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Theorem eqbrrdva 5626
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 5459 . . . 4 (𝐶 × 𝐷) ⊆ (V × V)
31, 2syl6ss 3901 . . 3 (𝜑𝐴 ⊆ (V × V))
4 df-rel 5450 . . 3 (Rel 𝐴𝐴 ⊆ (V × V))
53, 4sylibr 235 . 2 (𝜑 → Rel 𝐴)
6 eqbrrdva.2 . . . 4 (𝜑𝐵 ⊆ (𝐶 × 𝐷))
76, 2syl6ss 3901 . . 3 (𝜑𝐵 ⊆ (V × V))
8 df-rel 5450 . . 3 (Rel 𝐵𝐵 ⊆ (V × V))
97, 8sylibr 235 . 2 (𝜑 → Rel 𝐵)
101ssbrd 5005 . . . 4 (𝜑 → (𝑥𝐴𝑦𝑥(𝐶 × 𝐷)𝑦))
11 brxp 5489 . . . 4 (𝑥(𝐶 × 𝐷)𝑦 ↔ (𝑥𝐶𝑦𝐷))
1210, 11syl6ib 252 . . 3 (𝜑 → (𝑥𝐴𝑦 → (𝑥𝐶𝑦𝐷)))
136ssbrd 5005 . . . 4 (𝜑 → (𝑥𝐵𝑦𝑥(𝐶 × 𝐷)𝑦))
1413, 11syl6ib 252 . . 3 (𝜑 → (𝑥𝐵𝑦 → (𝑥𝐶𝑦𝐷)))
15 eqbrrdva.3 . . . 4 ((𝜑𝑥𝐶𝑦𝐷) → (𝑥𝐴𝑦𝑥𝐵𝑦))
16153expib 1115 . . 3 (𝜑 → ((𝑥𝐶𝑦𝐷) → (𝑥𝐴𝑦𝑥𝐵𝑦)))
1712, 14, 16pm5.21ndd 381 . 2 (𝜑 → (𝑥𝐴𝑦𝑥𝐵𝑦))
185, 9, 17eqbrrdv 5552 1 (𝜑𝐴 = 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396  w3a 1080   = wceq 1522  wcel 2081  Vcvv 3437  wss 3859   class class class wbr 4962   × cxp 5441  Rel wrel 5448
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1777  ax-4 1791  ax-5 1888  ax-6 1947  ax-7 1992  ax-8 2083  ax-9 2091  ax-10 2112  ax-11 2126  ax-12 2141  ax-ext 2769  ax-sep 5094  ax-nul 5101  ax-pr 5221
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3an 1082  df-tru 1525  df-ex 1762  df-nf 1766  df-sb 2043  df-clab 2776  df-cleq 2788  df-clel 2863  df-nfc 2935  df-ral 3110  df-rex 3111  df-rab 3114  df-v 3439  df-dif 3862  df-un 3864  df-in 3866  df-ss 3874  df-nul 4212  df-if 4382  df-sn 4473  df-pr 4475  df-op 4479  df-br 4963  df-opab 5025  df-xp 5449  df-rel 5450
This theorem is referenced by:  metustsym  22848
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