Theorem resmptf 6031

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 6029	. 2 ⊢ (𝐵 ⊆ 𝐴 → ((𝑦 ∈ 𝐴 ↦ ⦋𝑦 / 𝑥⦌𝐶) ↾ 𝐵) = (𝑦 ∈ 𝐵 ↦ ⦋𝑦 / 𝑥⦌𝐶))
2		resmptf.a	. . . 4 ⊢ Ⅎ𝑥𝐴
3		nfcv 2927	. . . 4 ⊢ Ⅎ𝑦𝐴
4		nfcv 2927	. . . 4 ⊢ Ⅎ𝑦𝐶
5		nfcsb1v 3879	. . . 4 ⊢ Ⅎ𝑥⦋𝑦 / 𝑥⦌𝐶
6		csbeq1a 3869	. . . 4 ⊢ (𝑥 = 𝑦 → 𝐶 = ⦋𝑦 / 𝑥⦌𝐶)
7	2, 3, 4, 5, 6	cbvmptf 5204	. . 3 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐶) = (𝑦 ∈ 𝐴 ↦ ⦋𝑦 / 𝑥⦌𝐶)
8	7	reseq1i 5964	. 2 ⊢ ((𝑥 ∈ 𝐴 ↦ 𝐶) ↾ 𝐵) = ((𝑦 ∈ 𝐴 ↦ ⦋𝑦 / 𝑥⦌𝐶) ↾ 𝐵)
9		resmptf.b	. . 3 ⊢ Ⅎ𝑥𝐵
10		nfcv 2927	. . 3 ⊢ Ⅎ𝑦𝐵
11	9, 10, 4, 5, 6	cbvmptf 5204	. 2 ⊢ (𝑥 ∈ 𝐵 ↦ 𝐶) = (𝑦 ∈ 𝐵 ↦ ⦋𝑦 / 𝑥⦌𝐶)
12	1, 8, 11	3eqtr4g 2825	1 ⊢ (𝐵 ⊆ 𝐴 → ((𝑥 ∈ 𝐴 ↦ 𝐶) ↾ 𝐵) = (𝑥 ∈ 𝐵 ↦ 𝐶))

Syntax hints:   → wi 4   = wceq 1563  Ⅎwnfc 2912  ⦋csb 3855   ⊆ wss 3907   ↦ cmpt 5185   ↾ cres 5653

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2737  ax-sep 5250  ax-pr 5394

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-nf 1807  df-sb 2094  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-rab 3418  df-v 3459  df-sbc 3748  df-csb 3856  df-dif 3910  df-un 3912  df-in 3914  df-ss 3924  df-nul 4289  df-if 4484  df-sn 4586  df-pr 4588  df-op 4592  df-opab 5167  df-mpt 5186  df-xp 5657  df-rel 5658  df-res 5663

This theorem is referenced by:  esumval  34348  esumel  34349  esumsplit  34355  esumss  34374  limsupequzmpt2  46291  liminfequzmpt2  46364