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Theorem resmptf 4762
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 4760 . 2 (𝐵𝐴 → ((𝑦𝐴𝑦 / 𝑥𝐶) ↾ 𝐵) = (𝑦𝐵𝑦 / 𝑥𝐶))
2 resmptf.a . . . 4 𝑥𝐴
3 nfcv 2228 . . . 4 𝑦𝐴
4 nfcv 2228 . . . 4 𝑦𝐶
5 nfcsb1v 2963 . . . 4 𝑥𝑦 / 𝑥𝐶
6 csbeq1a 2941 . . . 4 (𝑥 = 𝑦𝐶 = 𝑦 / 𝑥𝐶)
72, 3, 4, 5, 6cbvmptf 3932 . . 3 (𝑥𝐴𝐶) = (𝑦𝐴𝑦 / 𝑥𝐶)
87reseq1i 4709 . 2 ((𝑥𝐴𝐶) ↾ 𝐵) = ((𝑦𝐴𝑦 / 𝑥𝐶) ↾ 𝐵)
9 resmptf.b . . 3 𝑥𝐵
10 nfcv 2228 . . 3 𝑦𝐵
119, 10, 4, 5, 6cbvmptf 3932 . 2 (𝑥𝐵𝐶) = (𝑦𝐵𝑦 / 𝑥𝐶)
121, 8, 113eqtr4g 2145 1 (𝐵𝐴 → ((𝑥𝐴𝐶) ↾ 𝐵) = (𝑥𝐵𝐶))
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1289  wnfc 2215  csb 2933  wss 2999  cmpt 3899  cres 4440
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3957  ax-pow 4009  ax-pr 4036
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-v 2621  df-sbc 2841  df-csb 2934  df-un 3003  df-in 3005  df-ss 3012  df-pw 3431  df-sn 3452  df-pr 3453  df-op 3455  df-opab 3900  df-mpt 3901  df-xp 4444  df-rel 4445  df-res 4450
This theorem is referenced by: (None)
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