Theorem eqbrrdva 5847

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	⊢ (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 5667	. . . 4 ⊢ (𝐶 × 𝐷) ⊆ (V × V)
3	1, 2	sstrdi 3943	. . 3 ⊢ (𝜑 → 𝐴 ⊆ (V × V))
4		df-rel 5658	. . 3 ⊢ (Rel 𝐴 ↔ 𝐴 ⊆ (V × V))
5	3, 4	sylibr 237	. 2 ⊢ (𝜑 → Rel 𝐴)
6		eqbrrdva.2	. . . 4 ⊢ (𝜑 → 𝐵 ⊆ (𝐶 × 𝐷))
7	6, 2	sstrdi 3943	. . 3 ⊢ (𝜑 → 𝐵 ⊆ (V × V))
8		df-rel 5658	. . 3 ⊢ (Rel 𝐵 ↔ 𝐵 ⊆ (V × V))
9	7, 8	sylibr 237	. 2 ⊢ (𝜑 → Rel 𝐵)
10	1	ssbrd 5148	. . . 4 ⊢ (𝜑 → (𝑥𝐴𝑦 → 𝑥(𝐶 × 𝐷)𝑦))
11		brxp 5700	. . . 4 ⊢ (𝑥(𝐶 × 𝐷)𝑦 ↔ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷))
12	10, 11	imbitrdi 254	. . 3 ⊢ (𝜑 → (𝑥𝐴𝑦 → (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷)))
13	6	ssbrd 5148	. . . 4 ⊢ (𝜑 → (𝑥𝐵𝑦 → 𝑥(𝐶 × 𝐷)𝑦))
14	13, 11	imbitrdi 254	. . 3 ⊢ (𝜑 → (𝑥𝐵𝑦 → (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷)))
15		eqbrrdva.3	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷) → (𝑥𝐴𝑦 ↔ 𝑥𝐵𝑦))
16	15	3expib 1140	. . 3 ⊢ (𝜑 → ((𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷) → (𝑥𝐴𝑦 ↔ 𝑥𝐵𝑦)))
17	12, 14, 16	pm5.21ndd 382	. 2 ⊢ (𝜑 → (𝑥𝐴𝑦 ↔ 𝑥𝐵𝑦))
18	5, 9, 17	eqbrrdv 5769	1 ⊢ (𝜑 → 𝐴 = 𝐵)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2145  Vcvv 3451   ⊆ wss 3899   class class class wbr 5103   × cxp 5649  Rel wrel 5656

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2733  ax-sep 5249  ax-pr 5391

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2740  df-cleq 2753  df-clel 2836  df-ral 3078  df-rex 3088  df-rab 3414  df-v 3453  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-br 5104  df-opab 5168  df-xp 5657  df-rel 5658

This theorem is used by:  metustsym  24874