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Theorem ovresd 6967
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 6966 . 2 ((𝐴𝑋𝐵𝑋) → (𝐴(𝐷 ↾ (𝑋 × 𝑋))𝐵) = (𝐴𝐷𝐵))
41, 2, 3syl2anc 696 1 (𝜑 → (𝐴(𝐷 ↾ (𝑋 × 𝑋))𝐵) = (𝐴𝐷𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1632  wcel 2139   × cxp 5264  cres 5268  (class class class)co 6814
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pr 5055
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-br 4805  df-opab 4865  df-xp 5272  df-res 5278  df-iota 6012  df-fv 6057  df-ov 6817
This theorem is referenced by:  sscres  16704  fullsubc  16731  fullresc  16732  funcres2c  16782  psmetres2  22340  xmetres2  22387  prdsdsf  22393  xpsdsval  22407  xmssym  22491  xmstri2  22492  mstri2  22493  xmstri  22494  mstri  22495  xmstri3  22496  mstri3  22497  msrtri  22498  tmsxpsval  22564  ngptgp  22661  nlmvscn  22712  nrginvrcn  22717  nghmcn  22770  cnmpt1ds  22866  cnmpt2ds  22867  ipcn  23265  caussi  23315  causs  23316  minveclem2  23417  minveclem3b  23419  minveclem3  23420  minveclem4  23423  minveclem6  23425  ftc1lem6  24023  ulmdvlem1  24373  abelth  24414  cxpcn3  24709  rlimcnp  24912  hhssnv  28451  madjusmdetlem3  30225  qqhcn  30365  qqhucn  30366  ftc1cnnc  33815  ismtyres  33938  isdrngo2  34088  rngchom  42495  ringchom  42541  irinitoringc  42597  rhmsubclem4  42617
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