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Theorem afvco2 40557
Description: Value of a function composition, analogous to fvco2 6230. (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 6230 . . . . 5 ((𝐺 Fn 𝐴𝑋𝐴) → ((𝐹𝐺)‘𝑋) = (𝐹‘(𝐺𝑋)))
21adantl 482 . . . 4 ((((𝐺𝑋) ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {(𝐺𝑋)})) ∧ (𝐺 Fn 𝐴𝑋𝐴)) → ((𝐹𝐺)‘𝑋) = (𝐹‘(𝐺𝑋)))
3 simpll 789 . . . . . 6 ((((𝐺𝑋) ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {(𝐺𝑋)})) ∧ (𝐺 Fn 𝐴𝑋𝐴)) → (𝐺𝑋) ∈ dom 𝐹)
4 df-fn 5850 . . . . . . . . 9 (𝐺 Fn 𝐴 ↔ (Fun 𝐺 ∧ dom 𝐺 = 𝐴))
5 simpll 789 . . . . . . . . . 10 (((Fun 𝐺 ∧ dom 𝐺 = 𝐴) ∧ 𝑋𝐴) → Fun 𝐺)
6 eleq2 2687 . . . . . . . . . . . . . 14 (𝐴 = dom 𝐺 → (𝑋𝐴𝑋 ∈ dom 𝐺))
76eqcoms 2629 . . . . . . . . . . . . 13 (dom 𝐺 = 𝐴 → (𝑋𝐴𝑋 ∈ dom 𝐺))
87biimpd 219 . . . . . . . . . . . 12 (dom 𝐺 = 𝐴 → (𝑋𝐴𝑋 ∈ dom 𝐺))
98adantl 482 . . . . . . . . . . 11 ((Fun 𝐺 ∧ dom 𝐺 = 𝐴) → (𝑋𝐴𝑋 ∈ dom 𝐺))
109imp 445 . . . . . . . . . 10 (((Fun 𝐺 ∧ dom 𝐺 = 𝐴) ∧ 𝑋𝐴) → 𝑋 ∈ dom 𝐺)
115, 10jca 554 . . . . . . . . 9 (((Fun 𝐺 ∧ dom 𝐺 = 𝐴) ∧ 𝑋𝐴) → (Fun 𝐺𝑋 ∈ dom 𝐺))
124, 11sylanb 489 . . . . . . . 8 ((𝐺 Fn 𝐴𝑋𝐴) → (Fun 𝐺𝑋 ∈ dom 𝐺))
1312adantl 482 . . . . . . 7 ((((𝐺𝑋) ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {(𝐺𝑋)})) ∧ (𝐺 Fn 𝐴𝑋𝐴)) → (Fun 𝐺𝑋 ∈ dom 𝐺))
14 dmfco 6229 . . . . . . 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 247 . . . . 5 ((((𝐺𝑋) ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {(𝐺𝑋)})) ∧ (𝐺 Fn 𝐴𝑋𝐴)) → 𝑋 ∈ dom (𝐹𝐺))
17 funcoressn 40508 . . . . 5 ((((𝐺𝑋) ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {(𝐺𝑋)})) ∧ (𝐺 Fn 𝐴𝑋𝐴)) → Fun ((𝐹𝐺) ↾ {𝑋}))
18 df-dfat 40497 . . . . . 6 ((𝐹𝐺) defAt 𝑋 ↔ (𝑋 ∈ dom (𝐹𝐺) ∧ Fun ((𝐹𝐺) ↾ {𝑋})))
19 afvfundmfveq 40519 . . . . . 6 ((𝐹𝐺) defAt 𝑋 → ((𝐹𝐺)'''𝑋) = ((𝐹𝐺)‘𝑋))
2018, 19sylbir 225 . . . . 5 ((𝑋 ∈ dom (𝐹𝐺) ∧ Fun ((𝐹𝐺) ↾ {𝑋})) → ((𝐹𝐺)'''𝑋) = ((𝐹𝐺)‘𝑋))
2116, 17, 20syl2anc 692 . . . 4 ((((𝐺𝑋) ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {(𝐺𝑋)})) ∧ (𝐺 Fn 𝐴𝑋𝐴)) → ((𝐹𝐺)'''𝑋) = ((𝐹𝐺)‘𝑋))
22 df-dfat 40497 . . . . . 6 (𝐹 defAt (𝐺𝑋) ↔ ((𝐺𝑋) ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {(𝐺𝑋)})))
23 afvfundmfveq 40519 . . . . . 6 (𝐹 defAt (𝐺𝑋) → (𝐹'''(𝐺𝑋)) = (𝐹‘(𝐺𝑋)))
2422, 23sylbir 225 . . . . 5 (((𝐺𝑋) ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {(𝐺𝑋)})) → (𝐹'''(𝐺𝑋)) = (𝐹‘(𝐺𝑋)))
2524adantr 481 . . . 4 ((((𝐺𝑋) ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {(𝐺𝑋)})) ∧ (𝐺 Fn 𝐴𝑋𝐴)) → (𝐹'''(𝐺𝑋)) = (𝐹‘(𝐺𝑋)))
262, 21, 253eqtr4d 2665 . . 3 ((((𝐺𝑋) ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {(𝐺𝑋)})) ∧ (𝐺 Fn 𝐴𝑋𝐴)) → ((𝐹𝐺)'''𝑋) = (𝐹'''(𝐺𝑋)))
27 ianor 509 . . . . . 6 (¬ ((𝐺𝑋) ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {(𝐺𝑋)})) ↔ (¬ (𝐺𝑋) ∈ dom 𝐹 ∨ ¬ Fun (𝐹 ↾ {(𝐺𝑋)})))
2814funfni 5949 . . . . . . . . . . 11 ((𝐺 Fn 𝐴𝑋𝐴) → (𝑋 ∈ dom (𝐹𝐺) ↔ (𝐺𝑋) ∈ dom 𝐹))
2928bicomd 213 . . . . . . . . . 10 ((𝐺 Fn 𝐴𝑋𝐴) → ((𝐺𝑋) ∈ dom 𝐹𝑋 ∈ dom (𝐹𝐺)))
3029notbid 308 . . . . . . . . 9 ((𝐺 Fn 𝐴𝑋𝐴) → (¬ (𝐺𝑋) ∈ dom 𝐹 ↔ ¬ 𝑋 ∈ dom (𝐹𝐺)))
3130biimpd 219 . . . . . . . 8 ((𝐺 Fn 𝐴𝑋𝐴) → (¬ (𝐺𝑋) ∈ dom 𝐹 → ¬ 𝑋 ∈ dom (𝐹𝐺)))
32 ndmafv 40521 . . . . . . . 8 𝑋 ∈ dom (𝐹𝐺) → ((𝐹𝐺)'''𝑋) = V)
3331, 32syl6com 37 . . . . . . 7 (¬ (𝐺𝑋) ∈ dom 𝐹 → ((𝐺 Fn 𝐴𝑋𝐴) → ((𝐹𝐺)'''𝑋) = V))
34 funressnfv 40509 . . . . . . . . . . . 12 (((𝑋 ∈ dom (𝐹𝐺) ∧ Fun ((𝐹𝐺) ↾ {𝑋})) ∧ (𝐺 Fn 𝐴𝑋𝐴)) → Fun (𝐹 ↾ {(𝐺𝑋)}))
3534ex 450 . . . . . . . . . . 11 ((𝑋 ∈ dom (𝐹𝐺) ∧ Fun ((𝐹𝐺) ↾ {𝑋})) → ((𝐺 Fn 𝐴𝑋𝐴) → Fun (𝐹 ↾ {(𝐺𝑋)})))
36 afvnfundmuv 40520 . . . . . . . . . . . 12 (¬ (𝐹𝐺) defAt 𝑋 → ((𝐹𝐺)'''𝑋) = V)
3718, 36sylnbir 321 . . . . . . . . . . 11 (¬ (𝑋 ∈ dom (𝐹𝐺) ∧ Fun ((𝐹𝐺) ↾ {𝑋})) → ((𝐹𝐺)'''𝑋) = V)
3835, 37nsyl4 156 . . . . . . . . . 10 (¬ ((𝐹𝐺)'''𝑋) = V → ((𝐺 Fn 𝐴𝑋𝐴) → Fun (𝐹 ↾ {(𝐺𝑋)})))
3938com12 32 . . . . . . . . 9 ((𝐺 Fn 𝐴𝑋𝐴) → (¬ ((𝐹𝐺)'''𝑋) = V → Fun (𝐹 ↾ {(𝐺𝑋)})))
4039con1d 139 . . . . . . . 8 ((𝐺 Fn 𝐴𝑋𝐴) → (¬ Fun (𝐹 ↾ {(𝐺𝑋)}) → ((𝐹𝐺)'''𝑋) = V))
4140com12 32 . . . . . . 7 (¬ Fun (𝐹 ↾ {(𝐺𝑋)}) → ((𝐺 Fn 𝐴𝑋𝐴) → ((𝐹𝐺)'''𝑋) = V))
4233, 41jaoi 394 . . . . . 6 ((¬ (𝐺𝑋) ∈ dom 𝐹 ∨ ¬ Fun (𝐹 ↾ {(𝐺𝑋)})) → ((𝐺 Fn 𝐴𝑋𝐴) → ((𝐹𝐺)'''𝑋) = V))
4327, 42sylbi 207 . . . . 5 (¬ ((𝐺𝑋) ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {(𝐺𝑋)})) → ((𝐺 Fn 𝐴𝑋𝐴) → ((𝐹𝐺)'''𝑋) = V))
4443imp 445 . . . 4 ((¬ ((𝐺𝑋) ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {(𝐺𝑋)})) ∧ (𝐺 Fn 𝐴𝑋𝐴)) → ((𝐹𝐺)'''𝑋) = V)
45 afvnfundmuv 40520 . . . . . . 7 𝐹 defAt (𝐺𝑋) → (𝐹'''(𝐺𝑋)) = V)
4622, 45sylnbir 321 . . . . . 6 (¬ ((𝐺𝑋) ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {(𝐺𝑋)})) → (𝐹'''(𝐺𝑋)) = V)
4746eqcomd 2627 . . . . 5 (¬ ((𝐺𝑋) ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {(𝐺𝑋)})) → V = (𝐹'''(𝐺𝑋)))
4847adantr 481 . . . 4 ((¬ ((𝐺𝑋) ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {(𝐺𝑋)})) ∧ (𝐺 Fn 𝐴𝑋𝐴)) → V = (𝐹'''(𝐺𝑋)))
4944, 48eqtrd 2655 . . 3 ((¬ ((𝐺𝑋) ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {(𝐺𝑋)})) ∧ (𝐺 Fn 𝐴𝑋𝐴)) → ((𝐹𝐺)'''𝑋) = (𝐹'''(𝐺𝑋)))
5026, 49pm2.61ian 830 . 2 ((𝐺 Fn 𝐴𝑋𝐴) → ((𝐹𝐺)'''𝑋) = (𝐹'''(𝐺𝑋)))
51 eqidd 2622 . . 3 ((𝐺 Fn 𝐴𝑋𝐴) → 𝐹 = 𝐹)
524, 9sylbi 207 . . . . . 6 (𝐺 Fn 𝐴 → (𝑋𝐴𝑋 ∈ dom 𝐺))
5352imp 445 . . . . 5 ((𝐺 Fn 𝐴𝑋𝐴) → 𝑋 ∈ dom 𝐺)
54 fnfun 5946 . . . . . . 7 (𝐺 Fn 𝐴 → Fun 𝐺)
55 funres 5887 . . . . . . 7 (Fun 𝐺 → Fun (𝐺 ↾ {𝑋}))
5654, 55syl 17 . . . . . 6 (𝐺 Fn 𝐴 → Fun (𝐺 ↾ {𝑋}))
5756adantr 481 . . . . 5 ((𝐺 Fn 𝐴𝑋𝐴) → Fun (𝐺 ↾ {𝑋}))
58 df-dfat 40497 . . . . . 6 (𝐺 defAt 𝑋 ↔ (𝑋 ∈ dom 𝐺 ∧ Fun (𝐺 ↾ {𝑋})))
59 afvfundmfveq 40519 . . . . . 6 (𝐺 defAt 𝑋 → (𝐺'''𝑋) = (𝐺𝑋))
6058, 59sylbir 225 . . . . 5 ((𝑋 ∈ dom 𝐺 ∧ Fun (𝐺 ↾ {𝑋})) → (𝐺'''𝑋) = (𝐺𝑋))
6153, 57, 60syl2anc 692 . . . 4 ((𝐺 Fn 𝐴𝑋𝐴) → (𝐺'''𝑋) = (𝐺𝑋))
6261eqcomd 2627 . . 3 ((𝐺 Fn 𝐴𝑋𝐴) → (𝐺𝑋) = (𝐺'''𝑋))
6351, 62afveq12d 40514 . 2 ((𝐺 Fn 𝐴𝑋𝐴) → (𝐹'''(𝐺𝑋)) = (𝐹'''(𝐺'''𝑋)))
6450, 63eqtrd 2655 1 ((𝐺 Fn 𝐴𝑋𝐴) → ((𝐹𝐺)'''𝑋) = (𝐹'''(𝐺'''𝑋)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wo 383  wa 384   = wceq 1480  wcel 1987  Vcvv 3186  {csn 4148  dom cdm 5074  cres 5076  ccom 5078  Fun wfun 5841   Fn wfn 5842  cfv 5847   defAt wdfat 40494  '''cafv 40495
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-sbc 3418  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-br 4614  df-opab 4674  df-id 4989  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-iota 5810  df-fun 5849  df-fn 5850  df-fv 5855  df-dfat 40497  df-afv 40498
This theorem is referenced by: (None)
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