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Theorem eqbrrdva 4533
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 4474 . . . 4 (𝐶 × 𝐷) ⊆ (V × V)
31, 2syl6ss 2985 . . 3 (𝜑𝐴 ⊆ (V × V))
4 df-rel 4380 . . 3 (Rel 𝐴𝐴 ⊆ (V × V))
53, 4sylibr 141 . 2 (𝜑 → Rel 𝐴)
6 eqbrrdva.2 . . . 4 (𝜑𝐵 ⊆ (𝐶 × 𝐷))
76, 2syl6ss 2985 . . 3 (𝜑𝐵 ⊆ (V × V))
8 df-rel 4380 . . 3 (Rel 𝐵𝐵 ⊆ (V × V))
97, 8sylibr 141 . 2 (𝜑 → Rel 𝐵)
101ssbrd 3833 . . . 4 (𝜑 → (𝑥𝐴𝑦𝑥(𝐶 × 𝐷)𝑦))
11 brxp 4403 . . . 4 (𝑥(𝐶 × 𝐷)𝑦 ↔ (𝑥𝐶𝑦𝐷))
1210, 11syl6ib 154 . . 3 (𝜑 → (𝑥𝐴𝑦 → (𝑥𝐶𝑦𝐷)))
136ssbrd 3833 . . . 4 (𝜑 → (𝑥𝐵𝑦𝑥(𝐶 × 𝐷)𝑦))
1413, 11syl6ib 154 . . 3 (𝜑 → (𝑥𝐵𝑦 → (𝑥𝐶𝑦𝐷)))
15 eqbrrdva.3 . . . 4 ((𝜑𝑥𝐶𝑦𝐷) → (𝑥𝐴𝑦𝑥𝐵𝑦))
16153expib 1118 . . 3 (𝜑 → ((𝑥𝐶𝑦𝐷) → (𝑥𝐴𝑦𝑥𝐵𝑦)))
1712, 14, 16pm5.21ndd 631 . 2 (𝜑 → (𝑥𝐴𝑦𝑥𝐵𝑦))
185, 9, 17eqbrrdv 4465 1 (𝜑𝐴 = 𝐵)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 101  wb 102  w3a 896   = wceq 1259  wcel 1409  Vcvv 2574  wss 2945   class class class wbr 3792   × cxp 4371  Rel wrel 4378
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-sep 3903  ax-pow 3955  ax-pr 3972
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-rex 2329  df-v 2576  df-un 2950  df-in 2952  df-ss 2959  df-pw 3389  df-sn 3409  df-pr 3410  df-op 3412  df-br 3793  df-opab 3847  df-xp 4379  df-rel 4380
This theorem is referenced by: (None)
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