Theorem resmptf 5063

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 5061	. 2 ⊢ (𝐵 ⊆ 𝐴 → ((𝑦 ∈ 𝐴 ↦ ⦋𝑦 / 𝑥⦌𝐶) ↾ 𝐵) = (𝑦 ∈ 𝐵 ↦ ⦋𝑦 / 𝑥⦌𝐶))
2		resmptf.a	. . . 4 ⊢ Ⅎ𝑥𝐴
3		nfcv 2374	. . . 4 ⊢ Ⅎ𝑦𝐴
4		nfcv 2374	. . . 4 ⊢ Ⅎ𝑦𝐶
5		nfcsb1v 3160	. . . 4 ⊢ Ⅎ𝑥⦋𝑦 / 𝑥⦌𝐶
6		csbeq1a 3136	. . . 4 ⊢ (𝑥 = 𝑦 → 𝐶 = ⦋𝑦 / 𝑥⦌𝐶)
7	2, 3, 4, 5, 6	cbvmptf 4183	. . 3 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐶) = (𝑦 ∈ 𝐴 ↦ ⦋𝑦 / 𝑥⦌𝐶)
8	7	reseq1i 5009	. 2 ⊢ ((𝑥 ∈ 𝐴 ↦ 𝐶) ↾ 𝐵) = ((𝑦 ∈ 𝐴 ↦ ⦋𝑦 / 𝑥⦌𝐶) ↾ 𝐵)
9		resmptf.b	. . 3 ⊢ Ⅎ𝑥𝐵
10		nfcv 2374	. . 3 ⊢ Ⅎ𝑦𝐵
11	9, 10, 4, 5, 6	cbvmptf 4183	. 2 ⊢ (𝑥 ∈ 𝐵 ↦ 𝐶) = (𝑦 ∈ 𝐵 ↦ ⦋𝑦 / 𝑥⦌𝐶)
12	1, 8, 11	3eqtr4g 2289	1 ⊢ (𝐵 ⊆ 𝐴 → ((𝑥 ∈ 𝐴 ↦ 𝐶) ↾ 𝐵) = (𝑥 ∈ 𝐵 ↦ 𝐶))

Syntax hints:   → wi 4   = wceq 1397  Ⅎwnfc 2361  ⦋csb 3127   ⊆ wss 3200   ↦ cmpt 4150   ↾ cres 4727

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ral 2515  df-rex 2516  df-v 2804  df-sbc 3032  df-csb 3128  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-opab 4151  df-mpt 4152  df-xp 4731  df-rel 4732  df-res 4737

This theorem is referenced by: (None)