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Theorem resmptf 5610
Description: Restriction of the mapping operation. (Contributed by Thierry Arnoux, 28-Mar-2017.)
Hypotheses
Ref Expression
resmptf.a 𝑥𝐴
resmptf.b 𝑥𝐵
Assertion
Ref Expression
resmptf (𝐵𝐴 → ((𝑥𝐴𝐶) ↾ 𝐵) = (𝑥𝐵𝐶))

Proof of Theorem resmptf
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 resmpt 5608 . 2 (𝐵𝐴 → ((𝑦𝐴𝑦 / 𝑥𝐶) ↾ 𝐵) = (𝑦𝐵𝑦 / 𝑥𝐶))
2 resmptf.a . . . 4 𝑥𝐴
3 nfcv 2903 . . . 4 𝑦𝐴
4 nfcv 2903 . . . 4 𝑦𝐶
5 nfcsb1v 3691 . . . 4 𝑥𝑦 / 𝑥𝐶
6 csbeq1a 3684 . . . 4 (𝑥 = 𝑦𝐶 = 𝑦 / 𝑥𝐶)
72, 3, 4, 5, 6cbvmptf 4901 . . 3 (𝑥𝐴𝐶) = (𝑦𝐴𝑦 / 𝑥𝐶)
87reseq1i 5548 . 2 ((𝑥𝐴𝐶) ↾ 𝐵) = ((𝑦𝐴𝑦 / 𝑥𝐶) ↾ 𝐵)
9 resmptf.b . . 3 𝑥𝐵
10 nfcv 2903 . . 3 𝑦𝐵
119, 10, 4, 5, 6cbvmptf 4901 . 2 (𝑥𝐵𝐶) = (𝑦𝐵𝑦 / 𝑥𝐶)
121, 8, 113eqtr4g 2820 1 (𝐵𝐴 → ((𝑥𝐴𝐶) ↾ 𝐵) = (𝑥𝐵𝐶))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1632  wnfc 2890  csb 3675  wss 3716  cmpt 4882  cres 5269
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1989  ax-6 2055  ax-7 2091  ax-9 2149  ax-10 2169  ax-11 2184  ax-12 2197  ax-13 2392  ax-ext 2741  ax-sep 4934  ax-nul 4942  ax-pr 5056
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2048  df-clab 2748  df-cleq 2754  df-clel 2757  df-nfc 2892  df-rab 3060  df-v 3343  df-sbc 3578  df-csb 3676  df-dif 3719  df-un 3721  df-in 3723  df-ss 3730  df-nul 4060  df-if 4232  df-sn 4323  df-pr 4325  df-op 4329  df-opab 4866  df-mpt 4883  df-xp 5273  df-rel 5274  df-res 5279
This theorem is referenced by:  esumval  30439  esumel  30440  esumsplit  30446  esumss  30465  limsupequzmpt2  40472  liminfequzmpt2  40545
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