Theorem resmptf 5793

Description: Restriction of the mapping operation. (Contributed by Thierry Arnoux, 28-Mar-2017.)

Hypotheses

Ref	Expression
resmptf.a	⊢ Ⅎ𝑥𝐴
resmptf.b	⊢ Ⅎ𝑥𝐵

Assertion

Ref	Expression
resmptf	⊢ (𝐵 ⊆ 𝐴 → ((𝑥 ∈ 𝐴 ↦ 𝐶) ↾ 𝐵) = (𝑥 ∈ 𝐵 ↦ 𝐶))

Proof of Theorem resmptf

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		resmpt 5791	. 2 ⊢ (𝐵 ⊆ 𝐴 → ((𝑦 ∈ 𝐴 ↦ ⦋𝑦 / 𝑥⦌𝐶) ↾ 𝐵) = (𝑦 ∈ 𝐵 ↦ ⦋𝑦 / 𝑥⦌𝐶))
2		resmptf.a	. . . 4 ⊢ Ⅎ𝑥𝐴
3		nfcv 2949	. . . 4 ⊢ Ⅎ𝑦𝐴
4		nfcv 2949	. . . 4 ⊢ Ⅎ𝑦𝐶
5		nfcsb1v 3837	. . . 4 ⊢ Ⅎ𝑥⦋𝑦 / 𝑥⦌𝐶
6		csbeq1a 3828	. . . 4 ⊢ (𝑥 = 𝑦 → 𝐶 = ⦋𝑦 / 𝑥⦌𝐶)
7	2, 3, 4, 5, 6	cbvmptf 5064	. . 3 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐶) = (𝑦 ∈ 𝐴 ↦ ⦋𝑦 / 𝑥⦌𝐶)
8	7	reseq1i 5735	. 2 ⊢ ((𝑥 ∈ 𝐴 ↦ 𝐶) ↾ 𝐵) = ((𝑦 ∈ 𝐴 ↦ ⦋𝑦 / 𝑥⦌𝐶) ↾ 𝐵)
9		resmptf.b	. . 3 ⊢ Ⅎ𝑥𝐵
10		nfcv 2949	. . 3 ⊢ Ⅎ𝑦𝐵
11	9, 10, 4, 5, 6	cbvmptf 5064	. 2 ⊢ (𝑥 ∈ 𝐵 ↦ 𝐶) = (𝑦 ∈ 𝐵 ↦ ⦋𝑦 / 𝑥⦌𝐶)
12	1, 8, 11	3eqtr4g 2856	1 ⊢ (𝐵 ⊆ 𝐴 → ((𝑥 ∈ 𝐴 ↦ 𝐶) ↾ 𝐵) = (𝑥 ∈ 𝐵 ↦ 𝐶))

Syntax hints:   → wi 4   = wceq 1522  Ⅎwnfc 2933  ⦋csb 3815   ⊆ wss 3863   ↦ cmpt 5045   ↾ cres 5450

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1777  ax-4 1791  ax-5 1888  ax-6 1947  ax-7 1992  ax-8 2083  ax-9 2091  ax-10 2112  ax-11 2126  ax-12 2141  ax-13 2344  ax-ext 2769  ax-sep 5099  ax-nul 5106  ax-pr 5226

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3an 1082  df-tru 1525  df-ex 1762  df-nf 1766  df-sb 2043  df-clab 2776  df-cleq 2788  df-clel 2863  df-nfc 2935  df-rab 3114  df-v 3439  df-sbc 3710  df-csb 3816  df-dif 3866  df-un 3868  df-in 3870  df-ss 3878  df-nul 4216  df-if 4386  df-sn 4477  df-pr 4479  df-op 4483  df-opab 5029  df-mpt 5046  df-xp 5454  df-rel 5455  df-res 5460

This theorem is referenced by:  esumval  30927  esumel  30928  esumsplit  30934  esumss  30953  limsupequzmpt2  41567  liminfequzmpt2  41640