Theorem afvco2 41256

Description: Value of a function composition, analogous to fvco2 6273. (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 6273	. . . . 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 790	. . . . . 6 ⊢ ((((𝐺‘𝑋) ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {(𝐺‘𝑋)})) ∧ (𝐺 Fn 𝐴 ∧ 𝑋 ∈ 𝐴)) → (𝐺‘𝑋) ∈ dom 𝐹)
4		df-fn 5891	. . . . . . . . 9 ⊢ (𝐺 Fn 𝐴 ↔ (Fun 𝐺 ∧ dom 𝐺 = 𝐴))
5		simpll 790	. . . . . . . . . 10 ⊢ (((Fun 𝐺 ∧ dom 𝐺 = 𝐴) ∧ 𝑋 ∈ 𝐴) → Fun 𝐺)
6		eleq2 2690	. . . . . . . . . . . . . 14 ⊢ (𝐴 = dom 𝐺 → (𝑋 ∈ 𝐴 ↔ 𝑋 ∈ dom 𝐺))
7	6	eqcoms 2630	. . . . . . . . . . . . 13 ⊢ (dom 𝐺 = 𝐴 → (𝑋 ∈ 𝐴 ↔ 𝑋 ∈ dom 𝐺))
8	7	biimpd 219	. . . . . . . . . . . 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 445	. . . . . . . . . 10 ⊢ (((Fun 𝐺 ∧ dom 𝐺 = 𝐴) ∧ 𝑋 ∈ 𝐴) → 𝑋 ∈ dom 𝐺)
11	5, 10	jca 554	. . . . . . . . 9 ⊢ (((Fun 𝐺 ∧ dom 𝐺 = 𝐴) ∧ 𝑋 ∈ 𝐴) → (Fun 𝐺 ∧ 𝑋 ∈ dom 𝐺))
12	4, 11	sylanb 489	. . . . . . . 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 6272	. . . . . . 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 247	. . . . 5 ⊢ ((((𝐺‘𝑋) ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {(𝐺‘𝑋)})) ∧ (𝐺 Fn 𝐴 ∧ 𝑋 ∈ 𝐴)) → 𝑋 ∈ dom (𝐹 ∘ 𝐺))
17		funcoressn 41207	. . . . 5 ⊢ ((((𝐺‘𝑋) ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {(𝐺‘𝑋)})) ∧ (𝐺 Fn 𝐴 ∧ 𝑋 ∈ 𝐴)) → Fun ((𝐹 ∘ 𝐺) ↾ {𝑋}))
18		df-dfat 41196	. . . . . 6 ⊢ ((𝐹 ∘ 𝐺) defAt 𝑋 ↔ (𝑋 ∈ dom (𝐹 ∘ 𝐺) ∧ Fun ((𝐹 ∘ 𝐺) ↾ {𝑋})))
19		afvfundmfveq 41218	. . . . . 6 ⊢ ((𝐹 ∘ 𝐺) defAt 𝑋 → ((𝐹 ∘ 𝐺)'''𝑋) = ((𝐹 ∘ 𝐺)‘𝑋))
20	18, 19	sylbir 225	. . . . 5 ⊢ ((𝑋 ∈ dom (𝐹 ∘ 𝐺) ∧ Fun ((𝐹 ∘ 𝐺) ↾ {𝑋})) → ((𝐹 ∘ 𝐺)'''𝑋) = ((𝐹 ∘ 𝐺)‘𝑋))
21	16, 17, 20	syl2anc 693	. . . 4 ⊢ ((((𝐺‘𝑋) ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {(𝐺‘𝑋)})) ∧ (𝐺 Fn 𝐴 ∧ 𝑋 ∈ 𝐴)) → ((𝐹 ∘ 𝐺)'''𝑋) = ((𝐹 ∘ 𝐺)‘𝑋))
22		df-dfat 41196	. . . . . 6 ⊢ (𝐹 defAt (𝐺‘𝑋) ↔ ((𝐺‘𝑋) ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {(𝐺‘𝑋)})))
23		afvfundmfveq 41218	. . . . . 6 ⊢ (𝐹 defAt (𝐺‘𝑋) → (𝐹'''(𝐺‘𝑋)) = (𝐹‘(𝐺‘𝑋)))
24	22, 23	sylbir 225	. . . . 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 2666	. . 3 ⊢ ((((𝐺‘𝑋) ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {(𝐺‘𝑋)})) ∧ (𝐺 Fn 𝐴 ∧ 𝑋 ∈ 𝐴)) → ((𝐹 ∘ 𝐺)'''𝑋) = (𝐹'''(𝐺‘𝑋)))
27		ianor 509	. . . . . 6 ⊢ (¬ ((𝐺‘𝑋) ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {(𝐺‘𝑋)})) ↔ (¬ (𝐺‘𝑋) ∈ dom 𝐹 ∨ ¬ Fun (𝐹 ↾ {(𝐺‘𝑋)})))
28	14	funfni 5991	. . . . . . . . . . 11 ⊢ ((𝐺 Fn 𝐴 ∧ 𝑋 ∈ 𝐴) → (𝑋 ∈ dom (𝐹 ∘ 𝐺) ↔ (𝐺‘𝑋) ∈ dom 𝐹))
29	28	bicomd 213	. . . . . . . . . 10 ⊢ ((𝐺 Fn 𝐴 ∧ 𝑋 ∈ 𝐴) → ((𝐺‘𝑋) ∈ dom 𝐹 ↔ 𝑋 ∈ dom (𝐹 ∘ 𝐺)))
30	29	notbid 308	. . . . . . . . 9 ⊢ ((𝐺 Fn 𝐴 ∧ 𝑋 ∈ 𝐴) → (¬ (𝐺‘𝑋) ∈ dom 𝐹 ↔ ¬ 𝑋 ∈ dom (𝐹 ∘ 𝐺)))
31	30	biimpd 219	. . . . . . . 8 ⊢ ((𝐺 Fn 𝐴 ∧ 𝑋 ∈ 𝐴) → (¬ (𝐺‘𝑋) ∈ dom 𝐹 → ¬ 𝑋 ∈ dom (𝐹 ∘ 𝐺)))
32		ndmafv 41220	. . . . . . . 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 41208	. . . . . . . . . . . 12 ⊢ (((𝑋 ∈ dom (𝐹 ∘ 𝐺) ∧ Fun ((𝐹 ∘ 𝐺) ↾ {𝑋})) ∧ (𝐺 Fn 𝐴 ∧ 𝑋 ∈ 𝐴)) → Fun (𝐹 ↾ {(𝐺‘𝑋)}))
35	34	ex 450	. . . . . . . . . . 11 ⊢ ((𝑋 ∈ dom (𝐹 ∘ 𝐺) ∧ Fun ((𝐹 ∘ 𝐺) ↾ {𝑋})) → ((𝐺 Fn 𝐴 ∧ 𝑋 ∈ 𝐴) → Fun (𝐹 ↾ {(𝐺‘𝑋)})))
36		afvnfundmuv 41219	. . . . . . . . . . . 12 ⊢ (¬ (𝐹 ∘ 𝐺) defAt 𝑋 → ((𝐹 ∘ 𝐺)'''𝑋) = V)
37	18, 36	sylnbir 321	. . . . . . . . . . 11 ⊢ (¬ (𝑋 ∈ dom (𝐹 ∘ 𝐺) ∧ Fun ((𝐹 ∘ 𝐺) ↾ {𝑋})) → ((𝐹 ∘ 𝐺)'''𝑋) = V)
38	35, 37	nsyl4 156	. . . . . . . . . 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 139	. . . . . . . 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 394	. . . . . 6 ⊢ ((¬ (𝐺‘𝑋) ∈ dom 𝐹 ∨ ¬ Fun (𝐹 ↾ {(𝐺‘𝑋)})) → ((𝐺 Fn 𝐴 ∧ 𝑋 ∈ 𝐴) → ((𝐹 ∘ 𝐺)'''𝑋) = V))
43	27, 42	sylbi 207	. . . . 5 ⊢ (¬ ((𝐺‘𝑋) ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {(𝐺‘𝑋)})) → ((𝐺 Fn 𝐴 ∧ 𝑋 ∈ 𝐴) → ((𝐹 ∘ 𝐺)'''𝑋) = V))
44	43	imp 445	. . . 4 ⊢ ((¬ ((𝐺‘𝑋) ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {(𝐺‘𝑋)})) ∧ (𝐺 Fn 𝐴 ∧ 𝑋 ∈ 𝐴)) → ((𝐹 ∘ 𝐺)'''𝑋) = V)
45		afvnfundmuv 41219	. . . . . . 7 ⊢ (¬ 𝐹 defAt (𝐺‘𝑋) → (𝐹'''(𝐺‘𝑋)) = V)
46	22, 45	sylnbir 321	. . . . . 6 ⊢ (¬ ((𝐺‘𝑋) ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {(𝐺‘𝑋)})) → (𝐹'''(𝐺‘𝑋)) = V)
47	46	eqcomd 2628	. . . . 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 2656	. . 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 831	. 2 ⊢ ((𝐺 Fn 𝐴 ∧ 𝑋 ∈ 𝐴) → ((𝐹 ∘ 𝐺)'''𝑋) = (𝐹'''(𝐺‘𝑋)))
51		eqidd 2623	. . 3 ⊢ ((𝐺 Fn 𝐴 ∧ 𝑋 ∈ 𝐴) → 𝐹 = 𝐹)
52	4, 9	sylbi 207	. . . . . 6 ⊢ (𝐺 Fn 𝐴 → (𝑋 ∈ 𝐴 → 𝑋 ∈ dom 𝐺))
53	52	imp 445	. . . . 5 ⊢ ((𝐺 Fn 𝐴 ∧ 𝑋 ∈ 𝐴) → 𝑋 ∈ dom 𝐺)
54		fnfun 5988	. . . . . . 7 ⊢ (𝐺 Fn 𝐴 → Fun 𝐺)
55		funres 5929	. . . . . . 7 ⊢ (Fun 𝐺 → Fun (𝐺 ↾ {𝑋}))
56	54, 55	syl 17	. . . . . 6 ⊢ (𝐺 Fn 𝐴 → Fun (𝐺 ↾ {𝑋}))
57	56	adantr 481	. . . . 5 ⊢ ((𝐺 Fn 𝐴 ∧ 𝑋 ∈ 𝐴) → Fun (𝐺 ↾ {𝑋}))
58		df-dfat 41196	. . . . . 6 ⊢ (𝐺 defAt 𝑋 ↔ (𝑋 ∈ dom 𝐺 ∧ Fun (𝐺 ↾ {𝑋})))
59		afvfundmfveq 41218	. . . . . 6 ⊢ (𝐺 defAt 𝑋 → (𝐺'''𝑋) = (𝐺‘𝑋))
60	58, 59	sylbir 225	. . . . 5 ⊢ ((𝑋 ∈ dom 𝐺 ∧ Fun (𝐺 ↾ {𝑋})) → (𝐺'''𝑋) = (𝐺‘𝑋))
61	53, 57, 60	syl2anc 693	. . . 4 ⊢ ((𝐺 Fn 𝐴 ∧ 𝑋 ∈ 𝐴) → (𝐺'''𝑋) = (𝐺‘𝑋))
62	61	eqcomd 2628	. . 3 ⊢ ((𝐺 Fn 𝐴 ∧ 𝑋 ∈ 𝐴) → (𝐺‘𝑋) = (𝐺'''𝑋))
63	51, 62	afveq12d 41213	. 2 ⊢ ((𝐺 Fn 𝐴 ∧ 𝑋 ∈ 𝐴) → (𝐹'''(𝐺‘𝑋)) = (𝐹'''(𝐺'''𝑋)))
64	50, 63	eqtrd 2656	1 ⊢ ((𝐺 Fn 𝐴 ∧ 𝑋 ∈ 𝐴) → ((𝐹 ∘ 𝐺)'''𝑋) = (𝐹'''(𝐺'''𝑋)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 196   ∨ wo 383   ∧ wa 384   = wceq 1483   ∈ wcel 1990  Vcvv 3200  {csn 4177  dom cdm 5114   ↾ cres 5116   ∘ ccom 5118  Fun wfun 5882   Fn wfn 5883  ‘cfv 5888   defAt wdfat 41193  '''cafv 41194

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-fv 5896  df-dfat 41196  df-afv 41197

This theorem is referenced by: (None)