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Mirrors > Home > MPE Home > Th. List > eqbrrdva | Structured version Visualization version GIF version |
Description: Deduction from extensionality principle for relations, given an equivalence only on the relation domain and range. (Contributed by Thierry Arnoux, 28-Nov-2017.) |
Ref | Expression |
---|---|
eqbrrdva.1 | ⊢ (𝜑 → 𝐴 ⊆ (𝐶 × 𝐷)) |
eqbrrdva.2 | ⊢ (𝜑 → 𝐵 ⊆ (𝐶 × 𝐷)) |
eqbrrdva.3 | ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷) → (𝑥𝐴𝑦 ↔ 𝑥𝐵𝑦)) |
Ref | Expression |
---|---|
eqbrrdva | ⊢ (𝜑 → 𝐴 = 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqbrrdva.1 | . . . 4 ⊢ (𝜑 → 𝐴 ⊆ (𝐶 × 𝐷)) | |
2 | xpss 5567 | . . . 4 ⊢ (𝐶 × 𝐷) ⊆ (V × V) | |
3 | 1, 2 | sstrdi 3913 | . . 3 ⊢ (𝜑 → 𝐴 ⊆ (V × V)) |
4 | df-rel 5558 | . . 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 3913 | . . 3 ⊢ (𝜑 → 𝐵 ⊆ (V × V)) |
8 | df-rel 5558 | . . 3 ⊢ (Rel 𝐵 ↔ 𝐵 ⊆ (V × V)) | |
9 | 7, 8 | sylibr 237 | . 2 ⊢ (𝜑 → Rel 𝐵) |
10 | 1 | ssbrd 5096 | . . . 4 ⊢ (𝜑 → (𝑥𝐴𝑦 → 𝑥(𝐶 × 𝐷)𝑦)) |
11 | brxp 5598 | . . . 4 ⊢ (𝑥(𝐶 × 𝐷)𝑦 ↔ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷)) | |
12 | 10, 11 | syl6ib 254 | . . 3 ⊢ (𝜑 → (𝑥𝐴𝑦 → (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷))) |
13 | 6 | ssbrd 5096 | . . . 4 ⊢ (𝜑 → (𝑥𝐵𝑦 → 𝑥(𝐶 × 𝐷)𝑦)) |
14 | 13, 11 | syl6ib 254 | . . 3 ⊢ (𝜑 → (𝑥𝐵𝑦 → (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷))) |
15 | eqbrrdva.3 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷) → (𝑥𝐴𝑦 ↔ 𝑥𝐵𝑦)) | |
16 | 15 | 3expib 1124 | . . 3 ⊢ (𝜑 → ((𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷) → (𝑥𝐴𝑦 ↔ 𝑥𝐵𝑦))) |
17 | 12, 14, 16 | pm5.21ndd 384 | . 2 ⊢ (𝜑 → (𝑥𝐴𝑦 ↔ 𝑥𝐵𝑦)) |
18 | 5, 9, 17 | eqbrrdv 5663 | 1 ⊢ (𝜑 → 𝐴 = 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 ∧ w3a 1089 = wceq 1543 ∈ wcel 2110 Vcvv 3408 ⊆ wss 3866 class class class wbr 5053 × cxp 5549 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-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-sb 2071 df-clab 2715 df-cleq 2729 df-clel 2816 df-ral 3066 df-rex 3067 df-rab 3070 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-br 5054 df-opab 5116 df-xp 5557 df-rel 5558 |
This theorem is referenced by: metustsym 23453 |
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