Theorem resmptf 5992

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 5990	. 2 ⊢ (𝐵 ⊆ 𝐴 → ((𝑦 ∈ 𝐴 ↦ ⦋𝑦 / 𝑥⦌𝐶) ↾ 𝐵) = (𝑦 ∈ 𝐵 ↦ ⦋𝑦 / 𝑥⦌𝐶))
2		resmptf.a	. . . 4 ⊢ Ⅎ𝑥𝐴
3		nfcv 2895	. . . 4 ⊢ Ⅎ𝑦𝐴
4		nfcv 2895	. . . 4 ⊢ Ⅎ𝑦𝐶
5		nfcsb1v 3870	. . . 4 ⊢ Ⅎ𝑥⦋𝑦 / 𝑥⦌𝐶
6		csbeq1a 3860	. . . 4 ⊢ (𝑥 = 𝑦 → 𝐶 = ⦋𝑦 / 𝑥⦌𝐶)
7	2, 3, 4, 5, 6	cbvmptf 5193	. . 3 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐶) = (𝑦 ∈ 𝐴 ↦ ⦋𝑦 / 𝑥⦌𝐶)
8	7	reseq1i 5928	. 2 ⊢ ((𝑥 ∈ 𝐴 ↦ 𝐶) ↾ 𝐵) = ((𝑦 ∈ 𝐴 ↦ ⦋𝑦 / 𝑥⦌𝐶) ↾ 𝐵)
9		resmptf.b	. . 3 ⊢ Ⅎ𝑥𝐵
10		nfcv 2895	. . 3 ⊢ Ⅎ𝑦𝐵
11	9, 10, 4, 5, 6	cbvmptf 5193	. 2 ⊢ (𝑥 ∈ 𝐵 ↦ 𝐶) = (𝑦 ∈ 𝐵 ↦ ⦋𝑦 / 𝑥⦌𝐶)
12	1, 8, 11	3eqtr4g 2793	1 ⊢ (𝐵 ⊆ 𝐴 → ((𝑥 ∈ 𝐴 ↦ 𝐶) ↾ 𝐵) = (𝑥 ∈ 𝐵 ↦ 𝐶))

Syntax hints:   → wi 4   = wceq 1541  Ⅎwnfc 2880  ⦋csb 3846   ⊆ wss 3898   ↦ cmpt 5174   ↾ cres 5621

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2705  ax-sep 5236  ax-nul 5246  ax-pr 5372

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-clab 2712  df-cleq 2725  df-clel 2808  df-nfc 2882  df-rab 3397  df-v 3439  df-sbc 3738  df-csb 3847  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4283  df-if 4475  df-sn 4576  df-pr 4578  df-op 4582  df-opab 5156  df-mpt 5175  df-xp 5625  df-rel 5626  df-res 5631

This theorem is referenced by:  esumval  34080  esumel  34081  esumsplit  34087  esumss  34106  limsupequzmpt2  45840  liminfequzmpt2  45913