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Theorem resmptf 5414
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 5412 . 2 (𝐵𝐴 → ((𝑦𝐴𝑦 / 𝑥𝐶) ↾ 𝐵) = (𝑦𝐵𝑦 / 𝑥𝐶))
2 resmptf.a . . . 4 𝑥𝐴
3 nfcv 2767 . . . 4 𝑦𝐴
4 nfcv 2767 . . . 4 𝑦𝐶
5 nfcsb1v 3535 . . . 4 𝑥𝑦 / 𝑥𝐶
6 csbeq1a 3528 . . . 4 (𝑥 = 𝑦𝐶 = 𝑦 / 𝑥𝐶)
72, 3, 4, 5, 6cbvmptf 4713 . . 3 (𝑥𝐴𝐶) = (𝑦𝐴𝑦 / 𝑥𝐶)
87reseq1i 5356 . 2 ((𝑥𝐴𝐶) ↾ 𝐵) = ((𝑦𝐴𝑦 / 𝑥𝐶) ↾ 𝐵)
9 resmptf.b . . 3 𝑥𝐵
10 nfcv 2767 . . 3 𝑦𝐵
119, 10, 4, 5, 6cbvmptf 4713 . 2 (𝑥𝐵𝐶) = (𝑦𝐵𝑦 / 𝑥𝐶)
121, 8, 113eqtr4g 2685 1 (𝐵𝐴 → ((𝑥𝐴𝐶) ↾ 𝐵) = (𝑥𝐵𝐶))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1480  wnfc 2754  csb 3519  wss 3560  cmpt 4678  cres 5081
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1841  ax-6 1890  ax-7 1937  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606  ax-sep 4746  ax-nul 4754  ax-pr 4872
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1883  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-rab 2921  df-v 3193  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3897  df-if 4064  df-sn 4154  df-pr 4156  df-op 4160  df-opab 4679  df-mpt 4680  df-xp 5085  df-rel 5086  df-res 5091
This theorem is referenced by:  esumval  29881  esumel  29882  esumsplit  29888  esumss  29907  limsupequzmpt2  39341
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