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Theorem ovresd 6051
Description: Lemma for converting metric theorems to metric space theorems. (Contributed by Mario Carneiro, 2-Oct-2015.)
Hypotheses
Ref Expression
ovresd.1 (𝜑𝐴𝑋)
ovresd.2 (𝜑𝐵𝑋)
Assertion
Ref Expression
ovresd (𝜑 → (𝐴(𝐷 ↾ (𝑋 × 𝑋))𝐵) = (𝐴𝐷𝐵))

Proof of Theorem ovresd
StepHypRef Expression
1 ovresd.1 . 2 (𝜑𝐴𝑋)
2 ovresd.2 . 2 (𝜑𝐵𝑋)
3 ovres 6050 . 2 ((𝐴𝑋𝐵𝑋) → (𝐴(𝐷 ↾ (𝑋 × 𝑋))𝐵) = (𝐴𝐷𝐵))
41, 2, 3syl2anc 411 1 (𝜑 → (𝐴(𝐷 ↾ (𝑋 × 𝑋))𝐵) = (𝐴𝐷𝐵))
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1364  wcel 2164   × cxp 4653  cres 4657  (class class class)co 5910
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2167  ax-ext 2175  ax-sep 4147  ax-pow 4203  ax-pr 4238
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-v 2762  df-un 3157  df-in 3159  df-ss 3166  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-br 4030  df-opab 4091  df-xp 4661  df-res 4667  df-iota 5207  df-fv 5254  df-ov 5913
This theorem is referenced by:  psmetres2  14472  xmetres2  14518  xmssym  14608  xmstri2  14609  mstri2  14610  xmstri  14611  mstri  14612  xmstri3  14613  mstri3  14614  msrtri  14615  limcimolemlt  14789  cnplimcim  14792  cnplimclemr  14794  limccnpcntop  14800  limccnp2lem  14801
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