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Theorem afvco2 43719
 Description: Value of a function composition, analogous to fvco2 6739. (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 6739 . . . . 5 ((𝐺 Fn 𝐴𝑋𝐴) → ((𝐹𝐺)‘𝑋) = (𝐹‘(𝐺𝑋)))
21adantl 485 . . . 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 6331 . . . . . . . . 9 (𝐺 Fn 𝐴 ↔ (Fun 𝐺 ∧ dom 𝐺 = 𝐴))
5 simpll 766 . . . . . . . . . 10 (((Fun 𝐺 ∧ dom 𝐺 = 𝐴) ∧ 𝑋𝐴) → Fun 𝐺)
6 eleq2 2881 . . . . . . . . . . . . . 14 (𝐴 = dom 𝐺 → (𝑋𝐴𝑋 ∈ dom 𝐺))
76eqcoms 2809 . . . . . . . . . . . . 13 (dom 𝐺 = 𝐴 → (𝑋𝐴𝑋 ∈ dom 𝐺))
87biimpd 232 . . . . . . . . . . . 12 (dom 𝐺 = 𝐴 → (𝑋𝐴𝑋 ∈ dom 𝐺))
98adantl 485 . . . . . . . . . . 11 ((Fun 𝐺 ∧ dom 𝐺 = 𝐴) → (𝑋𝐴𝑋 ∈ dom 𝐺))
109imp 410 . . . . . . . . . 10 (((Fun 𝐺 ∧ dom 𝐺 = 𝐴) ∧ 𝑋𝐴) → 𝑋 ∈ dom 𝐺)
115, 10jca 515 . . . . . . . . 9 (((Fun 𝐺 ∧ dom 𝐺 = 𝐴) ∧ 𝑋𝐴) → (Fun 𝐺𝑋 ∈ dom 𝐺))
124, 11sylanb 584 . . . . . . . 8 ((𝐺 Fn 𝐴𝑋𝐴) → (Fun 𝐺𝑋 ∈ dom 𝐺))
1312adantl 485 . . . . . . 7 ((((𝐺𝑋) ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {(𝐺𝑋)})) ∧ (𝐺 Fn 𝐴𝑋𝐴)) → (Fun 𝐺𝑋 ∈ dom 𝐺))
14 dmfco 6738 . . . . . . 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 260 . . . . 5 ((((𝐺𝑋) ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {(𝐺𝑋)})) ∧ (𝐺 Fn 𝐴𝑋𝐴)) → 𝑋 ∈ dom (𝐹𝐺))
17 funcoressn 43621 . . . . 5 ((((𝐺𝑋) ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {(𝐺𝑋)})) ∧ (𝐺 Fn 𝐴𝑋𝐴)) → Fun ((𝐹𝐺) ↾ {𝑋}))
18 df-dfat 43662 . . . . . 6 ((𝐹𝐺) defAt 𝑋 ↔ (𝑋 ∈ dom (𝐹𝐺) ∧ Fun ((𝐹𝐺) ↾ {𝑋})))
19 afvfundmfveq 43681 . . . . . 6 ((𝐹𝐺) defAt 𝑋 → ((𝐹𝐺)'''𝑋) = ((𝐹𝐺)‘𝑋))
2018, 19sylbir 238 . . . . 5 ((𝑋 ∈ dom (𝐹𝐺) ∧ Fun ((𝐹𝐺) ↾ {𝑋})) → ((𝐹𝐺)'''𝑋) = ((𝐹𝐺)‘𝑋))
2116, 17, 20syl2anc 587 . . . 4 ((((𝐺𝑋) ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {(𝐺𝑋)})) ∧ (𝐺 Fn 𝐴𝑋𝐴)) → ((𝐹𝐺)'''𝑋) = ((𝐹𝐺)‘𝑋))
22 df-dfat 43662 . . . . . 6 (𝐹 defAt (𝐺𝑋) ↔ ((𝐺𝑋) ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {(𝐺𝑋)})))
23 afvfundmfveq 43681 . . . . . 6 (𝐹 defAt (𝐺𝑋) → (𝐹'''(𝐺𝑋)) = (𝐹‘(𝐺𝑋)))
2422, 23sylbir 238 . . . . 5 (((𝐺𝑋) ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {(𝐺𝑋)})) → (𝐹'''(𝐺𝑋)) = (𝐹‘(𝐺𝑋)))
2524adantr 484 . . . 4 ((((𝐺𝑋) ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {(𝐺𝑋)})) ∧ (𝐺 Fn 𝐴𝑋𝐴)) → (𝐹'''(𝐺𝑋)) = (𝐹‘(𝐺𝑋)))
262, 21, 253eqtr4d 2846 . . 3 ((((𝐺𝑋) ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {(𝐺𝑋)})) ∧ (𝐺 Fn 𝐴𝑋𝐴)) → ((𝐹𝐺)'''𝑋) = (𝐹'''(𝐺𝑋)))
27 ianor 979 . . . . . 6 (¬ ((𝐺𝑋) ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {(𝐺𝑋)})) ↔ (¬ (𝐺𝑋) ∈ dom 𝐹 ∨ ¬ Fun (𝐹 ↾ {(𝐺𝑋)})))
2814funfni 6432 . . . . . . . . . . 11 ((𝐺 Fn 𝐴𝑋𝐴) → (𝑋 ∈ dom (𝐹𝐺) ↔ (𝐺𝑋) ∈ dom 𝐹))
2928bicomd 226 . . . . . . . . . 10 ((𝐺 Fn 𝐴𝑋𝐴) → ((𝐺𝑋) ∈ dom 𝐹𝑋 ∈ dom (𝐹𝐺)))
3029notbid 321 . . . . . . . . 9 ((𝐺 Fn 𝐴𝑋𝐴) → (¬ (𝐺𝑋) ∈ dom 𝐹 ↔ ¬ 𝑋 ∈ dom (𝐹𝐺)))
3130biimpd 232 . . . . . . . 8 ((𝐺 Fn 𝐴𝑋𝐴) → (¬ (𝐺𝑋) ∈ dom 𝐹 → ¬ 𝑋 ∈ dom (𝐹𝐺)))
32 ndmafv 43683 . . . . . . . 8 𝑋 ∈ dom (𝐹𝐺) → ((𝐹𝐺)'''𝑋) = V)
3331, 32syl6com 37 . . . . . . 7 (¬ (𝐺𝑋) ∈ dom 𝐹 → ((𝐺 Fn 𝐴𝑋𝐴) → ((𝐹𝐺)'''𝑋) = V))
34 funressnfv 43622 . . . . . . . . . . . 12 (((𝑋 ∈ dom (𝐹𝐺) ∧ Fun ((𝐹𝐺) ↾ {𝑋})) ∧ (𝐺 Fn 𝐴𝑋𝐴)) → Fun (𝐹 ↾ {(𝐺𝑋)}))
3534ex 416 . . . . . . . . . . 11 ((𝑋 ∈ dom (𝐹𝐺) ∧ Fun ((𝐹𝐺) ↾ {𝑋})) → ((𝐺 Fn 𝐴𝑋𝐴) → Fun (𝐹 ↾ {(𝐺𝑋)})))
36 afvnfundmuv 43682 . . . . . . . . . . . 12 (¬ (𝐹𝐺) defAt 𝑋 → ((𝐹𝐺)'''𝑋) = V)
3718, 36sylnbir 334 . . . . . . . . . . 11 (¬ (𝑋 ∈ dom (𝐹𝐺) ∧ Fun ((𝐹𝐺) ↾ {𝑋})) → ((𝐹𝐺)'''𝑋) = V)
3835, 37nsyl4 161 . . . . . . . . . 10 (¬ ((𝐹𝐺)'''𝑋) = V → ((𝐺 Fn 𝐴𝑋𝐴) → Fun (𝐹 ↾ {(𝐺𝑋)})))
3938com12 32 . . . . . . . . 9 ((𝐺 Fn 𝐴𝑋𝐴) → (¬ ((𝐹𝐺)'''𝑋) = V → Fun (𝐹 ↾ {(𝐺𝑋)})))
4039con1d 147 . . . . . . . 8 ((𝐺 Fn 𝐴𝑋𝐴) → (¬ Fun (𝐹 ↾ {(𝐺𝑋)}) → ((𝐹𝐺)'''𝑋) = V))
4140com12 32 . . . . . . 7 (¬ Fun (𝐹 ↾ {(𝐺𝑋)}) → ((𝐺 Fn 𝐴𝑋𝐴) → ((𝐹𝐺)'''𝑋) = V))
4233, 41jaoi 854 . . . . . 6 ((¬ (𝐺𝑋) ∈ dom 𝐹 ∨ ¬ Fun (𝐹 ↾ {(𝐺𝑋)})) → ((𝐺 Fn 𝐴𝑋𝐴) → ((𝐹𝐺)'''𝑋) = V))
4327, 42sylbi 220 . . . . 5 (¬ ((𝐺𝑋) ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {(𝐺𝑋)})) → ((𝐺 Fn 𝐴𝑋𝐴) → ((𝐹𝐺)'''𝑋) = V))
4443imp 410 . . . 4 ((¬ ((𝐺𝑋) ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {(𝐺𝑋)})) ∧ (𝐺 Fn 𝐴𝑋𝐴)) → ((𝐹𝐺)'''𝑋) = V)
45 afvnfundmuv 43682 . . . . . . 7 𝐹 defAt (𝐺𝑋) → (𝐹'''(𝐺𝑋)) = V)
4622, 45sylnbir 334 . . . . . 6 (¬ ((𝐺𝑋) ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {(𝐺𝑋)})) → (𝐹'''(𝐺𝑋)) = V)
4746eqcomd 2807 . . . . 5 (¬ ((𝐺𝑋) ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {(𝐺𝑋)})) → V = (𝐹'''(𝐺𝑋)))
4847adantr 484 . . . 4 ((¬ ((𝐺𝑋) ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {(𝐺𝑋)})) ∧ (𝐺 Fn 𝐴𝑋𝐴)) → V = (𝐹'''(𝐺𝑋)))
4944, 48eqtrd 2836 . . 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 2802 . . 3 ((𝐺 Fn 𝐴𝑋𝐴) → 𝐹 = 𝐹)
524, 9sylbi 220 . . . . . 6 (𝐺 Fn 𝐴 → (𝑋𝐴𝑋 ∈ dom 𝐺))
5352imp 410 . . . . 5 ((𝐺 Fn 𝐴𝑋𝐴) → 𝑋 ∈ dom 𝐺)
54 fnfun 6427 . . . . . . 7 (𝐺 Fn 𝐴 → Fun 𝐺)
55 funres 6370 . . . . . . 7 (Fun 𝐺 → Fun (𝐺 ↾ {𝑋}))
5654, 55syl 17 . . . . . 6 (𝐺 Fn 𝐴 → Fun (𝐺 ↾ {𝑋}))
5756adantr 484 . . . . 5 ((𝐺 Fn 𝐴𝑋𝐴) → Fun (𝐺 ↾ {𝑋}))
58 df-dfat 43662 . . . . . 6 (𝐺 defAt 𝑋 ↔ (𝑋 ∈ dom 𝐺 ∧ Fun (𝐺 ↾ {𝑋})))
59 afvfundmfveq 43681 . . . . . 6 (𝐺 defAt 𝑋 → (𝐺'''𝑋) = (𝐺𝑋))
6058, 59sylbir 238 . . . . 5 ((𝑋 ∈ dom 𝐺 ∧ Fun (𝐺 ↾ {𝑋})) → (𝐺'''𝑋) = (𝐺𝑋))
6153, 57, 60syl2anc 587 . . . 4 ((𝐺 Fn 𝐴𝑋𝐴) → (𝐺'''𝑋) = (𝐺𝑋))
6261eqcomd 2807 . . 3 ((𝐺 Fn 𝐴𝑋𝐴) → (𝐺𝑋) = (𝐺'''𝑋))
6351, 62afveq12d 43676 . 2 ((𝐺 Fn 𝐴𝑋𝐴) → (𝐹'''(𝐺𝑋)) = (𝐹'''(𝐺'''𝑋)))
6450, 63eqtrd 2836 1 ((𝐺 Fn 𝐴𝑋𝐴) → ((𝐹𝐺)'''𝑋) = (𝐹'''(𝐺'''𝑋)))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 399   ∨ wo 844   = wceq 1538   ∈ wcel 2112  Vcvv 3444  {csn 4528  dom cdm 5523   ↾ cres 5525   ∘ ccom 5527  Fun wfun 6322   Fn wfn 6323  ‘cfv 6328   defAt wdfat 43659  '''cafv 43660 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 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-ral 3114  df-rex 3115  df-rab 3118  df-v 3446  df-sbc 3724  df-csb 3832  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4247  df-if 4429  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4804  df-int 4842  df-br 5034  df-opab 5096  df-id 5428  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-iota 6287  df-fun 6330  df-fn 6331  df-fv 6336  df-aiota 43629  df-dfat 43662  df-afv 43663 This theorem is referenced by: (None)
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