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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 5987 | . . . 4 ⊢ Rel ◡{〈〈𝑦, 𝑧〉, 𝑥〉 ∣ 𝜑} | |
2 | brcnvrabga.2 | . . . . 5 ⊢ 𝑅 = ◡{〈〈𝑦, 𝑧〉, 𝑥〉 ∣ 𝜑} | |
3 | 2 | releqi 5664 | . . . 4 ⊢ (Rel 𝑅 ↔ Rel ◡{〈〈𝑦, 𝑧〉, 𝑥〉 ∣ 𝜑}) |
4 | 1, 3 | mpbir 234 | . . 3 ⊢ Rel 𝑅 |
5 | 4 | relbrcnv 5990 | . 2 ⊢ (〈𝐵, 𝐶〉◡𝑅𝐴 ↔ 𝐴𝑅〈𝐵, 𝐶〉) |
6 | brrabga.1 | . . . . 5 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ∧ 𝑧 = 𝐶) → (𝜑 ↔ 𝜓)) | |
7 | 6 | 3coml 1129 | . . . 4 ⊢ ((𝑦 = 𝐵 ∧ 𝑧 = 𝐶 ∧ 𝑥 = 𝐴) → (𝜑 ↔ 𝜓)) |
8 | 2 | cnveqi 5758 | . . . . 5 ⊢ ◡𝑅 = ◡◡{〈〈𝑦, 𝑧〉, 𝑥〉 ∣ 𝜑} |
9 | reloprab 7289 | . . . . . 6 ⊢ Rel {〈〈𝑦, 𝑧〉, 𝑥〉 ∣ 𝜑} | |
10 | dfrel2 6067 | . . . . . 6 ⊢ (Rel {〈〈𝑦, 𝑧〉, 𝑥〉 ∣ 𝜑} ↔ ◡◡{〈〈𝑦, 𝑧〉, 𝑥〉 ∣ 𝜑} = {〈〈𝑦, 𝑧〉, 𝑥〉 ∣ 𝜑}) | |
11 | 9, 10 | mpbi 233 | . . . . 5 ⊢ ◡◡{〈〈𝑦, 𝑧〉, 𝑥〉 ∣ 𝜑} = {〈〈𝑦, 𝑧〉, 𝑥〉 ∣ 𝜑} |
12 | 8, 11 | eqtri 2766 | . . . 4 ⊢ ◡𝑅 = {〈〈𝑦, 𝑧〉, 𝑥〉 ∣ 𝜑} |
13 | 7, 12 | brrabga 36243 | . . 3 ⊢ ((𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋 ∧ 𝐴 ∈ 𝑉) → (〈𝐵, 𝐶〉◡𝑅𝐴 ↔ 𝜓)) |
14 | 13 | 3comr 1127 | . 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → (〈𝐵, 𝐶〉◡𝑅𝐴 ↔ 𝜓)) |
15 | 5, 14 | bitr3id 288 | 1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → (𝐴𝑅〈𝐵, 𝐶〉 ↔ 𝜓)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ w3a 1089 = wceq 1543 ∈ wcel 2111 〈cop 4562 class class class wbr 5068 ◡ccnv 5565 Rel wrel 5571 {coprab 7233 |
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 2113 ax-9 2121 ax-11 2159 ax-12 2176 ax-ext 2709 ax-sep 5207 ax-nul 5214 ax-pr 5337 |
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 2072 df-clab 2716 df-cleq 2730 df-clel 2817 df-ral 3067 df-rex 3068 df-rab 3071 df-v 3423 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4253 df-if 4455 df-sn 4557 df-pr 4559 df-op 4563 df-br 5069 df-opab 5131 df-xp 5572 df-rel 5573 df-cnv 5574 df-oprab 7236 |
This theorem is referenced by: brredunds 36506 |
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