Theorem eqbrrdva 5880

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 5701	. . . 4 ⊢ (𝐶 × 𝐷) ⊆ (V × V)
3	1, 2	sstrdi 3996	. . 3 ⊢ (𝜑 → 𝐴 ⊆ (V × V))
4		df-rel 5692	. . 3 ⊢ (Rel 𝐴 ↔ 𝐴 ⊆ (V × V))
5	3, 4	sylibr 234	. 2 ⊢ (𝜑 → Rel 𝐴)
6		eqbrrdva.2	. . . 4 ⊢ (𝜑 → 𝐵 ⊆ (𝐶 × 𝐷))
7	6, 2	sstrdi 3996	. . 3 ⊢ (𝜑 → 𝐵 ⊆ (V × V))
8		df-rel 5692	. . 3 ⊢ (Rel 𝐵 ↔ 𝐵 ⊆ (V × V))
9	7, 8	sylibr 234	. 2 ⊢ (𝜑 → Rel 𝐵)
10	1	ssbrd 5186	. . . 4 ⊢ (𝜑 → (𝑥𝐴𝑦 → 𝑥(𝐶 × 𝐷)𝑦))
11		brxp 5734	. . . 4 ⊢ (𝑥(𝐶 × 𝐷)𝑦 ↔ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷))
12	10, 11	imbitrdi 251	. . 3 ⊢ (𝜑 → (𝑥𝐴𝑦 → (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷)))
13	6	ssbrd 5186	. . . 4 ⊢ (𝜑 → (𝑥𝐵𝑦 → 𝑥(𝐶 × 𝐷)𝑦))
14	13, 11	imbitrdi 251	. . 3 ⊢ (𝜑 → (𝑥𝐵𝑦 → (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷)))
15		eqbrrdva.3	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷) → (𝑥𝐴𝑦 ↔ 𝑥𝐵𝑦))
16	15	3expib 1123	. . 3 ⊢ (𝜑 → ((𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷) → (𝑥𝐴𝑦 ↔ 𝑥𝐵𝑦)))
17	12, 14, 16	pm5.21ndd 379	. 2 ⊢ (𝜑 → (𝑥𝐴𝑦 ↔ 𝑥𝐵𝑦))
18	5, 9, 17	eqbrrdv 5803	1 ⊢ (𝜑 → 𝐴 = 𝐵)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1087   = wceq 1540   ∈ wcel 2108  Vcvv 3480   ⊆ wss 3951   class class class wbr 5143   × cxp 5683  Rel wrel 5690

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-br 5144  df-opab 5206  df-xp 5691  df-rel 5692

This theorem is referenced by:  metustsym  24568