Theorem resmpt 4955

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 4954	. 2 ⊢ (𝐵 ⊆ 𝐴 → ({⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐶)} ↾ 𝐵) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐵 ∧ 𝑦 = 𝐶)})
2		df-mpt 4066	. . 3 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐶) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐶)}
3	2	reseq1i 4903	. 2 ⊢ ((𝑥 ∈ 𝐴 ↦ 𝐶) ↾ 𝐵) = ({⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐶)} ↾ 𝐵)
4		df-mpt 4066	. 2 ⊢ (𝑥 ∈ 𝐵 ↦ 𝐶) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐵 ∧ 𝑦 = 𝐶)}
5	1, 3, 4	3eqtr4g 2235	1 ⊢ (𝐵 ⊆ 𝐴 → ((𝑥 ∈ 𝐴 ↦ 𝐶) ↾ 𝐵) = (𝑥 ∈ 𝐵 ↦ 𝐶))

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1353   ∈ wcel 2148   ⊆ wss 3129  {copab 4063   ↦ cmpt 4064   ↾ cres 4628

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4121  ax-pow 4174  ax-pr 4209

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2739  df-un 3133  df-in 3135  df-ss 3142  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-opab 4065  df-mpt 4066  df-xp 4632  df-rel 4633  df-res 4638

This theorem is referenced by:  resmpt3  4956  resmptf  4957  resmptd  4958  f1stres  6159  f2ndres  6160  tposss  6246  dftpos2  6261  dftpos4  6263  djuf1olemr  7052  fisumss  11399  isumclim3  11430  expcnv  11511  fprodssdc  11597  tgrest  13639  cnmptid  13751  dvidlemap  14130  dvcnp2cntop  14133  dvmulxxbr  14136  dvcoapbr  14141  dvrecap  14147