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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 6122 | . . . 4 ⊢ Rel ◡{〈〈𝑦, 𝑧〉, 𝑥〉 ∣ 𝜑} | |
| 2 | brcnvrabga.2 | . . . . 5 ⊢ 𝑅 = ◡{〈〈𝑦, 𝑧〉, 𝑥〉 ∣ 𝜑} | |
| 3 | 2 | releqi 5787 | . . . 4 ⊢ (Rel 𝑅 ↔ Rel ◡{〈〈𝑦, 𝑧〉, 𝑥〉 ∣ 𝜑}) | 
| 4 | 1, 3 | mpbir 231 | . . 3 ⊢ Rel 𝑅 | 
| 5 | 4 | relbrcnv 6125 | . 2 ⊢ (〈𝐵, 𝐶〉◡𝑅𝐴 ↔ 𝐴𝑅〈𝐵, 𝐶〉) | 
| 6 | brrabga.1 | . . . . 5 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ∧ 𝑧 = 𝐶) → (𝜑 ↔ 𝜓)) | |
| 7 | 6 | 3coml 1128 | . . . 4 ⊢ ((𝑦 = 𝐵 ∧ 𝑧 = 𝐶 ∧ 𝑥 = 𝐴) → (𝜑 ↔ 𝜓)) | 
| 8 | 2 | cnveqi 5885 | . . . . 5 ⊢ ◡𝑅 = ◡◡{〈〈𝑦, 𝑧〉, 𝑥〉 ∣ 𝜑} | 
| 9 | reloprab 7492 | . . . . . 6 ⊢ Rel {〈〈𝑦, 𝑧〉, 𝑥〉 ∣ 𝜑} | |
| 10 | dfrel2 6209 | . . . . . 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 2765 | . . . 4 ⊢ ◡𝑅 = {〈〈𝑦, 𝑧〉, 𝑥〉 ∣ 𝜑} | 
| 13 | 7, 12 | brrabga 38342 | . . 3 ⊢ ((𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋 ∧ 𝐴 ∈ 𝑉) → (〈𝐵, 𝐶〉◡𝑅𝐴 ↔ 𝜓)) | 
| 14 | 13 | 3comr 1126 | . 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 1087 = wceq 1540 ∈ wcel 2108 〈cop 4632 class class class wbr 5143 ◡ccnv 5684 Rel wrel 5690 {coprab 7432 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-br 5144 df-opab 5206 df-xp 5691 df-rel 5692 df-cnv 5693 df-oprab 7435 | 
| This theorem is referenced by: brredunds 38627 | 
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