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Theorem eqbrrdva 5439
Description: Deduction from extensionality principle for relations, given an equivalence only on the relation's 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 5274 . . . 4 (𝐶 × 𝐷) ⊆ (V × V)
31, 2syl6ss 3748 . . 3 (𝜑𝐴 ⊆ (V × V))
4 df-rel 5265 . . 3 (Rel 𝐴𝐴 ⊆ (V × V))
53, 4sylibr 224 . 2 (𝜑 → Rel 𝐴)
6 eqbrrdva.2 . . . 4 (𝜑𝐵 ⊆ (𝐶 × 𝐷))
76, 2syl6ss 3748 . . 3 (𝜑𝐵 ⊆ (V × V))
8 df-rel 5265 . . 3 (Rel 𝐵𝐵 ⊆ (V × V))
97, 8sylibr 224 . 2 (𝜑 → Rel 𝐵)
101ssbrd 4839 . . . 4 (𝜑 → (𝑥𝐴𝑦𝑥(𝐶 × 𝐷)𝑦))
11 brxp 5296 . . . 4 (𝑥(𝐶 × 𝐷)𝑦 ↔ (𝑥𝐶𝑦𝐷))
1210, 11syl6ib 241 . . 3 (𝜑 → (𝑥𝐴𝑦 → (𝑥𝐶𝑦𝐷)))
136ssbrd 4839 . . . 4 (𝜑 → (𝑥𝐵𝑦𝑥(𝐶 × 𝐷)𝑦))
1413, 11syl6ib 241 . . 3 (𝜑 → (𝑥𝐵𝑦 → (𝑥𝐶𝑦𝐷)))
15 eqbrrdva.3 . . . 4 ((𝜑𝑥𝐶𝑦𝐷) → (𝑥𝐴𝑦𝑥𝐵𝑦))
16153expib 1116 . . 3 (𝜑 → ((𝑥𝐶𝑦𝐷) → (𝑥𝐴𝑦𝑥𝐵𝑦)))
1712, 14, 16pm5.21ndd 368 . 2 (𝜑 → (𝑥𝐴𝑦𝑥𝐵𝑦))
185, 9, 17eqbrrdv 5366 1 (𝜑𝐴 = 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383  w3a 1072   = wceq 1624  wcel 2131  Vcvv 3332  wss 3707   class class class wbr 4796   × cxp 5256  Rel wrel 5263
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1863  ax-4 1878  ax-5 1980  ax-6 2046  ax-7 2082  ax-9 2140  ax-10 2160  ax-11 2175  ax-12 2188  ax-13 2383  ax-ext 2732  ax-sep 4925  ax-nul 4933  ax-pr 5047
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1627  df-ex 1846  df-nf 1851  df-sb 2039  df-clab 2739  df-cleq 2745  df-clel 2748  df-nfc 2883  df-ral 3047  df-rex 3048  df-rab 3051  df-v 3334  df-dif 3710  df-un 3712  df-in 3714  df-ss 3721  df-nul 4051  df-if 4223  df-sn 4314  df-pr 4316  df-op 4320  df-br 4797  df-opab 4857  df-xp 5264  df-rel 5265
This theorem is referenced by:  metustsym  22553
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