Theorem ofmpteq 7644

Description: Value of a pointwise operation on two functions defined using maps-to notation. (Contributed by Stefan O'Rear, 5-Oct-2014.)

Assertion

Ref	Expression
ofmpteq	⊢ ((𝐴 ∈ 𝑉 ∧ (𝑥 ∈ 𝐴 ↦ 𝐵) Fn 𝐴 ∧ (𝑥 ∈ 𝐴 ↦ 𝐶) Fn 𝐴) → ((𝑥 ∈ 𝐴 ↦ 𝐵) ∘f 𝑅(𝑥 ∈ 𝐴 ↦ 𝐶)) = (𝑥 ∈ 𝐴 ↦ (𝐵𝑅𝐶)))

Distinct variable groups:   𝑥,𝐴   𝑥,𝑅

Allowed substitution hints:   𝐵(𝑥)   𝐶(𝑥)   𝑉(𝑥)

Proof of Theorem ofmpteq

Dummy variable 𝑎 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		simp1 1142	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝑥 ∈ 𝐴 ↦ 𝐵) Fn 𝐴 ∧ (𝑥 ∈ 𝐴 ↦ 𝐶) Fn 𝐴) → 𝐴 ∈ 𝑉)
2		simpr 485	. . . 4 ⊢ (((𝐴 ∈ 𝑉 ∧ (𝑥 ∈ 𝐴 ↦ 𝐵) Fn 𝐴 ∧ (𝑥 ∈ 𝐴 ↦ 𝐶) Fn 𝐴) ∧ 𝑎 ∈ 𝐴) → 𝑎 ∈ 𝐴)
3		simpl2 1199	. . . . 5 ⊢ (((𝐴 ∈ 𝑉 ∧ (𝑥 ∈ 𝐴 ↦ 𝐵) Fn 𝐴 ∧ (𝑥 ∈ 𝐴 ↦ 𝐶) Fn 𝐴) ∧ 𝑎 ∈ 𝐴) → (𝑥 ∈ 𝐴 ↦ 𝐵) Fn 𝐴)
4		eqid 2739	. . . . . 6 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐴 ↦ 𝐵)
5	4	mptfng 6625	. . . . 5 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ V ↔ (𝑥 ∈ 𝐴 ↦ 𝐵) Fn 𝐴)
6	3, 5	sylibr 235	. . . 4 ⊢ (((𝐴 ∈ 𝑉 ∧ (𝑥 ∈ 𝐴 ↦ 𝐵) Fn 𝐴 ∧ (𝑥 ∈ 𝐴 ↦ 𝐶) Fn 𝐴) ∧ 𝑎 ∈ 𝐴) → ∀𝑥 ∈ 𝐴 𝐵 ∈ V)
7		nfcsb1v 3855	. . . . . 6 ⊢ Ⅎ𝑥⦋𝑎 / 𝑥⦌𝐵
8	7	nfel1 2917	. . . . 5 ⊢ Ⅎ𝑥⦋𝑎 / 𝑥⦌𝐵 ∈ V
9		csbeq1a 3845	. . . . . 6 ⊢ (𝑥 = 𝑎 → 𝐵 = ⦋𝑎 / 𝑥⦌𝐵)
10	9	eleq1d 2824	. . . . 5 ⊢ (𝑥 = 𝑎 → (𝐵 ∈ V ↔ ⦋𝑎 / 𝑥⦌𝐵 ∈ V))
11	8, 10	rspc 3548	. . . 4 ⊢ (𝑎 ∈ 𝐴 → (∀𝑥 ∈ 𝐴 𝐵 ∈ V → ⦋𝑎 / 𝑥⦌𝐵 ∈ V))
12	2, 6, 11	sylc 65	. . 3 ⊢ (((𝐴 ∈ 𝑉 ∧ (𝑥 ∈ 𝐴 ↦ 𝐵) Fn 𝐴 ∧ (𝑥 ∈ 𝐴 ↦ 𝐶) Fn 𝐴) ∧ 𝑎 ∈ 𝐴) → ⦋𝑎 / 𝑥⦌𝐵 ∈ V)
13		simpl3 1200	. . . . 5 ⊢ (((𝐴 ∈ 𝑉 ∧ (𝑥 ∈ 𝐴 ↦ 𝐵) Fn 𝐴 ∧ (𝑥 ∈ 𝐴 ↦ 𝐶) Fn 𝐴) ∧ 𝑎 ∈ 𝐴) → (𝑥 ∈ 𝐴 ↦ 𝐶) Fn 𝐴)
14		eqid 2739	. . . . . 6 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐶) = (𝑥 ∈ 𝐴 ↦ 𝐶)
15	14	mptfng 6625	. . . . 5 ⊢ (∀𝑥 ∈ 𝐴 𝐶 ∈ V ↔ (𝑥 ∈ 𝐴 ↦ 𝐶) Fn 𝐴)
16	13, 15	sylibr 235	. . . 4 ⊢ (((𝐴 ∈ 𝑉 ∧ (𝑥 ∈ 𝐴 ↦ 𝐵) Fn 𝐴 ∧ (𝑥 ∈ 𝐴 ↦ 𝐶) Fn 𝐴) ∧ 𝑎 ∈ 𝐴) → ∀𝑥 ∈ 𝐴 𝐶 ∈ V)
17		nfcsb1v 3855	. . . . . 6 ⊢ Ⅎ𝑥⦋𝑎 / 𝑥⦌𝐶
18	17	nfel1 2917	. . . . 5 ⊢ Ⅎ𝑥⦋𝑎 / 𝑥⦌𝐶 ∈ V
19		csbeq1a 3845	. . . . . 6 ⊢ (𝑥 = 𝑎 → 𝐶 = ⦋𝑎 / 𝑥⦌𝐶)
20	19	eleq1d 2824	. . . . 5 ⊢ (𝑥 = 𝑎 → (𝐶 ∈ V ↔ ⦋𝑎 / 𝑥⦌𝐶 ∈ V))
21	18, 20	rspc 3548	. . . 4 ⊢ (𝑎 ∈ 𝐴 → (∀𝑥 ∈ 𝐴 𝐶 ∈ V → ⦋𝑎 / 𝑥⦌𝐶 ∈ V))
22	2, 16, 21	sylc 65	. . 3 ⊢ (((𝐴 ∈ 𝑉 ∧ (𝑥 ∈ 𝐴 ↦ 𝐵) Fn 𝐴 ∧ (𝑥 ∈ 𝐴 ↦ 𝐶) Fn 𝐴) ∧ 𝑎 ∈ 𝐴) → ⦋𝑎 / 𝑥⦌𝐶 ∈ V)
23		nfcv 2901	. . . . 5 ⊢ Ⅎ𝑎𝐵
24	23, 7, 9	cbvmpt 5175	. . . 4 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑎 ∈ 𝐴 ↦ ⦋𝑎 / 𝑥⦌𝐵)
25	24	a1i 11	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝑥 ∈ 𝐴 ↦ 𝐵) Fn 𝐴 ∧ (𝑥 ∈ 𝐴 ↦ 𝐶) Fn 𝐴) → (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑎 ∈ 𝐴 ↦ ⦋𝑎 / 𝑥⦌𝐵))
26		nfcv 2901	. . . . 5 ⊢ Ⅎ𝑎𝐶
27	26, 17, 19	cbvmpt 5175	. . . 4 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐶) = (𝑎 ∈ 𝐴 ↦ ⦋𝑎 / 𝑥⦌𝐶)
28	27	a1i 11	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝑥 ∈ 𝐴 ↦ 𝐵) Fn 𝐴 ∧ (𝑥 ∈ 𝐴 ↦ 𝐶) Fn 𝐴) → (𝑥 ∈ 𝐴 ↦ 𝐶) = (𝑎 ∈ 𝐴 ↦ ⦋𝑎 / 𝑥⦌𝐶))
29	1, 12, 22, 25, 28	offval2 7641	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝑥 ∈ 𝐴 ↦ 𝐵) Fn 𝐴 ∧ (𝑥 ∈ 𝐴 ↦ 𝐶) Fn 𝐴) → ((𝑥 ∈ 𝐴 ↦ 𝐵) ∘f 𝑅(𝑥 ∈ 𝐴 ↦ 𝐶)) = (𝑎 ∈ 𝐴 ↦ (⦋𝑎 / 𝑥⦌𝐵𝑅⦋𝑎 / 𝑥⦌𝐶)))
30		nfcv 2901	. . 3 ⊢ Ⅎ𝑎(𝐵𝑅𝐶)
31		nfcv 2901	. . . 4 ⊢ Ⅎ𝑥𝑅
32	7, 31, 17	nfov 7387	. . 3 ⊢ Ⅎ𝑥(⦋𝑎 / 𝑥⦌𝐵𝑅⦋𝑎 / 𝑥⦌𝐶)
33	9, 19	oveq12d 7375	. . 3 ⊢ (𝑥 = 𝑎 → (𝐵𝑅𝐶) = (⦋𝑎 / 𝑥⦌𝐵𝑅⦋𝑎 / 𝑥⦌𝐶))
34	30, 32, 33	cbvmpt 5175	. 2 ⊢ (𝑥 ∈ 𝐴 ↦ (𝐵𝑅𝐶)) = (𝑎 ∈ 𝐴 ↦ (⦋𝑎 / 𝑥⦌𝐵𝑅⦋𝑎 / 𝑥⦌𝐶))
35	29, 34	eqtr4di 2792	1 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝑥 ∈ 𝐴 ↦ 𝐵) Fn 𝐴 ∧ (𝑥 ∈ 𝐴 ↦ 𝐶) Fn 𝐴) → ((𝑥 ∈ 𝐴 ↦ 𝐵) ∘f 𝑅(𝑥 ∈ 𝐴 ↦ 𝐶)) = (𝑥 ∈ 𝐴 ↦ (𝐵𝑅𝐶)))

Syntax hints:   → wi 4   ∧ wa 396   ∧ w3a 1092   = wceq 1547   ∈ wcel 2119  ∀wral 3053  Vcvv 3431  ⦋csb 3831   ↦ cmpt 5154   Fn wfn 6481  (class class class)co 7357   ∘_f cof 7619

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-rep 5200  ax-sep 5219  ax-nul 5229  ax-pr 5363

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-reu 3345  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 4263  df-if 4456  df-sn 4557  df-pr 4559  df-op 4563  df-uni 4840  df-iun 4924  df-br 5074  df-opab 5136  df-mpt 5155  df-id 5514  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-iota 6442  df-fun 6488  df-fn 6489  df-f 6490  df-f1 6491  df-fo 6492  df-f1o 6493  df-fv 6494  df-ov 7360  df-oprab 7361  df-mpo 7362  df-of 7621

This theorem is referenced by:  mdetrlin  22586  psrgsum  33741  esplyind  33768  mzpaddmpt  43199  mzpmulmpt  43200  mzpcompact2lem  43209