Theorem eqbrrdva 4523

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	⊢ (𝜑 → 𝐴 ⊆ (𝐶 × 𝐷))
eqbrrdva.2	⊢ (𝜑 → 𝐵 ⊆ (𝐶 × 𝐷))
eqbrrdva.3	⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷) → (𝑥𝐴𝑦 ↔ 𝑥𝐵𝑦))

Assertion

Ref	Expression
eqbrrdva	⊢ (𝜑 → 𝐴 = 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

Step	Hyp	Ref	Expression
1		eqbrrdva.1	. . . 4 ⊢ (𝜑 → 𝐴 ⊆ (𝐶 × 𝐷))
2		xpss 4464	. . . 4 ⊢ (𝐶 × 𝐷) ⊆ (V × V)
3	1, 2	syl6ss 3011	. . 3 ⊢ (𝜑 → 𝐴 ⊆ (V × V))
4		df-rel 4370	. . 3 ⊢ (Rel 𝐴 ↔ 𝐴 ⊆ (V × V))
5	3, 4	sylibr 132	. 2 ⊢ (𝜑 → Rel 𝐴)
6		eqbrrdva.2	. . . 4 ⊢ (𝜑 → 𝐵 ⊆ (𝐶 × 𝐷))
7	6, 2	syl6ss 3011	. . 3 ⊢ (𝜑 → 𝐵 ⊆ (V × V))
8		df-rel 4370	. . 3 ⊢ (Rel 𝐵 ↔ 𝐵 ⊆ (V × V))
9	7, 8	sylibr 132	. 2 ⊢ (𝜑 → Rel 𝐵)
10	1	ssbrd 3826	. . . 4 ⊢ (𝜑 → (𝑥𝐴𝑦 → 𝑥(𝐶 × 𝐷)𝑦))
11		brxp 4393	. . . 4 ⊢ (𝑥(𝐶 × 𝐷)𝑦 ↔ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷))
12	10, 11	syl6ib 159	. . 3 ⊢ (𝜑 → (𝑥𝐴𝑦 → (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷)))
13	6	ssbrd 3826	. . . 4 ⊢ (𝜑 → (𝑥𝐵𝑦 → 𝑥(𝐶 × 𝐷)𝑦))
14	13, 11	syl6ib 159	. . 3 ⊢ (𝜑 → (𝑥𝐵𝑦 → (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷)))
15		eqbrrdva.3	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷) → (𝑥𝐴𝑦 ↔ 𝑥𝐵𝑦))
16	15	3expib 1141	. . 3 ⊢ (𝜑 → ((𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷) → (𝑥𝐴𝑦 ↔ 𝑥𝐵𝑦)))
17	12, 14, 16	pm5.21ndd 653	. 2 ⊢ (𝜑 → (𝑥𝐴𝑦 ↔ 𝑥𝐵𝑦))
18	5, 9, 17	eqbrrdv 4455	1 ⊢ (𝜑 → 𝐴 = 𝐵)

Syntax hints:   → wi 4   ∧ wa 102   ↔ wb 103   ∧ w3a 919   = wceq 1284   ∈ wcel 1433  Vcvv 2601   ⊆ wss 2973   class class class wbr 3785   × cxp 4361  Rel wrel 4368

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-sep 3896  ax-pow 3948  ax-pr 3964

This theorem depends on definitions:  df-bi 115  df-3an 921  df-tru 1287  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ral 2353  df-rex 2354  df-v 2603  df-un 2977  df-in 2979  df-ss 2986  df-pw 3384  df-sn 3404  df-pr 3405  df-op 3407  df-br 3786  df-opab 3840  df-xp 4369  df-rel 4370

This theorem is referenced by: (None)