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Theorem resmptf 4919
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 4917 . 2 (𝐵𝐴 → ((𝑦𝐴𝑦 / 𝑥𝐶) ↾ 𝐵) = (𝑦𝐵𝑦 / 𝑥𝐶))
2 resmptf.a . . . 4 𝑥𝐴
3 nfcv 2299 . . . 4 𝑦𝐴
4 nfcv 2299 . . . 4 𝑦𝐶
5 nfcsb1v 3064 . . . 4 𝑥𝑦 / 𝑥𝐶
6 csbeq1a 3040 . . . 4 (𝑥 = 𝑦𝐶 = 𝑦 / 𝑥𝐶)
72, 3, 4, 5, 6cbvmptf 4061 . . 3 (𝑥𝐴𝐶) = (𝑦𝐴𝑦 / 𝑥𝐶)
87reseq1i 4865 . 2 ((𝑥𝐴𝐶) ↾ 𝐵) = ((𝑦𝐴𝑦 / 𝑥𝐶) ↾ 𝐵)
9 resmptf.b . . 3 𝑥𝐵
10 nfcv 2299 . . 3 𝑦𝐵
119, 10, 4, 5, 6cbvmptf 4061 . 2 (𝑥𝐵𝐶) = (𝑦𝐵𝑦 / 𝑥𝐶)
121, 8, 113eqtr4g 2215 1 (𝐵𝐴 → ((𝑥𝐴𝐶) ↾ 𝐵) = (𝑥𝐵𝐶))
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1335  wnfc 2286  csb 3031  wss 3102  cmpt 4028  cres 4591
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-14 2131  ax-ext 2139  ax-sep 4085  ax-pow 4138  ax-pr 4172
This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1338  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ral 2440  df-rex 2441  df-v 2714  df-sbc 2938  df-csb 3032  df-un 3106  df-in 3108  df-ss 3115  df-pw 3546  df-sn 3567  df-pr 3568  df-op 3570  df-opab 4029  df-mpt 4030  df-xp 4595  df-rel 4596  df-res 4601
This theorem is referenced by: (None)
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