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Mathbox for Peter Mazsa |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > brcnvrabga | Structured version Visualization version GIF version |
Description: The law of concretion for the converse of operation class abstraction. (Contributed by Peter Mazsa, 25-Oct-2022.) |
Ref | Expression |
---|---|
brrabga.1 | ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ∧ 𝑧 = 𝐶) → (𝜑 ↔ 𝜓)) |
brcnvrabga.2 | ⊢ 𝑅 = ◡{〈〈𝑦, 𝑧〉, 𝑥〉 ∣ 𝜑} |
Ref | Expression |
---|---|
brcnvrabga | ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → (𝐴𝑅〈𝐵, 𝐶〉 ↔ 𝜓)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | relcnv 6125 | . . . 4 ⊢ Rel ◡{〈〈𝑦, 𝑧〉, 𝑥〉 ∣ 𝜑} | |
2 | brcnvrabga.2 | . . . . 5 ⊢ 𝑅 = ◡{〈〈𝑦, 𝑧〉, 𝑥〉 ∣ 𝜑} | |
3 | 2 | releqi 5790 | . . . 4 ⊢ (Rel 𝑅 ↔ Rel ◡{〈〈𝑦, 𝑧〉, 𝑥〉 ∣ 𝜑}) |
4 | 1, 3 | mpbir 231 | . . 3 ⊢ Rel 𝑅 |
5 | 4 | relbrcnv 6128 | . 2 ⊢ (〈𝐵, 𝐶〉◡𝑅𝐴 ↔ 𝐴𝑅〈𝐵, 𝐶〉) |
6 | brrabga.1 | . . . . 5 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ∧ 𝑧 = 𝐶) → (𝜑 ↔ 𝜓)) | |
7 | 6 | 3coml 1126 | . . . 4 ⊢ ((𝑦 = 𝐵 ∧ 𝑧 = 𝐶 ∧ 𝑥 = 𝐴) → (𝜑 ↔ 𝜓)) |
8 | 2 | cnveqi 5888 | . . . . 5 ⊢ ◡𝑅 = ◡◡{〈〈𝑦, 𝑧〉, 𝑥〉 ∣ 𝜑} |
9 | reloprab 7492 | . . . . . 6 ⊢ Rel {〈〈𝑦, 𝑧〉, 𝑥〉 ∣ 𝜑} | |
10 | dfrel2 6211 | . . . . . 6 ⊢ (Rel {〈〈𝑦, 𝑧〉, 𝑥〉 ∣ 𝜑} ↔ ◡◡{〈〈𝑦, 𝑧〉, 𝑥〉 ∣ 𝜑} = {〈〈𝑦, 𝑧〉, 𝑥〉 ∣ 𝜑}) | |
11 | 9, 10 | mpbi 230 | . . . . 5 ⊢ ◡◡{〈〈𝑦, 𝑧〉, 𝑥〉 ∣ 𝜑} = {〈〈𝑦, 𝑧〉, 𝑥〉 ∣ 𝜑} |
12 | 8, 11 | eqtri 2763 | . . . 4 ⊢ ◡𝑅 = {〈〈𝑦, 𝑧〉, 𝑥〉 ∣ 𝜑} |
13 | 7, 12 | brrabga 38323 | . . 3 ⊢ ((𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋 ∧ 𝐴 ∈ 𝑉) → (〈𝐵, 𝐶〉◡𝑅𝐴 ↔ 𝜓)) |
14 | 13 | 3comr 1124 | . 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → (〈𝐵, 𝐶〉◡𝑅𝐴 ↔ 𝜓)) |
15 | 5, 14 | bitr3id 285 | 1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → (𝐴𝑅〈𝐵, 𝐶〉 ↔ 𝜓)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 ∧ w3a 1086 = wceq 1537 ∈ wcel 2106 〈cop 4637 class class class wbr 5148 ◡ccnv 5688 Rel wrel 5694 {coprab 7432 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-br 5149 df-opab 5211 df-xp 5695 df-rel 5696 df-cnv 5697 df-oprab 7435 |
This theorem is referenced by: brredunds 38608 |
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