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Theorem afvco2 41762
Description: Value of a function composition, analogous to fvco2 6435. (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 6435 . . . . 5 ((𝐺 Fn 𝐴𝑋𝐴) → ((𝐹𝐺)‘𝑋) = (𝐹‘(𝐺𝑋)))
21adantl 473 . . . 4 ((((𝐺𝑋) ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {(𝐺𝑋)})) ∧ (𝐺 Fn 𝐴𝑋𝐴)) → ((𝐹𝐺)‘𝑋) = (𝐹‘(𝐺𝑋)))
3 simpll 807 . . . . . 6 ((((𝐺𝑋) ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {(𝐺𝑋)})) ∧ (𝐺 Fn 𝐴𝑋𝐴)) → (𝐺𝑋) ∈ dom 𝐹)
4 df-fn 6052 . . . . . . . . 9 (𝐺 Fn 𝐴 ↔ (Fun 𝐺 ∧ dom 𝐺 = 𝐴))
5 simpll 807 . . . . . . . . . 10 (((Fun 𝐺 ∧ dom 𝐺 = 𝐴) ∧ 𝑋𝐴) → Fun 𝐺)
6 eleq2 2828 . . . . . . . . . . . . . 14 (𝐴 = dom 𝐺 → (𝑋𝐴𝑋 ∈ dom 𝐺))
76eqcoms 2768 . . . . . . . . . . . . 13 (dom 𝐺 = 𝐴 → (𝑋𝐴𝑋 ∈ dom 𝐺))
87biimpd 219 . . . . . . . . . . . 12 (dom 𝐺 = 𝐴 → (𝑋𝐴𝑋 ∈ dom 𝐺))
98adantl 473 . . . . . . . . . . 11 ((Fun 𝐺 ∧ dom 𝐺 = 𝐴) → (𝑋𝐴𝑋 ∈ dom 𝐺))
109imp 444 . . . . . . . . . 10 (((Fun 𝐺 ∧ dom 𝐺 = 𝐴) ∧ 𝑋𝐴) → 𝑋 ∈ dom 𝐺)
115, 10jca 555 . . . . . . . . 9 (((Fun 𝐺 ∧ dom 𝐺 = 𝐴) ∧ 𝑋𝐴) → (Fun 𝐺𝑋 ∈ dom 𝐺))
124, 11sylanb 490 . . . . . . . 8 ((𝐺 Fn 𝐴𝑋𝐴) → (Fun 𝐺𝑋 ∈ dom 𝐺))
1312adantl 473 . . . . . . 7 ((((𝐺𝑋) ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {(𝐺𝑋)})) ∧ (𝐺 Fn 𝐴𝑋𝐴)) → (Fun 𝐺𝑋 ∈ dom 𝐺))
14 dmfco 6434 . . . . . . 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 41713 . . . . 5 ((((𝐺𝑋) ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {(𝐺𝑋)})) ∧ (𝐺 Fn 𝐴𝑋𝐴)) → Fun ((𝐹𝐺) ↾ {𝑋}))
18 df-dfat 41702 . . . . . 6 ((𝐹𝐺) defAt 𝑋 ↔ (𝑋 ∈ dom (𝐹𝐺) ∧ Fun ((𝐹𝐺) ↾ {𝑋})))
19 afvfundmfveq 41724 . . . . . 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 696 . . . 4 ((((𝐺𝑋) ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {(𝐺𝑋)})) ∧ (𝐺 Fn 𝐴𝑋𝐴)) → ((𝐹𝐺)'''𝑋) = ((𝐹𝐺)‘𝑋))
22 df-dfat 41702 . . . . . 6 (𝐹 defAt (𝐺𝑋) ↔ ((𝐺𝑋) ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {(𝐺𝑋)})))
23 afvfundmfveq 41724 . . . . . 6 (𝐹 defAt (𝐺𝑋) → (𝐹'''(𝐺𝑋)) = (𝐹‘(𝐺𝑋)))
2422, 23sylbir 225 . . . . 5 (((𝐺𝑋) ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {(𝐺𝑋)})) → (𝐹'''(𝐺𝑋)) = (𝐹‘(𝐺𝑋)))
2524adantr 472 . . . 4 ((((𝐺𝑋) ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {(𝐺𝑋)})) ∧ (𝐺 Fn 𝐴𝑋𝐴)) → (𝐹'''(𝐺𝑋)) = (𝐹‘(𝐺𝑋)))
262, 21, 253eqtr4d 2804 . . 3 ((((𝐺𝑋) ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {(𝐺𝑋)})) ∧ (𝐺 Fn 𝐴𝑋𝐴)) → ((𝐹𝐺)'''𝑋) = (𝐹'''(𝐺𝑋)))
27 ianor 510 . . . . . 6 (¬ ((𝐺𝑋) ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {(𝐺𝑋)})) ↔ (¬ (𝐺𝑋) ∈ dom 𝐹 ∨ ¬ Fun (𝐹 ↾ {(𝐺𝑋)})))
2814funfni 6152 . . . . . . . . . . 11 ((𝐺 Fn 𝐴𝑋𝐴) → (𝑋 ∈ dom (𝐹𝐺) ↔ (𝐺𝑋) ∈ dom 𝐹))
2928bicomd 213 . . . . . . . . . 10 ((𝐺 Fn 𝐴𝑋𝐴) → ((𝐺𝑋) ∈ dom 𝐹𝑋 ∈ dom (𝐹𝐺)))
3029notbid 307 . . . . . . . . 9 ((𝐺 Fn 𝐴𝑋𝐴) → (¬ (𝐺𝑋) ∈ dom 𝐹 ↔ ¬ 𝑋 ∈ dom (𝐹𝐺)))
3130biimpd 219 . . . . . . . 8 ((𝐺 Fn 𝐴𝑋𝐴) → (¬ (𝐺𝑋) ∈ dom 𝐹 → ¬ 𝑋 ∈ dom (𝐹𝐺)))
32 ndmafv 41726 . . . . . . . 8 𝑋 ∈ dom (𝐹𝐺) → ((𝐹𝐺)'''𝑋) = V)
3331, 32syl6com 37 . . . . . . 7 (¬ (𝐺𝑋) ∈ dom 𝐹 → ((𝐺 Fn 𝐴𝑋𝐴) → ((𝐹𝐺)'''𝑋) = V))
34 funressnfv 41714 . . . . . . . . . . . 12 (((𝑋 ∈ dom (𝐹𝐺) ∧ Fun ((𝐹𝐺) ↾ {𝑋})) ∧ (𝐺 Fn 𝐴𝑋𝐴)) → Fun (𝐹 ↾ {(𝐺𝑋)}))
3534ex 449 . . . . . . . . . . 11 ((𝑋 ∈ dom (𝐹𝐺) ∧ Fun ((𝐹𝐺) ↾ {𝑋})) → ((𝐺 Fn 𝐴𝑋𝐴) → Fun (𝐹 ↾ {(𝐺𝑋)})))
36 afvnfundmuv 41725 . . . . . . . . . . . 12 (¬ (𝐹𝐺) defAt 𝑋 → ((𝐹𝐺)'''𝑋) = V)
3718, 36sylnbir 320 . . . . . . . . . . 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 393 . . . . . 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 444 . . . 4 ((¬ ((𝐺𝑋) ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {(𝐺𝑋)})) ∧ (𝐺 Fn 𝐴𝑋𝐴)) → ((𝐹𝐺)'''𝑋) = V)
45 afvnfundmuv 41725 . . . . . . 7 𝐹 defAt (𝐺𝑋) → (𝐹'''(𝐺𝑋)) = V)
4622, 45sylnbir 320 . . . . . 6 (¬ ((𝐺𝑋) ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {(𝐺𝑋)})) → (𝐹'''(𝐺𝑋)) = V)
4746eqcomd 2766 . . . . 5 (¬ ((𝐺𝑋) ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {(𝐺𝑋)})) → V = (𝐹'''(𝐺𝑋)))
4847adantr 472 . . . 4 ((¬ ((𝐺𝑋) ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {(𝐺𝑋)})) ∧ (𝐺 Fn 𝐴𝑋𝐴)) → V = (𝐹'''(𝐺𝑋)))
4944, 48eqtrd 2794 . . 3 ((¬ ((𝐺𝑋) ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {(𝐺𝑋)})) ∧ (𝐺 Fn 𝐴𝑋𝐴)) → ((𝐹𝐺)'''𝑋) = (𝐹'''(𝐺𝑋)))
5026, 49pm2.61ian 866 . 2 ((𝐺 Fn 𝐴𝑋𝐴) → ((𝐹𝐺)'''𝑋) = (𝐹'''(𝐺𝑋)))
51 eqidd 2761 . . 3 ((𝐺 Fn 𝐴𝑋𝐴) → 𝐹 = 𝐹)
524, 9sylbi 207 . . . . . 6 (𝐺 Fn 𝐴 → (𝑋𝐴𝑋 ∈ dom 𝐺))
5352imp 444 . . . . 5 ((𝐺 Fn 𝐴𝑋𝐴) → 𝑋 ∈ dom 𝐺)
54 fnfun 6149 . . . . . . 7 (𝐺 Fn 𝐴 → Fun 𝐺)
55 funres 6090 . . . . . . 7 (Fun 𝐺 → Fun (𝐺 ↾ {𝑋}))
5654, 55syl 17 . . . . . 6 (𝐺 Fn 𝐴 → Fun (𝐺 ↾ {𝑋}))
5756adantr 472 . . . . 5 ((𝐺 Fn 𝐴𝑋𝐴) → Fun (𝐺 ↾ {𝑋}))
58 df-dfat 41702 . . . . . 6 (𝐺 defAt 𝑋 ↔ (𝑋 ∈ dom 𝐺 ∧ Fun (𝐺 ↾ {𝑋})))
59 afvfundmfveq 41724 . . . . . 6 (𝐺 defAt 𝑋 → (𝐺'''𝑋) = (𝐺𝑋))
6058, 59sylbir 225 . . . . 5 ((𝑋 ∈ dom 𝐺 ∧ Fun (𝐺 ↾ {𝑋})) → (𝐺'''𝑋) = (𝐺𝑋))
6153, 57, 60syl2anc 696 . . . 4 ((𝐺 Fn 𝐴𝑋𝐴) → (𝐺'''𝑋) = (𝐺𝑋))
6261eqcomd 2766 . . 3 ((𝐺 Fn 𝐴𝑋𝐴) → (𝐺𝑋) = (𝐺'''𝑋))
6351, 62afveq12d 41719 . 2 ((𝐺 Fn 𝐴𝑋𝐴) → (𝐹'''(𝐺𝑋)) = (𝐹'''(𝐺'''𝑋)))
6450, 63eqtrd 2794 1 ((𝐺 Fn 𝐴𝑋𝐴) → ((𝐹𝐺)'''𝑋) = (𝐹'''(𝐺'''𝑋)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wo 382  wa 383   = wceq 1632  wcel 2139  Vcvv 3340  {csn 4321  dom cdm 5266  cres 5268  ccom 5270  Fun wfun 6043   Fn wfn 6044  cfv 6049   defAt wdfat 41699  '''cafv 41700
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-sbc 3577  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-br 4805  df-opab 4865  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-iota 6012  df-fun 6051  df-fn 6052  df-fv 6057  df-dfat 41702  df-afv 41703
This theorem is referenced by: (None)
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