Theorem afvco2 47973

Description: Value of a function composition, analogous to fvco2 6982. (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 6982	. . . . 5 ⊢ ((𝐺 Fn 𝐴 ∧ 𝑋 ∈ 𝐴) → ((𝐹 ∘ 𝐺)‘𝑋) = (𝐹‘(𝐺‘𝑋)))
2	1	adantl 487	. . . 4 ⊢ ((((𝐺‘𝑋) ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {(𝐺‘𝑋)})) ∧ (𝐺 Fn 𝐴 ∧ 𝑋 ∈ 𝐴)) → ((𝐹 ∘ 𝐺)‘𝑋) = (𝐹‘(𝐺‘𝑋)))
3		simpll 779	. . . . . 6 ⊢ ((((𝐺‘𝑋) ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {(𝐺‘𝑋)})) ∧ (𝐺 Fn 𝐴 ∧ 𝑋 ∈ 𝐴)) → (𝐺‘𝑋) ∈ dom 𝐹)
4		df-fn 6543	. . . . . . . . 9 ⊢ (𝐺 Fn 𝐴 ↔ (Fun 𝐺 ∧ dom 𝐺 = 𝐴))
5		simpll 779	. . . . . . . . . 10 ⊢ (((Fun 𝐺 ∧ dom 𝐺 = 𝐴) ∧ 𝑋 ∈ 𝐴) → Fun 𝐺)
6		eleq2 2854	. . . . . . . . . . . . . 14 ⊢ (𝐴 = dom 𝐺 → (𝑋 ∈ 𝐴 ↔ 𝑋 ∈ dom 𝐺))
7	6	eqcoms 2773	. . . . . . . . . . . . 13 ⊢ (dom 𝐺 = 𝐴 → (𝑋 ∈ 𝐴 ↔ 𝑋 ∈ dom 𝐺))
8	7	biimpd 232	. . . . . . . . . . . 12 ⊢ (dom 𝐺 = 𝐴 → (𝑋 ∈ 𝐴 → 𝑋 ∈ dom 𝐺))
9	8	adantl 487	. . . . . . . . . . 11 ⊢ ((Fun 𝐺 ∧ dom 𝐺 = 𝐴) → (𝑋 ∈ 𝐴 → 𝑋 ∈ dom 𝐺))
10	9	imp 412	. . . . . . . . . 10 ⊢ (((Fun 𝐺 ∧ dom 𝐺 = 𝐴) ∧ 𝑋 ∈ 𝐴) → 𝑋 ∈ dom 𝐺)
11	5, 10	jca 521	. . . . . . . . 9 ⊢ (((Fun 𝐺 ∧ dom 𝐺 = 𝐴) ∧ 𝑋 ∈ 𝐴) → (Fun 𝐺 ∧ 𝑋 ∈ dom 𝐺))
12	4, 11	sylanb 593	. . . . . . . 8 ⊢ ((𝐺 Fn 𝐴 ∧ 𝑋 ∈ 𝐴) → (Fun 𝐺 ∧ 𝑋 ∈ dom 𝐺))
13	12	adantl 487	. . . . . . 7 ⊢ ((((𝐺‘𝑋) ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {(𝐺‘𝑋)})) ∧ (𝐺 Fn 𝐴 ∧ 𝑋 ∈ 𝐴)) → (Fun 𝐺 ∧ 𝑋 ∈ dom 𝐺))
14		dmfco 6981	. . . . . . 7 ⊢ ((Fun 𝐺 ∧ 𝑋 ∈ dom 𝐺) → (𝑋 ∈ dom (𝐹 ∘ 𝐺) ↔ (𝐺‘𝑋) ∈ dom 𝐹))
15	13, 14	syl 18	. . . . . 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 260	. . . . 5 ⊢ ((((𝐺‘𝑋) ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {(𝐺‘𝑋)})) ∧ (𝐺 Fn 𝐴 ∧ 𝑋 ∈ 𝐴)) → 𝑋 ∈ dom (𝐹 ∘ 𝐺))
17		funcoressn 47839	. . . . 5 ⊢ ((((𝐺‘𝑋) ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {(𝐺‘𝑋)})) ∧ (𝐺 Fn 𝐴 ∧ 𝑋 ∈ 𝐴)) → Fun ((𝐹 ∘ 𝐺) ↾ {𝑋}))
18		df-dfat 47916	. . . . . 6 ⊢ ((𝐹 ∘ 𝐺) defAt 𝑋 ↔ (𝑋 ∈ dom (𝐹 ∘ 𝐺) ∧ Fun ((𝐹 ∘ 𝐺) ↾ {𝑋})))
19		afvfundmfveq 47935	. . . . . 6 ⊢ ((𝐹 ∘ 𝐺) defAt 𝑋 → ((𝐹 ∘ 𝐺)'''𝑋) = ((𝐹 ∘ 𝐺)‘𝑋))
20	18, 19	sylbir 238	. . . . 5 ⊢ ((𝑋 ∈ dom (𝐹 ∘ 𝐺) ∧ Fun ((𝐹 ∘ 𝐺) ↾ {𝑋})) → ((𝐹 ∘ 𝐺)'''𝑋) = ((𝐹 ∘ 𝐺)‘𝑋))
21	16, 17, 20	syl2anc 596	. . . 4 ⊢ ((((𝐺‘𝑋) ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {(𝐺‘𝑋)})) ∧ (𝐺 Fn 𝐴 ∧ 𝑋 ∈ 𝐴)) → ((𝐹 ∘ 𝐺)'''𝑋) = ((𝐹 ∘ 𝐺)‘𝑋))
22		df-dfat 47916	. . . . . 6 ⊢ (𝐹 defAt (𝐺‘𝑋) ↔ ((𝐺‘𝑋) ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {(𝐺‘𝑋)})))
23		afvfundmfveq 47935	. . . . . 6 ⊢ (𝐹 defAt (𝐺‘𝑋) → (𝐹'''(𝐺‘𝑋)) = (𝐹‘(𝐺‘𝑋)))
24	22, 23	sylbir 238	. . . . 5 ⊢ (((𝐺‘𝑋) ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {(𝐺‘𝑋)})) → (𝐹'''(𝐺‘𝑋)) = (𝐹‘(𝐺‘𝑋)))
25	24	adantr 486	. . . 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 2810	. . 3 ⊢ ((((𝐺‘𝑋) ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {(𝐺‘𝑋)})) ∧ (𝐺 Fn 𝐴 ∧ 𝑋 ∈ 𝐴)) → ((𝐹 ∘ 𝐺)'''𝑋) = (𝐹'''(𝐺‘𝑋)))
27		ianor 997	. . . . . 6 ⊢ (¬ ((𝐺‘𝑋) ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {(𝐺‘𝑋)})) ↔ (¬ (𝐺‘𝑋) ∈ dom 𝐹 ∨ ¬ Fun (𝐹 ↾ {(𝐺‘𝑋)})))
28	14	funfni 6645	. . . . . . . . . . 11 ⊢ ((𝐺 Fn 𝐴 ∧ 𝑋 ∈ 𝐴) → (𝑋 ∈ dom (𝐹 ∘ 𝐺) ↔ (𝐺‘𝑋) ∈ dom 𝐹))
29	28	bicomd 226	. . . . . . . . . 10 ⊢ ((𝐺 Fn 𝐴 ∧ 𝑋 ∈ 𝐴) → ((𝐺‘𝑋) ∈ dom 𝐹 ↔ 𝑋 ∈ dom (𝐹 ∘ 𝐺)))
30	29	notbid 321	. . . . . . . . 9 ⊢ ((𝐺 Fn 𝐴 ∧ 𝑋 ∈ 𝐴) → (¬ (𝐺‘𝑋) ∈ dom 𝐹 ↔ ¬ 𝑋 ∈ dom (𝐹 ∘ 𝐺)))
31	30	biimpd 232	. . . . . . . 8 ⊢ ((𝐺 Fn 𝐴 ∧ 𝑋 ∈ 𝐴) → (¬ (𝐺‘𝑋) ∈ dom 𝐹 → ¬ 𝑋 ∈ dom (𝐹 ∘ 𝐺)))
32		ndmafv 47937	. . . . . . . 8 ⊢ (¬ 𝑋 ∈ dom (𝐹 ∘ 𝐺) → ((𝐹 ∘ 𝐺)'''𝑋) = V)
33	31, 32	syl6com 38	. . . . . . 7 ⊢ (¬ (𝐺‘𝑋) ∈ dom 𝐹 → ((𝐺 Fn 𝐴 ∧ 𝑋 ∈ 𝐴) → ((𝐹 ∘ 𝐺)'''𝑋) = V))
34		funressnfv 47840	. . . . . . . . . . . 12 ⊢ (((𝑋 ∈ dom (𝐹 ∘ 𝐺) ∧ Fun ((𝐹 ∘ 𝐺) ↾ {𝑋})) ∧ (𝐺 Fn 𝐴 ∧ 𝑋 ∈ 𝐴)) → Fun (𝐹 ↾ {(𝐺‘𝑋)}))
35	34	ex 418	. . . . . . . . . . 11 ⊢ ((𝑋 ∈ dom (𝐹 ∘ 𝐺) ∧ Fun ((𝐹 ∘ 𝐺) ↾ {𝑋})) → ((𝐺 Fn 𝐴 ∧ 𝑋 ∈ 𝐴) → Fun (𝐹 ↾ {(𝐺‘𝑋)})))
36		afvnfundmuv 47936	. . . . . . . . . . . 12 ⊢ (¬ (𝐹 ∘ 𝐺) defAt 𝑋 → ((𝐹 ∘ 𝐺)'''𝑋) = V)
37	18, 36	sylnbir 334	. . . . . . . . . . 11 ⊢ (¬ (𝑋 ∈ dom (𝐹 ∘ 𝐺) ∧ Fun ((𝐹 ∘ 𝐺) ↾ {𝑋})) → ((𝐹 ∘ 𝐺)'''𝑋) = V)
38	35, 37	nsyl4 159	. . . . . . . . . 10 ⊢ (¬ ((𝐹 ∘ 𝐺)'''𝑋) = V → ((𝐺 Fn 𝐴 ∧ 𝑋 ∈ 𝐴) → Fun (𝐹 ↾ {(𝐺‘𝑋)})))
39	38	com12 33	. . . . . . . . 9 ⊢ ((𝐺 Fn 𝐴 ∧ 𝑋 ∈ 𝐴) → (¬ ((𝐹 ∘ 𝐺)'''𝑋) = V → Fun (𝐹 ↾ {(𝐺‘𝑋)})))
40	39	con1d 146	. . . . . . . 8 ⊢ ((𝐺 Fn 𝐴 ∧ 𝑋 ∈ 𝐴) → (¬ Fun (𝐹 ↾ {(𝐺‘𝑋)}) → ((𝐹 ∘ 𝐺)'''𝑋) = V))
41	40	com12 33	. . . . . . 7 ⊢ (¬ Fun (𝐹 ↾ {(𝐺‘𝑋)}) → ((𝐺 Fn 𝐴 ∧ 𝑋 ∈ 𝐴) → ((𝐹 ∘ 𝐺)'''𝑋) = V))
42	33, 41	jaoi 871	. . . . . 6 ⊢ ((¬ (𝐺‘𝑋) ∈ dom 𝐹 ∨ ¬ Fun (𝐹 ↾ {(𝐺‘𝑋)})) → ((𝐺 Fn 𝐴 ∧ 𝑋 ∈ 𝐴) → ((𝐹 ∘ 𝐺)'''𝑋) = V))
43	27, 42	sylbi 220	. . . . 5 ⊢ (¬ ((𝐺‘𝑋) ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {(𝐺‘𝑋)})) → ((𝐺 Fn 𝐴 ∧ 𝑋 ∈ 𝐴) → ((𝐹 ∘ 𝐺)'''𝑋) = V))
44	43	imp 412	. . . 4 ⊢ ((¬ ((𝐺‘𝑋) ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {(𝐺‘𝑋)})) ∧ (𝐺 Fn 𝐴 ∧ 𝑋 ∈ 𝐴)) → ((𝐹 ∘ 𝐺)'''𝑋) = V)
45		afvnfundmuv 47936	. . . . . . 7 ⊢ (¬ 𝐹 defAt (𝐺‘𝑋) → (𝐹'''(𝐺‘𝑋)) = V)
46	22, 45	sylnbir 334	. . . . . 6 ⊢ (¬ ((𝐺‘𝑋) ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {(𝐺‘𝑋)})) → (𝐹'''(𝐺‘𝑋)) = V)
47	46	eqcomd 2771	. . . . 5 ⊢ (¬ ((𝐺‘𝑋) ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {(𝐺‘𝑋)})) → V = (𝐹'''(𝐺‘𝑋)))
48	47	adantr 486	. . . 4 ⊢ ((¬ ((𝐺‘𝑋) ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {(𝐺‘𝑋)})) ∧ (𝐺 Fn 𝐴 ∧ 𝑋 ∈ 𝐴)) → V = (𝐹'''(𝐺‘𝑋)))
49	44, 48	eqtrd 2800	. . 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 824	. 2 ⊢ ((𝐺 Fn 𝐴 ∧ 𝑋 ∈ 𝐴) → ((𝐹 ∘ 𝐺)'''𝑋) = (𝐹'''(𝐺‘𝑋)))
51		eqidd 2766	. . 3 ⊢ ((𝐺 Fn 𝐴 ∧ 𝑋 ∈ 𝐴) → 𝐹 = 𝐹)
52	4, 9	sylbi 220	. . . . . 6 ⊢ (𝐺 Fn 𝐴 → (𝑋 ∈ 𝐴 → 𝑋 ∈ dom 𝐺))
53	52	imp 412	. . . . 5 ⊢ ((𝐺 Fn 𝐴 ∧ 𝑋 ∈ 𝐴) → 𝑋 ∈ dom 𝐺)
54		fnfun 6639	. . . . . . 7 ⊢ (𝐺 Fn 𝐴 → Fun 𝐺)
55	54	funresd 6583	. . . . . 6 ⊢ (𝐺 Fn 𝐴 → Fun (𝐺 ↾ {𝑋}))
56	55	adantr 486	. . . . 5 ⊢ ((𝐺 Fn 𝐴 ∧ 𝑋 ∈ 𝐴) → Fun (𝐺 ↾ {𝑋}))
57		df-dfat 47916	. . . . . 6 ⊢ (𝐺 defAt 𝑋 ↔ (𝑋 ∈ dom 𝐺 ∧ Fun (𝐺 ↾ {𝑋})))
58		afvfundmfveq 47935	. . . . . 6 ⊢ (𝐺 defAt 𝑋 → (𝐺'''𝑋) = (𝐺‘𝑋))
59	57, 58	sylbir 238	. . . . 5 ⊢ ((𝑋 ∈ dom 𝐺 ∧ Fun (𝐺 ↾ {𝑋})) → (𝐺'''𝑋) = (𝐺‘𝑋))
60	53, 56, 59	syl2anc 596	. . . 4 ⊢ ((𝐺 Fn 𝐴 ∧ 𝑋 ∈ 𝐴) → (𝐺'''𝑋) = (𝐺‘𝑋))
61	60	eqcomd 2771	. . 3 ⊢ ((𝐺 Fn 𝐴 ∧ 𝑋 ∈ 𝐴) → (𝐺‘𝑋) = (𝐺'''𝑋))
62	51, 61	afveq12d 47930	. 2 ⊢ ((𝐺 Fn 𝐴 ∧ 𝑋 ∈ 𝐴) → (𝐹'''(𝐺‘𝑋)) = (𝐹'''(𝐺'''𝑋)))
63	50, 62	eqtrd 2800	1 ⊢ ((𝐺 Fn 𝐴 ∧ 𝑋 ∈ 𝐴) → ((𝐹 ∘ 𝐺)'''𝑋) = (𝐹'''(𝐺'''𝑋)))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   ∨ wo 861   = wceq 1570   ∈ wcel 2146  Vcvv 3457  {csn 4591  dom cdm 5663   ↾ cres 5665   ∘ ccom 5667  Fun wfun 6534   Fn wfn 6535  ‘cfv 6540   defAt wdfat 47913  '''cafv 47914

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2737  ax-sep 5259  ax-nul 5271  ax-pr 5406

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3082  df-rex 3092  df-rab 3419  df-v 3459  df-sbc 3747  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4287  df-if 4490  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4875  df-int 4915  df-br 5112  df-opab 5176  df-id 5558  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-iota 6496  df-fun 6542  df-fn 6543  df-fv 6548  df-aiota 47882  df-dfat 47916  df-afv 47917

This theorem is used by: (None)