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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 5069 | . . 3 ⊢ (〈𝐴, 𝐵〉𝑅𝐶 ↔ 〈〈𝐴, 𝐵〉, 𝐶〉 ∈ 𝑅) | |
2 | brrabga.2 | . . . 4 ⊢ 𝑅 = {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝜑} | |
3 | 2 | eleq2i 2906 | . . 3 ⊢ (〈〈𝐴, 𝐵〉, 𝐶〉 ∈ 𝑅 ↔ 〈〈𝐴, 𝐵〉, 𝐶〉 ∈ {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝜑}) |
4 | 1, 3 | bitri 277 | . 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 7263 | . 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → (〈〈𝐴, 𝐵〉, 𝐶〉 ∈ {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ 𝜑} ↔ 𝜓)) |
7 | 4, 6 | syl5bb 285 | 1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → (〈𝐴, 𝐵〉𝑅𝐶 ↔ 𝜓)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ w3a 1083 = wceq 1537 ∈ wcel 2114 〈cop 4575 class class class wbr 5068 {coprab 7159 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pr 5332 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-rab 3149 df-v 3498 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-sn 4570 df-pr 4572 df-op 4576 df-br 5069 df-oprab 7162 |
This theorem is referenced by: brcnvrabga 35601 |
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