Theorem eqrelrd2 32650

Description: A version of eqrelrdv2 5812 with explicit nonfree declarations. (Contributed by Thierry Arnoux, 28-Aug-2017.)

Hypotheses

Ref	Expression
eqrelrd2.1	⊢ Ⅎ𝑥𝜑
eqrelrd2.2	⊢ Ⅎ𝑦𝜑
eqrelrd2.3	⊢ Ⅎ𝑥𝐴
eqrelrd2.4	⊢ Ⅎ𝑦𝐴
eqrelrd2.5	⊢ Ⅎ𝑥𝐵
eqrelrd2.6	⊢ Ⅎ𝑦𝐵
eqrelrd2.7	⊢ (𝜑 → (⟨𝑥, 𝑦⟩ ∈ 𝐴 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐵))

Assertion

Ref	Expression
eqrelrd2	⊢ (((Rel 𝐴 ∧ Rel 𝐵) ∧ 𝜑) → 𝐴 = 𝐵)

Distinct variable group:   𝑥,𝑦

Allowed substitution hints:   𝜑(𝑥,𝑦)   𝐴(𝑥,𝑦)   𝐵(𝑥,𝑦)

Proof of Theorem eqrelrd2

Step	Hyp	Ref	Expression
1		eqrelrd2.1	. . . 4 ⊢ Ⅎ𝑥𝜑
2		eqrelrd2.2	. . . . 5 ⊢ Ⅎ𝑦𝜑
3		eqrelrd2.7	. . . . 5 ⊢ (𝜑 → (⟨𝑥, 𝑦⟩ ∈ 𝐴 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐵))
4	2, 3	alrimi 2213	. . . 4 ⊢ (𝜑 → ∀𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐴 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐵))
5	1, 4	alrimi 2213	. . 3 ⊢ (𝜑 → ∀𝑥∀𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐴 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐵))
6	5	adantl 481	. 2 ⊢ (((Rel 𝐴 ∧ Rel 𝐵) ∧ 𝜑) → ∀𝑥∀𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐴 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐵))
7		eqrelrd2.3	. . . . . 6 ⊢ Ⅎ𝑥𝐴
8		eqrelrd2.4	. . . . . 6 ⊢ Ⅎ𝑦𝐴
9		eqrelrd2.5	. . . . . 6 ⊢ Ⅎ𝑥𝐵
10		eqrelrd2.6	. . . . . 6 ⊢ Ⅎ𝑦𝐵
11	1, 2, 7, 8, 9, 10	ssrelf 32649	. . . . 5 ⊢ (Rel 𝐴 → (𝐴 ⊆ 𝐵 ↔ ∀𝑥∀𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐴 → ⟨𝑥, 𝑦⟩ ∈ 𝐵)))
12	1, 2, 9, 10, 7, 8	ssrelf 32649	. . . . 5 ⊢ (Rel 𝐵 → (𝐵 ⊆ 𝐴 ↔ ∀𝑥∀𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐵 → ⟨𝑥, 𝑦⟩ ∈ 𝐴)))
13	11, 12	bi2anan9 638	. . . 4 ⊢ ((Rel 𝐴 ∧ Rel 𝐵) → ((𝐴 ⊆ 𝐵 ∧ 𝐵 ⊆ 𝐴) ↔ (∀𝑥∀𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐴 → ⟨𝑥, 𝑦⟩ ∈ 𝐵) ∧ ∀𝑥∀𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐵 → ⟨𝑥, 𝑦⟩ ∈ 𝐴))))
14		eqss 4014	. . . 4 ⊢ (𝐴 = 𝐵 ↔ (𝐴 ⊆ 𝐵 ∧ 𝐵 ⊆ 𝐴))
15		2albiim 1890	. . . 4 ⊢ (∀𝑥∀𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐴 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐵) ↔ (∀𝑥∀𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐴 → ⟨𝑥, 𝑦⟩ ∈ 𝐵) ∧ ∀𝑥∀𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐵 → ⟨𝑥, 𝑦⟩ ∈ 𝐴)))
16	13, 14, 15	3bitr4g 314	. . 3 ⊢ ((Rel 𝐴 ∧ Rel 𝐵) → (𝐴 = 𝐵 ↔ ∀𝑥∀𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐴 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐵)))
17	16	adantr 480	. 2 ⊢ (((Rel 𝐴 ∧ Rel 𝐵) ∧ 𝜑) → (𝐴 = 𝐵 ↔ ∀𝑥∀𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐴 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐵)))
18	6, 17	mpbird 257	1 ⊢ (((Rel 𝐴 ∧ Rel 𝐵) ∧ 𝜑) → 𝐴 = 𝐵)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395  ∀wal 1537   = wceq 1539  Ⅎwnf 1782   ∈ wcel 2108  Ⅎwnfc 2890   ⊆ wss 3966  ⟨cop 4640  Rel wrel 5698

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5305  ax-nul 5315  ax-pr 5441

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ral 3062  df-v 3483  df-dif 3969  df-un 3971  df-ss 3983  df-nul 4343  df-if 4535  df-sn 4635  df-pr 4637  df-op 4641  df-opab 5214  df-xp 5699  df-rel 5700

This theorem is referenced by:  fpwrelmap  32765