Theorem ofmpteq 7736

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 1136	. . 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 1192	. . . . 5 ⊢ (((𝐴 ∈ 𝑉 ∧ (𝑥 ∈ 𝐴 ↦ 𝐵) Fn 𝐴 ∧ (𝑥 ∈ 𝐴 ↦ 𝐶) Fn 𝐴) ∧ 𝑎 ∈ 𝐴) → (𝑥 ∈ 𝐴 ↦ 𝐵) Fn 𝐴)
4		eqid 2740	. . . . . 6 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐴 ↦ 𝐵)
5	4	mptfng 6719	. . . . 5 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ V ↔ (𝑥 ∈ 𝐴 ↦ 𝐵) Fn 𝐴)
6	3, 5	sylibr 234	. . . 4 ⊢ (((𝐴 ∈ 𝑉 ∧ (𝑥 ∈ 𝐴 ↦ 𝐵) Fn 𝐴 ∧ (𝑥 ∈ 𝐴 ↦ 𝐶) Fn 𝐴) ∧ 𝑎 ∈ 𝐴) → ∀𝑥 ∈ 𝐴 𝐵 ∈ V)
7		nfcsb1v 3946	. . . . . 6 ⊢ Ⅎ𝑥⦋𝑎 / 𝑥⦌𝐵
8	7	nfel1 2925	. . . . 5 ⊢ Ⅎ𝑥⦋𝑎 / 𝑥⦌𝐵 ∈ V
9		csbeq1a 3935	. . . . . 6 ⊢ (𝑥 = 𝑎 → 𝐵 = ⦋𝑎 / 𝑥⦌𝐵)
10	9	eleq1d 2829	. . . . 5 ⊢ (𝑥 = 𝑎 → (𝐵 ∈ V ↔ ⦋𝑎 / 𝑥⦌𝐵 ∈ V))
11	8, 10	rspc 3623	. . . 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 1193	. . . . 5 ⊢ (((𝐴 ∈ 𝑉 ∧ (𝑥 ∈ 𝐴 ↦ 𝐵) Fn 𝐴 ∧ (𝑥 ∈ 𝐴 ↦ 𝐶) Fn 𝐴) ∧ 𝑎 ∈ 𝐴) → (𝑥 ∈ 𝐴 ↦ 𝐶) Fn 𝐴)
14		eqid 2740	. . . . . 6 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐶) = (𝑥 ∈ 𝐴 ↦ 𝐶)
15	14	mptfng 6719	. . . . 5 ⊢ (∀𝑥 ∈ 𝐴 𝐶 ∈ V ↔ (𝑥 ∈ 𝐴 ↦ 𝐶) Fn 𝐴)
16	13, 15	sylibr 234	. . . 4 ⊢ (((𝐴 ∈ 𝑉 ∧ (𝑥 ∈ 𝐴 ↦ 𝐵) Fn 𝐴 ∧ (𝑥 ∈ 𝐴 ↦ 𝐶) Fn 𝐴) ∧ 𝑎 ∈ 𝐴) → ∀𝑥 ∈ 𝐴 𝐶 ∈ V)
17		nfcsb1v 3946	. . . . . 6 ⊢ Ⅎ𝑥⦋𝑎 / 𝑥⦌𝐶
18	17	nfel1 2925	. . . . 5 ⊢ Ⅎ𝑥⦋𝑎 / 𝑥⦌𝐶 ∈ V
19		csbeq1a 3935	. . . . . 6 ⊢ (𝑥 = 𝑎 → 𝐶 = ⦋𝑎 / 𝑥⦌𝐶)
20	19	eleq1d 2829	. . . . 5 ⊢ (𝑥 = 𝑎 → (𝐶 ∈ V ↔ ⦋𝑎 / 𝑥⦌𝐶 ∈ V))
21	18, 20	rspc 3623	. . . 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 5277	. . . 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 5277	. . . 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 7734	. 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 7478	. . 3 ⊢ Ⅎ𝑥(⦋𝑎 / 𝑥⦌𝐵𝑅⦋𝑎 / 𝑥⦌𝐶)
33	9, 19	oveq12d 7466	. . 3 ⊢ (𝑥 = 𝑎 → (𝐵𝑅𝐶) = (⦋𝑎 / 𝑥⦌𝐵𝑅⦋𝑎 / 𝑥⦌𝐶))
34	30, 32, 33	cbvmpt 5277	. 2 ⊢ (𝑥 ∈ 𝐴 ↦ (𝐵𝑅𝐶)) = (𝑎 ∈ 𝐴 ↦ (⦋𝑎 / 𝑥⦌𝐵𝑅⦋𝑎 / 𝑥⦌𝐶))
35	29, 34	eqtr4di 2798	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 1087   = wceq 1537   ∈ wcel 2108  ∀wral 3067  Vcvv 3488  ⦋csb 3921   ↦ cmpt 5249   Fn wfn 6568  (class class class)co 7448   ∘_f cof 7712

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-rep 5303  ax-sep 5317  ax-nul 5324  ax-pr 5447

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-iun 5017  df-br 5167  df-opab 5229  df-mpt 5250  df-id 5593  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-ov 7451  df-oprab 7452  df-mpo 7453  df-of 7714

This theorem is referenced by:  mdetrlin  22629  mzpaddmpt  42697  mzpmulmpt  42698  mzpcompact2lem  42707