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Mirrors > Home > ILE Home > Th. List > eqbrrdva | GIF version |
Description: Deduction from extensionality principle for relations, given an equivalence only on the relation's 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 4706 | . . . 4 ⊢ (𝐶 × 𝐷) ⊆ (V × V) | |
3 | 1, 2 | sstrdi 3149 | . . 3 ⊢ (𝜑 → 𝐴 ⊆ (V × V)) |
4 | df-rel 4605 | . . 3 ⊢ (Rel 𝐴 ↔ 𝐴 ⊆ (V × V)) | |
5 | 3, 4 | sylibr 133 | . 2 ⊢ (𝜑 → Rel 𝐴) |
6 | eqbrrdva.2 | . . . 4 ⊢ (𝜑 → 𝐵 ⊆ (𝐶 × 𝐷)) | |
7 | 6, 2 | sstrdi 3149 | . . 3 ⊢ (𝜑 → 𝐵 ⊆ (V × V)) |
8 | df-rel 4605 | . . 3 ⊢ (Rel 𝐵 ↔ 𝐵 ⊆ (V × V)) | |
9 | 7, 8 | sylibr 133 | . 2 ⊢ (𝜑 → Rel 𝐵) |
10 | 1 | ssbrd 4019 | . . . 4 ⊢ (𝜑 → (𝑥𝐴𝑦 → 𝑥(𝐶 × 𝐷)𝑦)) |
11 | brxp 4629 | . . . 4 ⊢ (𝑥(𝐶 × 𝐷)𝑦 ↔ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷)) | |
12 | 10, 11 | syl6ib 160 | . . 3 ⊢ (𝜑 → (𝑥𝐴𝑦 → (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷))) |
13 | 6 | ssbrd 4019 | . . . 4 ⊢ (𝜑 → (𝑥𝐵𝑦 → 𝑥(𝐶 × 𝐷)𝑦)) |
14 | 13, 11 | syl6ib 160 | . . 3 ⊢ (𝜑 → (𝑥𝐵𝑦 → (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷))) |
15 | eqbrrdva.3 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷) → (𝑥𝐴𝑦 ↔ 𝑥𝐵𝑦)) | |
16 | 15 | 3expib 1195 | . . 3 ⊢ (𝜑 → ((𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷) → (𝑥𝐴𝑦 ↔ 𝑥𝐵𝑦))) |
17 | 12, 14, 16 | pm5.21ndd 695 | . 2 ⊢ (𝜑 → (𝑥𝐴𝑦 ↔ 𝑥𝐵𝑦)) |
18 | 5, 9, 17 | eqbrrdv 4695 | 1 ⊢ (𝜑 → 𝐴 = 𝐵) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ↔ wb 104 ∧ w3a 967 = wceq 1342 ∈ wcel 2135 Vcvv 2721 ⊆ wss 3111 class class class wbr 3976 × cxp 4596 Rel wrel 4603 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-14 2138 ax-ext 2146 ax-sep 4094 ax-pow 4147 ax-pr 4181 |
This theorem depends on definitions: df-bi 116 df-3an 969 df-tru 1345 df-nf 1448 df-sb 1750 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ral 2447 df-rex 2448 df-v 2723 df-un 3115 df-in 3117 df-ss 3124 df-pw 3555 df-sn 3576 df-pr 3577 df-op 3579 df-br 3977 df-opab 4038 df-xp 4604 df-rel 4605 |
This theorem is referenced by: (None) |
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