Theorem ofmpteq 7546

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 1134	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝑥 ∈ 𝐴 ↦ 𝐵) Fn 𝐴 ∧ (𝑥 ∈ 𝐴 ↦ 𝐶) Fn 𝐴) → 𝐴 ∈ 𝑉)
2		simpr 484	. . . 4 ⊢ (((𝐴 ∈ 𝑉 ∧ (𝑥 ∈ 𝐴 ↦ 𝐵) Fn 𝐴 ∧ (𝑥 ∈ 𝐴 ↦ 𝐶) Fn 𝐴) ∧ 𝑎 ∈ 𝐴) → 𝑎 ∈ 𝐴)
3		simpl2 1190	. . . . 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 6568	. . . . 5 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ V ↔ (𝑥 ∈ 𝐴 ↦ 𝐵) Fn 𝐴)
6	3, 5	sylibr 233	. . . 4 ⊢ (((𝐴 ∈ 𝑉 ∧ (𝑥 ∈ 𝐴 ↦ 𝐵) Fn 𝐴 ∧ (𝑥 ∈ 𝐴 ↦ 𝐶) Fn 𝐴) ∧ 𝑎 ∈ 𝐴) → ∀𝑥 ∈ 𝐴 𝐵 ∈ V)
7		nfcsb1v 3861	. . . . . 6 ⊢ Ⅎ𝑥⦋𝑎 / 𝑥⦌𝐵
8	7	nfel1 2924	. . . . 5 ⊢ Ⅎ𝑥⦋𝑎 / 𝑥⦌𝐵 ∈ V
9		csbeq1a 3850	. . . . . 6 ⊢ (𝑥 = 𝑎 → 𝐵 = ⦋𝑎 / 𝑥⦌𝐵)
10	9	eleq1d 2824	. . . . 5 ⊢ (𝑥 = 𝑎 → (𝐵 ∈ V ↔ ⦋𝑎 / 𝑥⦌𝐵 ∈ V))
11	8, 10	rspc 3547	. . . 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 1191	. . . . 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 6568	. . . . 5 ⊢ (∀𝑥 ∈ 𝐴 𝐶 ∈ V ↔ (𝑥 ∈ 𝐴 ↦ 𝐶) Fn 𝐴)
16	13, 15	sylibr 233	. . . 4 ⊢ (((𝐴 ∈ 𝑉 ∧ (𝑥 ∈ 𝐴 ↦ 𝐵) Fn 𝐴 ∧ (𝑥 ∈ 𝐴 ↦ 𝐶) Fn 𝐴) ∧ 𝑎 ∈ 𝐴) → ∀𝑥 ∈ 𝐴 𝐶 ∈ V)
17		nfcsb1v 3861	. . . . . 6 ⊢ Ⅎ𝑥⦋𝑎 / 𝑥⦌𝐶
18	17	nfel1 2924	. . . . 5 ⊢ Ⅎ𝑥⦋𝑎 / 𝑥⦌𝐶 ∈ V
19		csbeq1a 3850	. . . . . 6 ⊢ (𝑥 = 𝑎 → 𝐶 = ⦋𝑎 / 𝑥⦌𝐶)
20	19	eleq1d 2824	. . . . 5 ⊢ (𝑥 = 𝑎 → (𝐶 ∈ V ↔ ⦋𝑎 / 𝑥⦌𝐶 ∈ V))
21	18, 20	rspc 3547	. . . 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 2908	. . . . 5 ⊢ Ⅎ𝑎𝐵
24	23, 7, 9	cbvmpt 5189	. . . 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 2908	. . . . 5 ⊢ Ⅎ𝑎𝐶
27	26, 17, 19	cbvmpt 5189	. . . 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 7544	. 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 2908	. . 3 ⊢ Ⅎ𝑎(𝐵𝑅𝐶)
31		nfcv 2908	. . . 4 ⊢ Ⅎ𝑥𝑅
32	7, 31, 17	nfov 7298	. . 3 ⊢ Ⅎ𝑥(⦋𝑎 / 𝑥⦌𝐵𝑅⦋𝑎 / 𝑥⦌𝐶)
33	9, 19	oveq12d 7286	. . 3 ⊢ (𝑥 = 𝑎 → (𝐵𝑅𝐶) = (⦋𝑎 / 𝑥⦌𝐵𝑅⦋𝑎 / 𝑥⦌𝐶))
34	30, 32, 33	cbvmpt 5189	. 2 ⊢ (𝑥 ∈ 𝐴 ↦ (𝐵𝑅𝐶)) = (𝑎 ∈ 𝐴 ↦ (⦋𝑎 / 𝑥⦌𝐵𝑅⦋𝑎 / 𝑥⦌𝐶))
35	29, 34	eqtr4di 2797	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 395   ∧ w3a 1085   = wceq 1541   ∈ wcel 2109  ∀wral 3065  Vcvv 3430  ⦋csb 3836   ↦ cmpt 5161   Fn wfn 6425  (class class class)co 7268   ∘_f cof 7522

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1801  ax-4 1815  ax-5 1916  ax-6 1974  ax-7 2014  ax-8 2111  ax-9 2119  ax-10 2140  ax-11 2157  ax-12 2174  ax-ext 2710  ax-rep 5213  ax-sep 5226  ax-nul 5233  ax-pr 5355

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1544  df-fal 1554  df-ex 1786  df-nf 1790  df-sb 2071  df-mo 2541  df-eu 2570  df-clab 2717  df-cleq 2731  df-clel 2817  df-nfc 2890  df-ne 2945  df-ral 3070  df-rex 3071  df-reu 3072  df-rab 3074  df-v 3432  df-sbc 3720  df-csb 3837  df-dif 3894  df-un 3896  df-in 3898  df-ss 3908  df-nul 4262  df-if 4465  df-sn 4567  df-pr 4569  df-op 4573  df-uni 4845  df-iun 4931  df-br 5079  df-opab 5141  df-mpt 5162  df-id 5488  df-xp 5594  df-rel 5595  df-cnv 5596  df-co 5597  df-dm 5598  df-rn 5599  df-res 5600  df-ima 5601  df-iota 6388  df-fun 6432  df-fn 6433  df-f 6434  df-f1 6435  df-fo 6436  df-f1o 6437  df-fv 6438  df-ov 7271  df-oprab 7272  df-mpo 7273  df-of 7524

This theorem is referenced by:  mdetrlin  21732  mzpaddmpt  40543  mzpmulmpt  40544  mzpcompact2lem  40553