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Mirrors > Home > MPE Home > Th. List > eqbrrdiv | Structured version Visualization version GIF version |
Description: Deduction from extensionality principle for relations. (Contributed by Rodolfo Medina, 10-Oct-2010.) |
Ref | Expression |
---|---|
eqbrrdiv.1 | ⊢ Rel 𝐴 |
eqbrrdiv.2 | ⊢ Rel 𝐵 |
eqbrrdiv.3 | ⊢ (𝜑 → (𝑥𝐴𝑦 ↔ 𝑥𝐵𝑦)) |
Ref | Expression |
---|---|
eqbrrdiv | ⊢ (𝜑 → 𝐴 = 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqbrrdiv.1 | . 2 ⊢ Rel 𝐴 | |
2 | eqbrrdiv.2 | . 2 ⊢ Rel 𝐵 | |
3 | eqbrrdiv.3 | . . 3 ⊢ (𝜑 → (𝑥𝐴𝑦 ↔ 𝑥𝐵𝑦)) | |
4 | df-br 5054 | . . 3 ⊢ (𝑥𝐴𝑦 ↔ 〈𝑥, 𝑦〉 ∈ 𝐴) | |
5 | df-br 5054 | . . 3 ⊢ (𝑥𝐵𝑦 ↔ 〈𝑥, 𝑦〉 ∈ 𝐵) | |
6 | 3, 4, 5 | 3bitr3g 316 | . 2 ⊢ (𝜑 → (〈𝑥, 𝑦〉 ∈ 𝐴 ↔ 〈𝑥, 𝑦〉 ∈ 𝐵)) |
7 | 1, 2, 6 | eqrelrdv 5662 | 1 ⊢ (𝜑 → 𝐴 = 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 = wceq 1543 ∈ wcel 2110 〈cop 4547 class class class wbr 5053 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 |
This theorem depends on definitions: df-bi 210 df-an 400 df-tru 1546 df-ex 1788 df-sb 2071 df-clab 2715 df-cleq 2729 df-clel 2816 df-v 3410 df-in 3873 df-ss 3883 df-br 5054 df-opab 5116 df-xp 5557 df-rel 5558 |
This theorem is referenced by: funcpropd 17407 fullpropd 17427 fthpropd 17428 dvres 24808 eqfunresadj 33454 |
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