Theorem ovresd 6699

Description: Lemma for converting metric theorems to metric space theorems. (Contributed by Mario Carneiro, 2-Oct-2015.)

Hypotheses

Ref	Expression
ovresd.1	⊢ (𝜑 → 𝐴 ∈ 𝑋)
ovresd.2	⊢ (𝜑 → 𝐵 ∈ 𝑋)

Assertion

Ref	Expression
ovresd	⊢ (𝜑 → (𝐴(𝐷 ↾ (𝑋 × 𝑋))𝐵) = (𝐴𝐷𝐵))

Proof of Theorem ovresd

Step	Hyp	Ref	Expression
1		ovresd.1	. 2 ⊢ (𝜑 → 𝐴 ∈ 𝑋)
2		ovresd.2	. 2 ⊢ (𝜑 → 𝐵 ∈ 𝑋)
3		ovres 6698	. 2 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴(𝐷 ↾ (𝑋 × 𝑋))𝐵) = (𝐴𝐷𝐵))
4	1, 2, 3	syl2anc 691	1 ⊢ (𝜑 → (𝐴(𝐷 ↾ (𝑋 × 𝑋))𝐵) = (𝐴𝐷𝐵))

Syntax hints:   → wi 4   = wceq 1475   ∈ wcel 1977   × cxp 5036   ↾ cres 5040  (class class class)co 6549

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-xp 5044  df-res 5050  df-iota 5768  df-fv 5812  df-ov 6552

This theorem is referenced by:  sscres  16306  fullsubc  16333  fullresc  16334  funcres2c  16384  psmetres2  21929  xmetres2  21976  prdsdsf  21982  xpsdsval  21996  xmssym  22080  xmstri2  22081  mstri2  22082  xmstri  22083  mstri  22084  xmstri3  22085  mstri3  22086  msrtri  22087  tmsxpsval  22153  ngptgp  22250  nlmvscn  22301  nrginvrcn  22306  nghmcn  22359  cnmpt1ds  22453  cnmpt2ds  22454  ipcn  22853  caussi  22903  causs  22904  minveclem2  23005  minveclem3b  23007  minveclem3  23008  minveclem4  23011  minveclem6  23013  ftc1lem6  23608  ulmdvlem1  23958  abelth  23999  cxpcn3  24289  rlimcnp  24492  hhssnv  27505  madjusmdetlem3  29223  qqhcn  29363  qqhucn  29364  ftc1cnnc  32654  ismtyres  32777  isdrngo2  32927  rngchom  41759  ringchom  41805  irinitoringc  41861  rhmsubclem4  41881