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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 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) | 
| Colors of variables: wff setvar class | 
| 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 | 
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