 45828 . 2 ((𝐺 Fn 𝐴𝑋𝐴) → (𝐹'''(𝐺𝑋)) = (𝐹'''(𝐺'''𝑋)))
6350, 62eqtrd 2773 1 ((𝐺 Fn 𝐴𝑋𝐴) → ((𝐹𝐺)'''𝑋) = (𝐹'''(𝐺'''𝑋)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 397  wo 846   = wceq 1542  wcel 2107  Vcvv 3475  {csn 4628  dom cdm 5676  cres 5678  ccom 5680  Fun wfun 6535   Fn wfn 6536  cfv 6541   defAt wdfat 45811  '''cafv 45812
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 2704  ax-sep 5299  ax-nul 5306  ax-pr 5427
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 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-int 4951  df-br 5149  df-opab 5211  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-iota 6493  df-fun 6543  df-fn 6544  df-fv 6549  df-aiota 45780  df-dfat 45814  df-afv 45815
This theorem is referenced by: (None)
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