Theorem eqbrrdva 5862

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 5685	. . . 4 ⊢ (𝐶 × 𝐷) ⊆ (V × V)
3	1, 2	sstrdi 3989	. . 3 ⊢ (𝜑 → 𝐴 ⊆ (V × V))
4		df-rel 5676	. . 3 ⊢ (Rel 𝐴 ↔ 𝐴 ⊆ (V × V))
5	3, 4	sylibr 233	. 2 ⊢ (𝜑 → Rel 𝐴)
6		eqbrrdva.2	. . . 4 ⊢ (𝜑 → 𝐵 ⊆ (𝐶 × 𝐷))
7	6, 2	sstrdi 3989	. . 3 ⊢ (𝜑 → 𝐵 ⊆ (V × V))
8		df-rel 5676	. . 3 ⊢ (Rel 𝐵 ↔ 𝐵 ⊆ (V × V))
9	7, 8	sylibr 233	. 2 ⊢ (𝜑 → Rel 𝐵)
10	1	ssbrd 5184	. . . 4 ⊢ (𝜑 → (𝑥𝐴𝑦 → 𝑥(𝐶 × 𝐷)𝑦))
11		brxp 5718	. . . 4 ⊢ (𝑥(𝐶 × 𝐷)𝑦 ↔ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷))
12	10, 11	imbitrdi 250	. . 3 ⊢ (𝜑 → (𝑥𝐴𝑦 → (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷)))
13	6	ssbrd 5184	. . . 4 ⊢ (𝜑 → (𝑥𝐵𝑦 → 𝑥(𝐶 × 𝐷)𝑦))
14	13, 11	imbitrdi 250	. . 3 ⊢ (𝜑 → (𝑥𝐵𝑦 → (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷)))
15		eqbrrdva.3	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷) → (𝑥𝐴𝑦 ↔ 𝑥𝐵𝑦))
16	15	3expib 1119	. . 3 ⊢ (𝜑 → ((𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷) → (𝑥𝐴𝑦 ↔ 𝑥𝐵𝑦)))
17	12, 14, 16	pm5.21ndd 379	. 2 ⊢ (𝜑 → (𝑥𝐴𝑦 ↔ 𝑥𝐵𝑦))
18	5, 9, 17	eqbrrdv 5786	1 ⊢ (𝜑 → 𝐴 = 𝐵)

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098  Vcvv 3468   ⊆ wss 3943   class class class wbr 5141   × cxp 5667  Rel wrel 5674

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pr 5420

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2704  df-cleq 2718  df-clel 2804  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-br 5142  df-opab 5204  df-xp 5675  df-rel 5676

This theorem is referenced by:  metustsym  24414