Theorem resmpt 5091

Description: Restriction of the mapping operation. (Contributed by Mario Carneiro, 15-Jul-2013.)

Assertion

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

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Allowed substitution hint:   𝐶(𝑥)

Proof of Theorem resmpt

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		resopab2 5090	. 2 ⊢ (𝐵 ⊆ 𝐴 → ({⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐶)} ↾ 𝐵) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐵 ∧ 𝑦 = 𝐶)})
2		df-mpt 4178	. . 3 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐶) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐶)}
3	2	reseq1i 5039	. 2 ⊢ ((𝑥 ∈ 𝐴 ↦ 𝐶) ↾ 𝐵) = ({⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐶)} ↾ 𝐵)
4		df-mpt 4178	. 2 ⊢ (𝑥 ∈ 𝐵 ↦ 𝐶) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐵 ∧ 𝑦 = 𝐶)}
5	1, 3, 4	3eqtr4g 2292	1 ⊢ (𝐵 ⊆ 𝐴 → ((𝑥 ∈ 𝐴 ↦ 𝐶) ↾ 𝐵) = (𝑥 ∈ 𝐵 ↦ 𝐶))

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1398   ∈ wcel 2205   ⊆ wss 3214  {copab 4175   ↦ cmpt 4176   ↾ cres 4756

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2208  ax-ext 2216  ax-sep 4233  ax-pow 4292  ax-pr 4327

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ral 2527  df-rex 2528  df-v 2817  df-un 3218  df-in 3220  df-ss 3227  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-opab 4177  df-mpt 4178  df-xp 4760  df-rel 4761  df-res 4766

This theorem is referenced by:  resmpt3  5092  resmptf  5093  resmptd  5094  f1stres  6366  f2ndres  6367  tposss  6490  dftpos2  6505  dftpos4  6507  djuf1olemr  7358  fisumss  12106  isumclim3  12137  expcnv  12218  fprodssdc  12304  conjsubg  14033  gsumfzfsumlemm  14864  tgrest  15163  cnmptid  15275  hovercncf  15640  dvidlemap  15685  dvidrelem  15686  dvidsslem  15687  dvcnp2cntop  15693  dvmulxxbr  15696  dvcoapbr  15701  dvrecap  15707