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Mirrors > Home > MPE Home > Th. List > eqrelrdv | Structured version Visualization version GIF version |
Description: Deduce equality of relations from equivalence of membership. (Contributed by Rodolfo Medina, 10-Oct-2010.) |
Ref | Expression |
---|---|
eqrelrdv.1 | ⊢ Rel 𝐴 |
eqrelrdv.2 | ⊢ Rel 𝐵 |
eqrelrdv.3 | ⊢ (𝜑 → (〈𝑥, 𝑦〉 ∈ 𝐴 ↔ 〈𝑥, 𝑦〉 ∈ 𝐵)) |
Ref | Expression |
---|---|
eqrelrdv | ⊢ (𝜑 → 𝐴 = 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqrelrdv.3 | . . 3 ⊢ (𝜑 → (〈𝑥, 𝑦〉 ∈ 𝐴 ↔ 〈𝑥, 𝑦〉 ∈ 𝐵)) | |
2 | 1 | alrimivv 1936 | . 2 ⊢ (𝜑 → ∀𝑥∀𝑦(〈𝑥, 𝑦〉 ∈ 𝐴 ↔ 〈𝑥, 𝑦〉 ∈ 𝐵)) |
3 | eqrelrdv.1 | . . 3 ⊢ Rel 𝐴 | |
4 | eqrelrdv.2 | . . 3 ⊢ Rel 𝐵 | |
5 | eqrel 5655 | . . 3 ⊢ ((Rel 𝐴 ∧ Rel 𝐵) → (𝐴 = 𝐵 ↔ ∀𝑥∀𝑦(〈𝑥, 𝑦〉 ∈ 𝐴 ↔ 〈𝑥, 𝑦〉 ∈ 𝐵))) | |
6 | 3, 4, 5 | mp2an 692 | . 2 ⊢ (𝐴 = 𝐵 ↔ ∀𝑥∀𝑦(〈𝑥, 𝑦〉 ∈ 𝐴 ↔ 〈𝑥, 𝑦〉 ∈ 𝐵)) |
7 | 2, 6 | sylibr 237 | 1 ⊢ (𝜑 → 𝐴 = 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∀wal 1541 = wceq 1543 ∈ wcel 2110 〈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-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-opab 5116 df-xp 5557 df-rel 5558 |
This theorem is referenced by: eqbrrdiv 5664 fcnvres 6596 fmptco 6944 fpwwe2lem7 10251 fpwwe2lem11 10255 fsumcom2 15338 fprodcom2 15546 gsumcom2 19360 lgsquadlem1 26261 lgsquadlem2 26262 fmptcof2 30714 dfcnv2 30733 dih1dimatlem 39080 |
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