Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > eqrelrd2 | Structured version Visualization version GIF version |
Description: A version of eqrelrdv2 5665 with explicit nonfree declarations. (Contributed by Thierry Arnoux, 28-Aug-2017.) |
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)) |
Ref | Expression |
---|---|
eqrelrd2 | ⊢ (((Rel 𝐴 ∧ Rel 𝐵) ∧ 𝜑) → 𝐴 = 𝐵) |
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 2211 | . . . 4 ⊢ (𝜑 → ∀𝑦(〈𝑥, 𝑦〉 ∈ 𝐴 ↔ 〈𝑥, 𝑦〉 ∈ 𝐵)) |
5 | 1, 4 | alrimi 2211 | . . 3 ⊢ (𝜑 → ∀𝑥∀𝑦(〈𝑥, 𝑦〉 ∈ 𝐴 ↔ 〈𝑥, 𝑦〉 ∈ 𝐵)) |
6 | 5 | adantl 485 | . 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 30674 | . . . . 5 ⊢ (Rel 𝐴 → (𝐴 ⊆ 𝐵 ↔ ∀𝑥∀𝑦(〈𝑥, 𝑦〉 ∈ 𝐴 → 〈𝑥, 𝑦〉 ∈ 𝐵))) |
12 | 1, 2, 9, 10, 7, 8 | ssrelf 30674 | . . . . 5 ⊢ (Rel 𝐵 → (𝐵 ⊆ 𝐴 ↔ ∀𝑥∀𝑦(〈𝑥, 𝑦〉 ∈ 𝐵 → 〈𝑥, 𝑦〉 ∈ 𝐴))) |
13 | 11, 12 | bi2anan9 639 | . . . 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 3916 | . . . 4 ⊢ (𝐴 = 𝐵 ↔ (𝐴 ⊆ 𝐵 ∧ 𝐵 ⊆ 𝐴)) | |
15 | 2albiim 1898 | . . . 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 317 | . . 3 ⊢ ((Rel 𝐴 ∧ Rel 𝐵) → (𝐴 = 𝐵 ↔ ∀𝑥∀𝑦(〈𝑥, 𝑦〉 ∈ 𝐴 ↔ 〈𝑥, 𝑦〉 ∈ 𝐵))) |
17 | 16 | adantr 484 | . 2 ⊢ (((Rel 𝐴 ∧ Rel 𝐵) ∧ 𝜑) → (𝐴 = 𝐵 ↔ ∀𝑥∀𝑦(〈𝑥, 𝑦〉 ∈ 𝐴 ↔ 〈𝑥, 𝑦〉 ∈ 𝐵))) |
18 | 6, 17 | mpbird 260 | 1 ⊢ (((Rel 𝐴 ∧ Rel 𝐵) ∧ 𝜑) → 𝐴 = 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 ∀wal 1541 = wceq 1543 Ⅎwnf 1791 ∈ wcel 2110 Ⅎwnfc 2884 ⊆ wss 3866 〈cop 4547 Rel wrel 5556 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2708 ax-sep 5192 ax-nul 5199 ax-pr 5322 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2886 df-ral 3066 df-v 3410 df-dif 3869 df-un 3871 df-in 3873 df-ss 3883 df-nul 4238 df-if 4440 df-sn 4542 df-pr 4544 df-op 4548 df-opab 5116 df-xp 5557 df-rel 5558 |
This theorem is referenced by: fpwrelmap 30788 |
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