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Mirrors > Home > MPE Home > Th. List > Mathboxes > brrabga | Structured version Visualization version GIF version |
Description: The law of concretion for operation class abstraction. (Contributed by Peter Mazsa, 24-Oct-2022.) |
Ref | Expression |
---|---|
brrabga.1 | ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ∧ 𝑧 = 𝐶) → (𝜑 ↔ 𝜓)) |
brrabga.2 | ⊢ 𝑅 = {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝜑} |
Ref | Expression |
---|---|
brrabga | ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → (〈𝐴, 𝐵〉𝑅𝐶 ↔ 𝜓)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-br 5040 | . . 3 ⊢ (〈𝐴, 𝐵〉𝑅𝐶 ↔ 〈〈𝐴, 𝐵〉, 𝐶〉 ∈ 𝑅) | |
2 | brrabga.2 | . . . 4 ⊢ 𝑅 = {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝜑} | |
3 | 2 | eleq2i 2822 | . . 3 ⊢ (〈〈𝐴, 𝐵〉, 𝐶〉 ∈ 𝑅 ↔ 〈〈𝐴, 𝐵〉, 𝐶〉 ∈ {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝜑}) |
4 | 1, 3 | bitri 278 | . 2 ⊢ (〈𝐴, 𝐵〉𝑅𝐶 ↔ 〈〈𝐴, 𝐵〉, 𝐶〉 ∈ {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝜑}) |
5 | brrabga.1 | . . 3 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ∧ 𝑧 = 𝐶) → (𝜑 ↔ 𝜓)) | |
6 | 5 | eloprabga 7296 | . 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → (〈〈𝐴, 𝐵〉, 𝐶〉 ∈ {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝜑} ↔ 𝜓)) |
7 | 4, 6 | syl5bb 286 | 1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → (〈𝐴, 𝐵〉𝑅𝐶 ↔ 𝜓)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ w3a 1089 = wceq 1543 ∈ wcel 2112 〈cop 4533 class class class wbr 5039 {coprab 7192 |
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 2018 ax-8 2114 ax-9 2122 ax-12 2177 ax-ext 2708 ax-sep 5177 ax-nul 5184 ax-pr 5307 |
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 2073 df-clab 2715 df-cleq 2728 df-clel 2809 df-rab 3060 df-v 3400 df-dif 3856 df-un 3858 df-nul 4224 df-if 4426 df-sn 4528 df-pr 4530 df-op 4534 df-br 5040 df-oprab 7195 |
This theorem is referenced by: brcnvrabga 36163 |
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