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Theorem eqbrrdva 5883
Description: Deduction from extensionality principle for relations, given an equivalence only on the relation domain and range. (Contributed by Thierry Arnoux, 28-Nov-2017.)
Hypotheses
Ref Expression
eqbrrdva.1 (𝜑𝐴 ⊆ (𝐶 × 𝐷))
eqbrrdva.2 (𝜑𝐵 ⊆ (𝐶 × 𝐷))
eqbrrdva.3 ((𝜑𝑥𝐶𝑦𝐷) → (𝑥𝐴𝑦𝑥𝐵𝑦))
Assertion
Ref Expression
eqbrrdva (𝜑𝐴 = 𝐵)
Distinct variable groups:   𝑥,𝑦,𝐴   𝑥,𝐵,𝑦   𝜑,𝑥,𝑦
Allowed substitution hints:   𝐶(𝑥,𝑦)   𝐷(𝑥,𝑦)

Proof of Theorem eqbrrdva
StepHypRef Expression
1 eqbrrdva.1 . . . 4 (𝜑𝐴 ⊆ (𝐶 × 𝐷))
2 xpss 5705 . . . 4 (𝐶 × 𝐷) ⊆ (V × V)
31, 2sstrdi 4008 . . 3 (𝜑𝐴 ⊆ (V × V))
4 df-rel 5696 . . 3 (Rel 𝐴𝐴 ⊆ (V × V))
53, 4sylibr 234 . 2 (𝜑 → Rel 𝐴)
6 eqbrrdva.2 . . . 4 (𝜑𝐵 ⊆ (𝐶 × 𝐷))
76, 2sstrdi 4008 . . 3 (𝜑𝐵 ⊆ (V × V))
8 df-rel 5696 . . 3 (Rel 𝐵𝐵 ⊆ (V × V))
97, 8sylibr 234 . 2 (𝜑 → Rel 𝐵)
101ssbrd 5191 . . . 4 (𝜑 → (𝑥𝐴𝑦𝑥(𝐶 × 𝐷)𝑦))
11 brxp 5738 . . . 4 (𝑥(𝐶 × 𝐷)𝑦 ↔ (𝑥𝐶𝑦𝐷))
1210, 11imbitrdi 251 . . 3 (𝜑 → (𝑥𝐴𝑦 → (𝑥𝐶𝑦𝐷)))
136ssbrd 5191 . . . 4 (𝜑 → (𝑥𝐵𝑦𝑥(𝐶 × 𝐷)𝑦))
1413, 11imbitrdi 251 . . 3 (𝜑 → (𝑥𝐵𝑦 → (𝑥𝐶𝑦𝐷)))
15 eqbrrdva.3 . . . 4 ((𝜑𝑥𝐶𝑦𝐷) → (𝑥𝐴𝑦𝑥𝐵𝑦))
16153expib 1121 . . 3 (𝜑 → ((𝑥𝐶𝑦𝐷) → (𝑥𝐴𝑦𝑥𝐵𝑦)))
1712, 14, 16pm5.21ndd 379 . 2 (𝜑 → (𝑥𝐴𝑦𝑥𝐵𝑦))
185, 9, 17eqbrrdv 5806 1 (𝜑𝐴 = 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  w3a 1086   = wceq 1537  wcel 2106  Vcvv 3478  wss 3963   class class class wbr 5148   × cxp 5687  Rel wrel 5694
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-br 5149  df-opab 5211  df-xp 5695  df-rel 5696
This theorem is referenced by:  metustsym  24584
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