Theorem afvco2 47639

Description: Value of a function composition, analogous to fvco2 6924. (Contributed by Alexander van der Vekens, 23-Jul-2017.)

Assertion

Ref	Expression
afvco2	⊢ ((𝐺 Fn 𝐴 ∧ 𝑋 ∈ 𝐴) → ((𝐹 ∘ 𝐺)'''𝑋) = (𝐹'''(𝐺'''𝑋)))

Proof of Theorem afvco2

Step	Hyp	Ref	Expression
1		fvco2 6924	. . . . 5 ⊢ ((𝐺 Fn 𝐴 ∧ 𝑋 ∈ 𝐴) → ((𝐹 ∘ 𝐺)‘𝑋) = (𝐹‘(𝐺‘𝑋)))
2	1	adantl 482	. . . 4 ⊢ ((((𝐺‘𝑋) ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {(𝐺‘𝑋)})) ∧ (𝐺 Fn 𝐴 ∧ 𝑋 ∈ 𝐴)) → ((𝐹 ∘ 𝐺)‘𝑋) = (𝐹‘(𝐺‘𝑋)))
3		simpll 772	. . . . . 6 ⊢ ((((𝐺‘𝑋) ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {(𝐺‘𝑋)})) ∧ (𝐺 Fn 𝐴 ∧ 𝑋 ∈ 𝐴)) → (𝐺‘𝑋) ∈ dom 𝐹)
4		df-fn 6488	. . . . . . . . 9 ⊢ (𝐺 Fn 𝐴 ↔ (Fun 𝐺 ∧ dom 𝐺 = 𝐴))
5		simpll 772	. . . . . . . . . 10 ⊢ (((Fun 𝐺 ∧ dom 𝐺 = 𝐴) ∧ 𝑋 ∈ 𝐴) → Fun 𝐺)
6		eleq2 2828	. . . . . . . . . . . . . 14 ⊢ (𝐴 = dom 𝐺 → (𝑋 ∈ 𝐴 ↔ 𝑋 ∈ dom 𝐺))
7	6	eqcoms 2747	. . . . . . . . . . . . 13 ⊢ (dom 𝐺 = 𝐴 → (𝑋 ∈ 𝐴 ↔ 𝑋 ∈ dom 𝐺))
8	7	biimpd 230	. . . . . . . . . . . 12 ⊢ (dom 𝐺 = 𝐴 → (𝑋 ∈ 𝐴 → 𝑋 ∈ dom 𝐺))
9	8	adantl 482	. . . . . . . . . . 11 ⊢ ((Fun 𝐺 ∧ dom 𝐺 = 𝐴) → (𝑋 ∈ 𝐴 → 𝑋 ∈ dom 𝐺))
10	9	imp 407	. . . . . . . . . 10 ⊢ (((Fun 𝐺 ∧ dom 𝐺 = 𝐴) ∧ 𝑋 ∈ 𝐴) → 𝑋 ∈ dom 𝐺)
11	5, 10	jca 516	. . . . . . . . 9 ⊢ (((Fun 𝐺 ∧ dom 𝐺 = 𝐴) ∧ 𝑋 ∈ 𝐴) → (Fun 𝐺 ∧ 𝑋 ∈ dom 𝐺))
12	4, 11	sylanb 587	. . . . . . . 8 ⊢ ((𝐺 Fn 𝐴 ∧ 𝑋 ∈ 𝐴) → (Fun 𝐺 ∧ 𝑋 ∈ dom 𝐺))
13	12	adantl 482	. . . . . . 7 ⊢ ((((𝐺‘𝑋) ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {(𝐺‘𝑋)})) ∧ (𝐺 Fn 𝐴 ∧ 𝑋 ∈ 𝐴)) → (Fun 𝐺 ∧ 𝑋 ∈ dom 𝐺))
14		dmfco 6923	. . . . . . 7 ⊢ ((Fun 𝐺 ∧ 𝑋 ∈ dom 𝐺) → (𝑋 ∈ dom (𝐹 ∘ 𝐺) ↔ (𝐺‘𝑋) ∈ dom 𝐹))
15	13, 14	syl 17	. . . . . 6 ⊢ ((((𝐺‘𝑋) ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {(𝐺‘𝑋)})) ∧ (𝐺 Fn 𝐴 ∧ 𝑋 ∈ 𝐴)) → (𝑋 ∈ dom (𝐹 ∘ 𝐺) ↔ (𝐺‘𝑋) ∈ dom 𝐹))
16	3, 15	mpbird 258	. . . . 5 ⊢ ((((𝐺‘𝑋) ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {(𝐺‘𝑋)})) ∧ (𝐺 Fn 𝐴 ∧ 𝑋 ∈ 𝐴)) → 𝑋 ∈ dom (𝐹 ∘ 𝐺))
17		funcoressn 47505	. . . . 5 ⊢ ((((𝐺‘𝑋) ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {(𝐺‘𝑋)})) ∧ (𝐺 Fn 𝐴 ∧ 𝑋 ∈ 𝐴)) → Fun ((𝐹 ∘ 𝐺) ↾ {𝑋}))
18		df-dfat 47582	. . . . . 6 ⊢ ((𝐹 ∘ 𝐺) defAt 𝑋 ↔ (𝑋 ∈ dom (𝐹 ∘ 𝐺) ∧ Fun ((𝐹 ∘ 𝐺) ↾ {𝑋})))
19		afvfundmfveq 47601	. . . . . 6 ⊢ ((𝐹 ∘ 𝐺) defAt 𝑋 → ((𝐹 ∘ 𝐺)'''𝑋) = ((𝐹 ∘ 𝐺)‘𝑋))
20	18, 19	sylbir 236	. . . . 5 ⊢ ((𝑋 ∈ dom (𝐹 ∘ 𝐺) ∧ Fun ((𝐹 ∘ 𝐺) ↾ {𝑋})) → ((𝐹 ∘ 𝐺)'''𝑋) = ((𝐹 ∘ 𝐺)‘𝑋))
21	16, 17, 20	syl2anc 590	. . . 4 ⊢ ((((𝐺‘𝑋) ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {(𝐺‘𝑋)})) ∧ (𝐺 Fn 𝐴 ∧ 𝑋 ∈ 𝐴)) → ((𝐹 ∘ 𝐺)'''𝑋) = ((𝐹 ∘ 𝐺)‘𝑋))
22		df-dfat 47582	. . . . . 6 ⊢ (𝐹 defAt (𝐺‘𝑋) ↔ ((𝐺‘𝑋) ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {(𝐺‘𝑋)})))
23		afvfundmfveq 47601	. . . . . 6 ⊢ (𝐹 defAt (𝐺‘𝑋) → (𝐹'''(𝐺‘𝑋)) = (𝐹‘(𝐺‘𝑋)))
24	22, 23	sylbir 236	. . . . 5 ⊢ (((𝐺‘𝑋) ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {(𝐺‘𝑋)})) → (𝐹'''(𝐺‘𝑋)) = (𝐹‘(𝐺‘𝑋)))
25	24	adantr 481	. . . 4 ⊢ ((((𝐺‘𝑋) ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {(𝐺‘𝑋)})) ∧ (𝐺 Fn 𝐴 ∧ 𝑋 ∈ 𝐴)) → (𝐹'''(𝐺‘𝑋)) = (𝐹‘(𝐺‘𝑋)))
26	2, 21, 25	3eqtr4d 2784	. . 3 ⊢ ((((𝐺‘𝑋) ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {(𝐺‘𝑋)})) ∧ (𝐺 Fn 𝐴 ∧ 𝑋 ∈ 𝐴)) → ((𝐹 ∘ 𝐺)'''𝑋) = (𝐹'''(𝐺‘𝑋)))
27		ianor 989	. . . . . 6 ⊢ (¬ ((𝐺‘𝑋) ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {(𝐺‘𝑋)})) ↔ (¬ (𝐺‘𝑋) ∈ dom 𝐹 ∨ ¬ Fun (𝐹 ↾ {(𝐺‘𝑋)})))
28	14	funfni 6591	. . . . . . . . . . 11 ⊢ ((𝐺 Fn 𝐴 ∧ 𝑋 ∈ 𝐴) → (𝑋 ∈ dom (𝐹 ∘ 𝐺) ↔ (𝐺‘𝑋) ∈ dom 𝐹))
29	28	bicomd 224	. . . . . . . . . 10 ⊢ ((𝐺 Fn 𝐴 ∧ 𝑋 ∈ 𝐴) → ((𝐺‘𝑋) ∈ dom 𝐹 ↔ 𝑋 ∈ dom (𝐹 ∘ 𝐺)))
30	29	notbid 319	. . . . . . . . 9 ⊢ ((𝐺 Fn 𝐴 ∧ 𝑋 ∈ 𝐴) → (¬ (𝐺‘𝑋) ∈ dom 𝐹 ↔ ¬ 𝑋 ∈ dom (𝐹 ∘ 𝐺)))
31	30	biimpd 230	. . . . . . . 8 ⊢ ((𝐺 Fn 𝐴 ∧ 𝑋 ∈ 𝐴) → (¬ (𝐺‘𝑋) ∈ dom 𝐹 → ¬ 𝑋 ∈ dom (𝐹 ∘ 𝐺)))
32		ndmafv 47603	. . . . . . . 8 ⊢ (¬ 𝑋 ∈ dom (𝐹 ∘ 𝐺) → ((𝐹 ∘ 𝐺)'''𝑋) = V)
33	31, 32	syl6com 37	. . . . . . 7 ⊢ (¬ (𝐺‘𝑋) ∈ dom 𝐹 → ((𝐺 Fn 𝐴 ∧ 𝑋 ∈ 𝐴) → ((𝐹 ∘ 𝐺)'''𝑋) = V))
34		funressnfv 47506	. . . . . . . . . . . 12 ⊢ (((𝑋 ∈ dom (𝐹 ∘ 𝐺) ∧ Fun ((𝐹 ∘ 𝐺) ↾ {𝑋})) ∧ (𝐺 Fn 𝐴 ∧ 𝑋 ∈ 𝐴)) → Fun (𝐹 ↾ {(𝐺‘𝑋)}))
35	34	ex 413	. . . . . . . . . . 11 ⊢ ((𝑋 ∈ dom (𝐹 ∘ 𝐺) ∧ Fun ((𝐹 ∘ 𝐺) ↾ {𝑋})) → ((𝐺 Fn 𝐴 ∧ 𝑋 ∈ 𝐴) → Fun (𝐹 ↾ {(𝐺‘𝑋)})))
36		afvnfundmuv 47602	. . . . . . . . . . . 12 ⊢ (¬ (𝐹 ∘ 𝐺) defAt 𝑋 → ((𝐹 ∘ 𝐺)'''𝑋) = V)
37	18, 36	sylnbir 332	. . . . . . . . . . 11 ⊢ (¬ (𝑋 ∈ dom (𝐹 ∘ 𝐺) ∧ Fun ((𝐹 ∘ 𝐺) ↾ {𝑋})) → ((𝐹 ∘ 𝐺)'''𝑋) = V)
38	35, 37	nsyl4 158	. . . . . . . . . 10 ⊢ (¬ ((𝐹 ∘ 𝐺)'''𝑋) = V → ((𝐺 Fn 𝐴 ∧ 𝑋 ∈ 𝐴) → Fun (𝐹 ↾ {(𝐺‘𝑋)})))
39	38	com12 32	. . . . . . . . 9 ⊢ ((𝐺 Fn 𝐴 ∧ 𝑋 ∈ 𝐴) → (¬ ((𝐹 ∘ 𝐺)'''𝑋) = V → Fun (𝐹 ↾ {(𝐺‘𝑋)})))
40	39	con1d 145	. . . . . . . 8 ⊢ ((𝐺 Fn 𝐴 ∧ 𝑋 ∈ 𝐴) → (¬ Fun (𝐹 ↾ {(𝐺‘𝑋)}) → ((𝐹 ∘ 𝐺)'''𝑋) = V))
41	40	com12 32	. . . . . . 7 ⊢ (¬ Fun (𝐹 ↾ {(𝐺‘𝑋)}) → ((𝐺 Fn 𝐴 ∧ 𝑋 ∈ 𝐴) → ((𝐹 ∘ 𝐺)'''𝑋) = V))
42	33, 41	jaoi 863	. . . . . 6 ⊢ ((¬ (𝐺‘𝑋) ∈ dom 𝐹 ∨ ¬ Fun (𝐹 ↾ {(𝐺‘𝑋)})) → ((𝐺 Fn 𝐴 ∧ 𝑋 ∈ 𝐴) → ((𝐹 ∘ 𝐺)'''𝑋) = V))
43	27, 42	sylbi 218	. . . . 5 ⊢ (¬ ((𝐺‘𝑋) ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {(𝐺‘𝑋)})) → ((𝐺 Fn 𝐴 ∧ 𝑋 ∈ 𝐴) → ((𝐹 ∘ 𝐺)'''𝑋) = V))
44	43	imp 407	. . . 4 ⊢ ((¬ ((𝐺‘𝑋) ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {(𝐺‘𝑋)})) ∧ (𝐺 Fn 𝐴 ∧ 𝑋 ∈ 𝐴)) → ((𝐹 ∘ 𝐺)'''𝑋) = V)
45		afvnfundmuv 47602	. . . . . . 7 ⊢ (¬ 𝐹 defAt (𝐺‘𝑋) → (𝐹'''(𝐺‘𝑋)) = V)
46	22, 45	sylnbir 332	. . . . . 6 ⊢ (¬ ((𝐺‘𝑋) ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {(𝐺‘𝑋)})) → (𝐹'''(𝐺‘𝑋)) = V)
47	46	eqcomd 2745	. . . . 5 ⊢ (¬ ((𝐺‘𝑋) ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {(𝐺‘𝑋)})) → V = (𝐹'''(𝐺‘𝑋)))
48	47	adantr 481	. . . 4 ⊢ ((¬ ((𝐺‘𝑋) ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {(𝐺‘𝑋)})) ∧ (𝐺 Fn 𝐴 ∧ 𝑋 ∈ 𝐴)) → V = (𝐹'''(𝐺‘𝑋)))
49	44, 48	eqtrd 2774	. . 3 ⊢ ((¬ ((𝐺‘𝑋) ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {(𝐺‘𝑋)})) ∧ (𝐺 Fn 𝐴 ∧ 𝑋 ∈ 𝐴)) → ((𝐹 ∘ 𝐺)'''𝑋) = (𝐹'''(𝐺‘𝑋)))
50	26, 49	pm2.61ian 817	. 2 ⊢ ((𝐺 Fn 𝐴 ∧ 𝑋 ∈ 𝐴) → ((𝐹 ∘ 𝐺)'''𝑋) = (𝐹'''(𝐺‘𝑋)))
51		eqidd 2740	. . 3 ⊢ ((𝐺 Fn 𝐴 ∧ 𝑋 ∈ 𝐴) → 𝐹 = 𝐹)
52	4, 9	sylbi 218	. . . . . 6 ⊢ (𝐺 Fn 𝐴 → (𝑋 ∈ 𝐴 → 𝑋 ∈ dom 𝐺))
53	52	imp 407	. . . . 5 ⊢ ((𝐺 Fn 𝐴 ∧ 𝑋 ∈ 𝐴) → 𝑋 ∈ dom 𝐺)
54		fnfun 6585	. . . . . . 7 ⊢ (𝐺 Fn 𝐴 → Fun 𝐺)
55	54	funresd 6528	. . . . . 6 ⊢ (𝐺 Fn 𝐴 → Fun (𝐺 ↾ {𝑋}))
56	55	adantr 481	. . . . 5 ⊢ ((𝐺 Fn 𝐴 ∧ 𝑋 ∈ 𝐴) → Fun (𝐺 ↾ {𝑋}))
57		df-dfat 47582	. . . . . 6 ⊢ (𝐺 defAt 𝑋 ↔ (𝑋 ∈ dom 𝐺 ∧ Fun (𝐺 ↾ {𝑋})))
58		afvfundmfveq 47601	. . . . . 6 ⊢ (𝐺 defAt 𝑋 → (𝐺'''𝑋) = (𝐺‘𝑋))
59	57, 58	sylbir 236	. . . . 5 ⊢ ((𝑋 ∈ dom 𝐺 ∧ Fun (𝐺 ↾ {𝑋})) → (𝐺'''𝑋) = (𝐺‘𝑋))
60	53, 56, 59	syl2anc 590	. . . 4 ⊢ ((𝐺 Fn 𝐴 ∧ 𝑋 ∈ 𝐴) → (𝐺'''𝑋) = (𝐺‘𝑋))
61	60	eqcomd 2745	. . 3 ⊢ ((𝐺 Fn 𝐴 ∧ 𝑋 ∈ 𝐴) → (𝐺‘𝑋) = (𝐺'''𝑋))
62	51, 61	afveq12d 47596	. 2 ⊢ ((𝐺 Fn 𝐴 ∧ 𝑋 ∈ 𝐴) → (𝐹'''(𝐺‘𝑋)) = (𝐹'''(𝐺'''𝑋)))
63	50, 62	eqtrd 2774	1 ⊢ ((𝐺 Fn 𝐴 ∧ 𝑋 ∈ 𝐴) → ((𝐹 ∘ 𝐺)'''𝑋) = (𝐹'''(𝐺'''𝑋)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 207   ∧ wa 396   ∨ wo 853   = wceq 1547   ∈ wcel 2119  Vcvv 3431  {csn 4555  dom cdm 5618   ↾ cres 5620   ∘ ccom 5622  Fun wfun 6479   Fn wfn 6480  ‘cfv 6485   defAt wdfat 47579  '''cafv 47580

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2711  ax-sep 5218  ax-nul 5228  ax-pr 5362

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2718  df-cleq 2731  df-clel 2814  df-nfc 2888  df-ne 2935  df-ral 3054  df-rex 3064  df-rab 3392  df-v 3433  df-sbc 3724  df-csb 3832  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4262  df-if 4455  df-sn 4556  df-pr 4558  df-op 4562  df-uni 4839  df-int 4878  df-br 5073  df-opab 5135  df-id 5513  df-xp 5624  df-rel 5625  df-cnv 5626  df-co 5627  df-dm 5628  df-rn 5629  df-res 5630  df-ima 5631  df-iota 6441  df-fun 6487  df-fn 6488  df-fv 6493  df-aiota 47548  df-dfat 47582  df-afv 47583

This theorem is referenced by: (None)