Theorem ofmpteq 7716

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 1154	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝑥 ∈ 𝐴 ↦ 𝐵) Fn 𝐴 ∧ (𝑥 ∈ 𝐴 ↦ 𝐶) Fn 𝐴) → 𝐴 ∈ 𝑉)
2		simpr 490	. . . 4 ⊢ (((𝐴 ∈ 𝑉 ∧ (𝑥 ∈ 𝐴 ↦ 𝐵) Fn 𝐴 ∧ (𝑥 ∈ 𝐴 ↦ 𝐶) Fn 𝐴) ∧ 𝑎 ∈ 𝐴) → 𝑎 ∈ 𝐴)
3		simpl2 1211	. . . . 5 ⊢ (((𝐴 ∈ 𝑉 ∧ (𝑥 ∈ 𝐴 ↦ 𝐵) Fn 𝐴 ∧ (𝑥 ∈ 𝐴 ↦ 𝐶) Fn 𝐴) ∧ 𝑎 ∈ 𝐴) → (𝑥 ∈ 𝐴 ↦ 𝐵) Fn 𝐴)
4		eqid 2761	. . . . . 6 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐴 ↦ 𝐵)
5	4	mptfng 6678	. . . . 5 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ V ↔ (𝑥 ∈ 𝐴 ↦ 𝐵) Fn 𝐴)
6	3, 5	sylibr 237	. . . 4 ⊢ (((𝐴 ∈ 𝑉 ∧ (𝑥 ∈ 𝐴 ↦ 𝐵) Fn 𝐴 ∧ (𝑥 ∈ 𝐴 ↦ 𝐶) Fn 𝐴) ∧ 𝑎 ∈ 𝐴) → ∀𝑥 ∈ 𝐴 𝐵 ∈ V)
7		nfcsb1v 3871	. . . . . 6 ⊢ Ⅎ𝑥⦋𝑎 / 𝑥⦌𝐵
8	7	nfel1 2939	. . . . 5 ⊢ Ⅎ𝑥⦋𝑎 / 𝑥⦌𝐵 ∈ V
9		csbeq1a 3861	. . . . . 6 ⊢ (𝑥 = 𝑎 → 𝐵 = ⦋𝑎 / 𝑥⦌𝐵)
10	9	eleq1d 2846	. . . . 5 ⊢ (𝑥 = 𝑎 → (𝐵 ∈ V ↔ ⦋𝑎 / 𝑥⦌𝐵 ∈ V))
11	8, 10	rspc 3565	. . . 4 ⊢ (𝑎 ∈ 𝐴 → (∀𝑥 ∈ 𝐴 𝐵 ∈ V → ⦋𝑎 / 𝑥⦌𝐵 ∈ V))
12	2, 6, 11	sylc 66	. . 3 ⊢ (((𝐴 ∈ 𝑉 ∧ (𝑥 ∈ 𝐴 ↦ 𝐵) Fn 𝐴 ∧ (𝑥 ∈ 𝐴 ↦ 𝐶) Fn 𝐴) ∧ 𝑎 ∈ 𝐴) → ⦋𝑎 / 𝑥⦌𝐵 ∈ V)
13		simpl3 1212	. . . . 5 ⊢ (((𝐴 ∈ 𝑉 ∧ (𝑥 ∈ 𝐴 ↦ 𝐵) Fn 𝐴 ∧ (𝑥 ∈ 𝐴 ↦ 𝐶) Fn 𝐴) ∧ 𝑎 ∈ 𝐴) → (𝑥 ∈ 𝐴 ↦ 𝐶) Fn 𝐴)
14		eqid 2761	. . . . . 6 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐶) = (𝑥 ∈ 𝐴 ↦ 𝐶)
15	14	mptfng 6678	. . . . 5 ⊢ (∀𝑥 ∈ 𝐴 𝐶 ∈ V ↔ (𝑥 ∈ 𝐴 ↦ 𝐶) Fn 𝐴)
16	13, 15	sylibr 237	. . . 4 ⊢ (((𝐴 ∈ 𝑉 ∧ (𝑥 ∈ 𝐴 ↦ 𝐵) Fn 𝐴 ∧ (𝑥 ∈ 𝐴 ↦ 𝐶) Fn 𝐴) ∧ 𝑎 ∈ 𝐴) → ∀𝑥 ∈ 𝐴 𝐶 ∈ V)
17		nfcsb1v 3871	. . . . . 6 ⊢ Ⅎ𝑥⦋𝑎 / 𝑥⦌𝐶
18	17	nfel1 2939	. . . . 5 ⊢ Ⅎ𝑥⦋𝑎 / 𝑥⦌𝐶 ∈ V
19		csbeq1a 3861	. . . . . 6 ⊢ (𝑥 = 𝑎 → 𝐶 = ⦋𝑎 / 𝑥⦌𝐶)
20	19	eleq1d 2846	. . . . 5 ⊢ (𝑥 = 𝑎 → (𝐶 ∈ V ↔ ⦋𝑎 / 𝑥⦌𝐶 ∈ V))
21	18, 20	rspc 3565	. . . 4 ⊢ (𝑎 ∈ 𝐴 → (∀𝑥 ∈ 𝐴 𝐶 ∈ V → ⦋𝑎 / 𝑥⦌𝐶 ∈ V))
22	2, 16, 21	sylc 66	. . 3 ⊢ (((𝐴 ∈ 𝑉 ∧ (𝑥 ∈ 𝐴 ↦ 𝐵) Fn 𝐴 ∧ (𝑥 ∈ 𝐴 ↦ 𝐶) Fn 𝐴) ∧ 𝑎 ∈ 𝐴) → ⦋𝑎 / 𝑥⦌𝐶 ∈ V)
23		nfcv 2923	. . . . 5 ⊢ Ⅎ𝑎𝐵
24	23, 7, 9	cbvmpt 5207	. . . 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 2923	. . . . 5 ⊢ Ⅎ𝑎𝐶
27	26, 17, 19	cbvmpt 5207	. . . 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 7713	. 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 2923	. . 3 ⊢ Ⅎ𝑎(𝐵𝑅𝐶)
31		nfcv 2923	. . . 4 ⊢ Ⅎ𝑥𝑅
32	7, 31, 17	nfov 7450	. . 3 ⊢ Ⅎ𝑥(⦋𝑎 / 𝑥⦌𝐵𝑅⦋𝑎 / 𝑥⦌𝐶)
33	9, 19	oveq12d 7438	. . 3 ⊢ (𝑥 = 𝑎 → (𝐵𝑅𝐶) = (⦋𝑎 / 𝑥⦌𝐵𝑅⦋𝑎 / 𝑥⦌𝐶))
34	30, 32, 33	cbvmpt 5207	. 2 ⊢ (𝑥 ∈ 𝐴 ↦ (𝐵𝑅𝐶)) = (𝑎 ∈ 𝐴 ↦ (⦋𝑎 / 𝑥⦌𝐵𝑅⦋𝑎 / 𝑥⦌𝐶))
35	29, 34	eqtr4di 2814	1 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝑥 ∈ 𝐴 ↦ 𝐵) Fn 𝐴 ∧ (𝑥 ∈ 𝐴 ↦ 𝐶) Fn 𝐴) → ((𝑥 ∈ 𝐴 ↦ 𝐵) ∘f 𝑅(𝑥 ∈ 𝐴 ↦ 𝐶)) = (𝑥 ∈ 𝐴 ↦ (𝐵𝑅𝐶)))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2145  ∀wral 3077  Vcvv 3451  ⦋csb 3847   ↦ cmpt 5186   Fn wfn 6533  (class class class)co 7420   ∘_f cof 7691

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 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-rep 5232  ax-sep 5249  ax-nul 5260  ax-pr 5391

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 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3078  df-rex 3088  df-reu 3367  df-rab 3414  df-v 3453  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-id 5546  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-ov 7423  df-oprab 7424  df-mpo 7425  df-of 7693

This theorem is used by:  mdetrlin  22917  psrgsum  34180  esplyind  34207  mzpaddmpt  43751  mzpmulmpt  43752  mzpcompact2lem  43761