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Theorem ovresd 5583
 Description: Lemma for converting metric theorems to metric space theorems. (Contributed by Mario Carneiro, 2-Oct-2015.)
Hypotheses
Ref Expression
ovresd.1 (φA 𝑋)
ovresd.2 (φB 𝑋)
Assertion
Ref Expression
ovresd (φ → (A(𝐷 ↾ (𝑋 × 𝑋))B) = (A𝐷B))

Proof of Theorem ovresd
StepHypRef Expression
1 ovresd.1 . 2 (φA 𝑋)
2 ovresd.2 . 2 (φB 𝑋)
3 ovres 5582 . 2 ((A 𝑋 B 𝑋) → (A(𝐷 ↾ (𝑋 × 𝑋))B) = (A𝐷B))
41, 2, 3syl2anc 391 1 (φ → (A(𝐷 ↾ (𝑋 × 𝑋))B) = (A𝐷B))
 Colors of variables: wff set class Syntax hints:   → wi 4   = wceq 1242   ∈ wcel 1390   × cxp 4286   ↾ cres 4290  (class class class)co 5455 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935 This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-br 3756  df-opab 3810  df-xp 4294  df-res 4300  df-iota 4810  df-fv 4853  df-ov 5458 This theorem is referenced by: (None)
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