Theorem resmptf 6039

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 6037	. 2 ⊢ (𝐵 ⊆ 𝐴 → ((𝑦 ∈ 𝐴 ↦ ⦋𝑦 / 𝑥⦌𝐶) ↾ 𝐵) = (𝑦 ∈ 𝐵 ↦ ⦋𝑦 / 𝑥⦌𝐶))
2		resmptf.a	. . . 4 ⊢ Ⅎ𝑥𝐴
3		nfcv 2924	. . . 4 ⊢ Ⅎ𝑦𝐴
4		nfcv 2924	. . . 4 ⊢ Ⅎ𝑦𝐶
5		nfcsb1v 3874	. . . 4 ⊢ Ⅎ𝑥⦋𝑦 / 𝑥⦌𝐶
6		csbeq1a 3864	. . . 4 ⊢ (𝑥 = 𝑦 → 𝐶 = ⦋𝑦 / 𝑥⦌𝐶)
7	2, 3, 4, 5, 6	cbvmptf 5209	. . 3 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐶) = (𝑦 ∈ 𝐴 ↦ ⦋𝑦 / 𝑥⦌𝐶)
8	7	reseq1i 5972	. 2 ⊢ ((𝑥 ∈ 𝐴 ↦ 𝐶) ↾ 𝐵) = ((𝑦 ∈ 𝐴 ↦ ⦋𝑦 / 𝑥⦌𝐶) ↾ 𝐵)
9		resmptf.b	. . 3 ⊢ Ⅎ𝑥𝐵
10		nfcv 2924	. . 3 ⊢ Ⅎ𝑦𝐵
11	9, 10, 4, 5, 6	cbvmptf 5209	. 2 ⊢ (𝑥 ∈ 𝐵 ↦ 𝐶) = (𝑦 ∈ 𝐵 ↦ ⦋𝑦 / 𝑥⦌𝐶)
12	1, 8, 11	3eqtr4g 2822	1 ⊢ (𝐵 ⊆ 𝐴 → ((𝑥 ∈ 𝐴 ↦ 𝐶) ↾ 𝐵) = (𝑥 ∈ 𝐵 ↦ 𝐶))

This proof depends on syntax axioms:   → wi 4   = wceq 1570  Ⅎwnfc 2909  ⦋csb 3850   ⊆ wss 3902   ↦ cmpt 5190   ↾ cres 5661

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 2215  ax-ext 2734  ax-sep 5255  ax-pr 5402

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-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-rab 3415  df-v 3455  df-sbc 3743  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4283  df-if 4486  df-sn 4588  df-pr 4590  df-op 4594  df-opab 5172  df-mpt 5191  df-xp 5665  df-rel 5666  df-res 5671

This theorem is used by:  esumval  34564  esumel  34565  esumsplit  34571  esumss  34590  limsupequzmpt2  46554  liminfequzmpt2  46627