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Theorem resmpt 4599
 Description: Restriction of the mapping operation. (Contributed by Mario Carneiro, 15-Jul-2013.)
Assertion
Ref Expression
resmpt (BA → ((x A𝐶) ↾ B) = (x B𝐶))
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.
StepHypRef Expression
1 resopab2 4598 . 2 (BA → ({⟨x, y⟩ ∣ (x A y = 𝐶)} ↾ B) = {⟨x, y⟩ ∣ (x B y = 𝐶)})
2 df-mpt 3811 . . 3 (x A𝐶) = {⟨x, y⟩ ∣ (x A y = 𝐶)}
32reseq1i 4551 . 2 ((x A𝐶) ↾ B) = ({⟨x, y⟩ ∣ (x A y = 𝐶)} ↾ B)
4 df-mpt 3811 . 2 (x B𝐶) = {⟨x, y⟩ ∣ (x B y = 𝐶)}
51, 3, 43eqtr4g 2094 1 (BA → ((x A𝐶) ↾ B) = (x B𝐶))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97   = wceq 1242   ∈ wcel 1390   ⊆ wss 2911  {copab 3808   ↦ cmpt 3809   ↾ cres 4290 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935 This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-opab 3810  df-mpt 3811  df-xp 4294  df-rel 4295  df-res 4300 This theorem is referenced by:  resmpt3  4600  f1stres  5728  f2ndres  5729  tposss  5802  dftpos2  5817  dftpos4  5819
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