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Theorem resmptf 5993
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 5991 . 2 (𝐵𝐴 → ((𝑦𝐴𝑦 / 𝑥𝐶) ↾ 𝐵) = (𝑦𝐵𝑦 / 𝑥𝐶))
2 resmptf.a . . . 4 𝑥𝐴
3 nfcv 2907 . . . 4 𝑦𝐴
4 nfcv 2907 . . . 4 𝑦𝐶
5 nfcsb1v 3880 . . . 4 𝑥𝑦 / 𝑥𝐶
6 csbeq1a 3869 . . . 4 (𝑥 = 𝑦𝐶 = 𝑦 / 𝑥𝐶)
72, 3, 4, 5, 6cbvmptf 5214 . . 3 (𝑥𝐴𝐶) = (𝑦𝐴𝑦 / 𝑥𝐶)
87reseq1i 5933 . 2 ((𝑥𝐴𝐶) ↾ 𝐵) = ((𝑦𝐴𝑦 / 𝑥𝐶) ↾ 𝐵)
9 resmptf.b . . 3 𝑥𝐵
10 nfcv 2907 . . 3 𝑦𝐵
119, 10, 4, 5, 6cbvmptf 5214 . 2 (𝑥𝐵𝐶) = (𝑦𝐵𝑦 / 𝑥𝐶)
121, 8, 113eqtr4g 2801 1 (𝐵𝐴 → ((𝑥𝐴𝐶) ↾ 𝐵) = (𝑥𝐵𝐶))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1541  wnfc 2887  csb 3855  wss 3910  cmpt 5188  cres 5635
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-sep 5256  ax-nul 5263  ax-pr 5384
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4283  df-if 4487  df-sn 4587  df-pr 4589  df-op 4593  df-opab 5168  df-mpt 5189  df-xp 5639  df-rel 5640  df-res 5645
This theorem is referenced by:  esumval  32636  esumel  32637  esumsplit  32643  esumss  32662  limsupequzmpt2  43931  liminfequzmpt2  44004
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