Theorem afvco2 47188

Description: Value of a function composition, analogous to fvco2 7006. (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 7006	. . . . 5 ⊢ ((𝐺 Fn 𝐴 ∧ 𝑋 ∈ 𝐴) → ((𝐹 ∘ 𝐺)‘𝑋) = (𝐹‘(𝐺‘𝑋)))
2	1	adantl 481	. . . 4 ⊢ ((((𝐺‘𝑋) ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {(𝐺‘𝑋)})) ∧ (𝐺 Fn 𝐴 ∧ 𝑋 ∈ 𝐴)) → ((𝐹 ∘ 𝐺)‘𝑋) = (𝐹‘(𝐺‘𝑋)))
3		simpll 767	. . . . . 6 ⊢ ((((𝐺‘𝑋) ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {(𝐺‘𝑋)})) ∧ (𝐺 Fn 𝐴 ∧ 𝑋 ∈ 𝐴)) → (𝐺‘𝑋) ∈ dom 𝐹)
4		df-fn 6564	. . . . . . . . 9 ⊢ (𝐺 Fn 𝐴 ↔ (Fun 𝐺 ∧ dom 𝐺 = 𝐴))
5		simpll 767	. . . . . . . . . 10 ⊢ (((Fun 𝐺 ∧ dom 𝐺 = 𝐴) ∧ 𝑋 ∈ 𝐴) → Fun 𝐺)
6		eleq2 2830	. . . . . . . . . . . . . 14 ⊢ (𝐴 = dom 𝐺 → (𝑋 ∈ 𝐴 ↔ 𝑋 ∈ dom 𝐺))
7	6	eqcoms 2745	. . . . . . . . . . . . 13 ⊢ (dom 𝐺 = 𝐴 → (𝑋 ∈ 𝐴 ↔ 𝑋 ∈ dom 𝐺))
8	7	biimpd 229	. . . . . . . . . . . 12 ⊢ (dom 𝐺 = 𝐴 → (𝑋 ∈ 𝐴 → 𝑋 ∈ dom 𝐺))
9	8	adantl 481	. . . . . . . . . . 11 ⊢ ((Fun 𝐺 ∧ dom 𝐺 = 𝐴) → (𝑋 ∈ 𝐴 → 𝑋 ∈ dom 𝐺))
10	9	imp 406	. . . . . . . . . 10 ⊢ (((Fun 𝐺 ∧ dom 𝐺 = 𝐴) ∧ 𝑋 ∈ 𝐴) → 𝑋 ∈ dom 𝐺)
11	5, 10	jca 511	. . . . . . . . 9 ⊢ (((Fun 𝐺 ∧ dom 𝐺 = 𝐴) ∧ 𝑋 ∈ 𝐴) → (Fun 𝐺 ∧ 𝑋 ∈ dom 𝐺))
12	4, 11	sylanb 581	. . . . . . . 8 ⊢ ((𝐺 Fn 𝐴 ∧ 𝑋 ∈ 𝐴) → (Fun 𝐺 ∧ 𝑋 ∈ dom 𝐺))
13	12	adantl 481	. . . . . . 7 ⊢ ((((𝐺‘𝑋) ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {(𝐺‘𝑋)})) ∧ (𝐺 Fn 𝐴 ∧ 𝑋 ∈ 𝐴)) → (Fun 𝐺 ∧ 𝑋 ∈ dom 𝐺))
14		dmfco 7005	. . . . . . 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 257	. . . . 5 ⊢ ((((𝐺‘𝑋) ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {(𝐺‘𝑋)})) ∧ (𝐺 Fn 𝐴 ∧ 𝑋 ∈ 𝐴)) → 𝑋 ∈ dom (𝐹 ∘ 𝐺))
17		funcoressn 47054	. . . . 5 ⊢ ((((𝐺‘𝑋) ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {(𝐺‘𝑋)})) ∧ (𝐺 Fn 𝐴 ∧ 𝑋 ∈ 𝐴)) → Fun ((𝐹 ∘ 𝐺) ↾ {𝑋}))
18		df-dfat 47131	. . . . . 6 ⊢ ((𝐹 ∘ 𝐺) defAt 𝑋 ↔ (𝑋 ∈ dom (𝐹 ∘ 𝐺) ∧ Fun ((𝐹 ∘ 𝐺) ↾ {𝑋})))
19		afvfundmfveq 47150	. . . . . 6 ⊢ ((𝐹 ∘ 𝐺) defAt 𝑋 → ((𝐹 ∘ 𝐺)'''𝑋) = ((𝐹 ∘ 𝐺)‘𝑋))
20	18, 19	sylbir 235	. . . . 5 ⊢ ((𝑋 ∈ dom (𝐹 ∘ 𝐺) ∧ Fun ((𝐹 ∘ 𝐺) ↾ {𝑋})) → ((𝐹 ∘ 𝐺)'''𝑋) = ((𝐹 ∘ 𝐺)‘𝑋))
21	16, 17, 20	syl2anc 584	. . . 4 ⊢ ((((𝐺‘𝑋) ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {(𝐺‘𝑋)})) ∧ (𝐺 Fn 𝐴 ∧ 𝑋 ∈ 𝐴)) → ((𝐹 ∘ 𝐺)'''𝑋) = ((𝐹 ∘ 𝐺)‘𝑋))
22		df-dfat 47131	. . . . . 6 ⊢ (𝐹 defAt (𝐺‘𝑋) ↔ ((𝐺‘𝑋) ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {(𝐺‘𝑋)})))
23		afvfundmfveq 47150	. . . . . 6 ⊢ (𝐹 defAt (𝐺‘𝑋) → (𝐹'''(𝐺‘𝑋)) = (𝐹‘(𝐺‘𝑋)))
24	22, 23	sylbir 235	. . . . 5 ⊢ (((𝐺‘𝑋) ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {(𝐺‘𝑋)})) → (𝐹'''(𝐺‘𝑋)) = (𝐹‘(𝐺‘𝑋)))
25	24	adantr 480	. . . 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 2787	. . 3 ⊢ ((((𝐺‘𝑋) ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {(𝐺‘𝑋)})) ∧ (𝐺 Fn 𝐴 ∧ 𝑋 ∈ 𝐴)) → ((𝐹 ∘ 𝐺)'''𝑋) = (𝐹'''(𝐺‘𝑋)))
27		ianor 984	. . . . . 6 ⊢ (¬ ((𝐺‘𝑋) ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {(𝐺‘𝑋)})) ↔ (¬ (𝐺‘𝑋) ∈ dom 𝐹 ∨ ¬ Fun (𝐹 ↾ {(𝐺‘𝑋)})))
28	14	funfni 6674	. . . . . . . . . . 11 ⊢ ((𝐺 Fn 𝐴 ∧ 𝑋 ∈ 𝐴) → (𝑋 ∈ dom (𝐹 ∘ 𝐺) ↔ (𝐺‘𝑋) ∈ dom 𝐹))
29	28	bicomd 223	. . . . . . . . . 10 ⊢ ((𝐺 Fn 𝐴 ∧ 𝑋 ∈ 𝐴) → ((𝐺‘𝑋) ∈ dom 𝐹 ↔ 𝑋 ∈ dom (𝐹 ∘ 𝐺)))
30	29	notbid 318	. . . . . . . . 9 ⊢ ((𝐺 Fn 𝐴 ∧ 𝑋 ∈ 𝐴) → (¬ (𝐺‘𝑋) ∈ dom 𝐹 ↔ ¬ 𝑋 ∈ dom (𝐹 ∘ 𝐺)))
31	30	biimpd 229	. . . . . . . 8 ⊢ ((𝐺 Fn 𝐴 ∧ 𝑋 ∈ 𝐴) → (¬ (𝐺‘𝑋) ∈ dom 𝐹 → ¬ 𝑋 ∈ dom (𝐹 ∘ 𝐺)))
32		ndmafv 47152	. . . . . . . 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 47055	. . . . . . . . . . . 12 ⊢ (((𝑋 ∈ dom (𝐹 ∘ 𝐺) ∧ Fun ((𝐹 ∘ 𝐺) ↾ {𝑋})) ∧ (𝐺 Fn 𝐴 ∧ 𝑋 ∈ 𝐴)) → Fun (𝐹 ↾ {(𝐺‘𝑋)}))
35	34	ex 412	. . . . . . . . . . 11 ⊢ ((𝑋 ∈ dom (𝐹 ∘ 𝐺) ∧ Fun ((𝐹 ∘ 𝐺) ↾ {𝑋})) → ((𝐺 Fn 𝐴 ∧ 𝑋 ∈ 𝐴) → Fun (𝐹 ↾ {(𝐺‘𝑋)})))
36		afvnfundmuv 47151	. . . . . . . . . . . 12 ⊢ (¬ (𝐹 ∘ 𝐺) defAt 𝑋 → ((𝐹 ∘ 𝐺)'''𝑋) = V)
37	18, 36	sylnbir 331	. . . . . . . . . . 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 858	. . . . . 6 ⊢ ((¬ (𝐺‘𝑋) ∈ dom 𝐹 ∨ ¬ Fun (𝐹 ↾ {(𝐺‘𝑋)})) → ((𝐺 Fn 𝐴 ∧ 𝑋 ∈ 𝐴) → ((𝐹 ∘ 𝐺)'''𝑋) = V))
43	27, 42	sylbi 217	. . . . 5 ⊢ (¬ ((𝐺‘𝑋) ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {(𝐺‘𝑋)})) → ((𝐺 Fn 𝐴 ∧ 𝑋 ∈ 𝐴) → ((𝐹 ∘ 𝐺)'''𝑋) = V))
44	43	imp 406	. . . 4 ⊢ ((¬ ((𝐺‘𝑋) ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {(𝐺‘𝑋)})) ∧ (𝐺 Fn 𝐴 ∧ 𝑋 ∈ 𝐴)) → ((𝐹 ∘ 𝐺)'''𝑋) = V)
45		afvnfundmuv 47151	. . . . . . 7 ⊢ (¬ 𝐹 defAt (𝐺‘𝑋) → (𝐹'''(𝐺‘𝑋)) = V)
46	22, 45	sylnbir 331	. . . . . 6 ⊢ (¬ ((𝐺‘𝑋) ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {(𝐺‘𝑋)})) → (𝐹'''(𝐺‘𝑋)) = V)
47	46	eqcomd 2743	. . . . 5 ⊢ (¬ ((𝐺‘𝑋) ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {(𝐺‘𝑋)})) → V = (𝐹'''(𝐺‘𝑋)))
48	47	adantr 480	. . . 4 ⊢ ((¬ ((𝐺‘𝑋) ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {(𝐺‘𝑋)})) ∧ (𝐺 Fn 𝐴 ∧ 𝑋 ∈ 𝐴)) → V = (𝐹'''(𝐺‘𝑋)))
49	44, 48	eqtrd 2777	. . 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 812	. 2 ⊢ ((𝐺 Fn 𝐴 ∧ 𝑋 ∈ 𝐴) → ((𝐹 ∘ 𝐺)'''𝑋) = (𝐹'''(𝐺‘𝑋)))
51		eqidd 2738	. . 3 ⊢ ((𝐺 Fn 𝐴 ∧ 𝑋 ∈ 𝐴) → 𝐹 = 𝐹)
52	4, 9	sylbi 217	. . . . . 6 ⊢ (𝐺 Fn 𝐴 → (𝑋 ∈ 𝐴 → 𝑋 ∈ dom 𝐺))
53	52	imp 406	. . . . 5 ⊢ ((𝐺 Fn 𝐴 ∧ 𝑋 ∈ 𝐴) → 𝑋 ∈ dom 𝐺)
54		fnfun 6668	. . . . . . 7 ⊢ (𝐺 Fn 𝐴 → Fun 𝐺)
55	54	funresd 6609	. . . . . 6 ⊢ (𝐺 Fn 𝐴 → Fun (𝐺 ↾ {𝑋}))
56	55	adantr 480	. . . . 5 ⊢ ((𝐺 Fn 𝐴 ∧ 𝑋 ∈ 𝐴) → Fun (𝐺 ↾ {𝑋}))
57		df-dfat 47131	. . . . . 6 ⊢ (𝐺 defAt 𝑋 ↔ (𝑋 ∈ dom 𝐺 ∧ Fun (𝐺 ↾ {𝑋})))
58		afvfundmfveq 47150	. . . . . 6 ⊢ (𝐺 defAt 𝑋 → (𝐺'''𝑋) = (𝐺‘𝑋))
59	57, 58	sylbir 235	. . . . 5 ⊢ ((𝑋 ∈ dom 𝐺 ∧ Fun (𝐺 ↾ {𝑋})) → (𝐺'''𝑋) = (𝐺‘𝑋))
60	53, 56, 59	syl2anc 584	. . . 4 ⊢ ((𝐺 Fn 𝐴 ∧ 𝑋 ∈ 𝐴) → (𝐺'''𝑋) = (𝐺‘𝑋))
61	60	eqcomd 2743	. . 3 ⊢ ((𝐺 Fn 𝐴 ∧ 𝑋 ∈ 𝐴) → (𝐺‘𝑋) = (𝐺'''𝑋))
62	51, 61	afveq12d 47145	. 2 ⊢ ((𝐺 Fn 𝐴 ∧ 𝑋 ∈ 𝐴) → (𝐹'''(𝐺‘𝑋)) = (𝐹'''(𝐺'''𝑋)))
63	50, 62	eqtrd 2777	1 ⊢ ((𝐺 Fn 𝐴 ∧ 𝑋 ∈ 𝐴) → ((𝐹 ∘ 𝐺)'''𝑋) = (𝐹'''(𝐺'''𝑋)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∨ wo 848   = wceq 1540   ∈ wcel 2108  Vcvv 3480  {csn 4626  dom cdm 5685   ↾ cres 5687   ∘ ccom 5689  Fun wfun 6555   Fn wfn 6556  ‘cfv 6561   defAt wdfat 47128  '''cafv 47129

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-int 4947  df-br 5144  df-opab 5206  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-fv 6569  df-aiota 47097  df-dfat 47131  df-afv 47132

This theorem is referenced by: (None)