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Mirrors > Home > MPE Home > Th. List > brab2a | Structured version Visualization version GIF version |
Description: The law of concretion for a binary relation. Ordered pair membership in an ordered pair class abstraction. (Contributed by Mario Carneiro, 28-Apr-2015.) |
Ref | Expression |
---|---|
brab2a.1 | ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (𝜑 ↔ 𝜓)) |
brab2a.2 | ⊢ 𝑅 = {〈𝑥, 𝑦〉 ∣ ((𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷) ∧ 𝜑)} |
Ref | Expression |
---|---|
brab2a | ⊢ (𝐴𝑅𝐵 ↔ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) ∧ 𝜓)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | brab2a.2 | . . . 4 ⊢ 𝑅 = {〈𝑥, 𝑦〉 ∣ ((𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷) ∧ 𝜑)} | |
2 | opabssxp 5350 | . . . 4 ⊢ {〈𝑥, 𝑦〉 ∣ ((𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷) ∧ 𝜑)} ⊆ (𝐶 × 𝐷) | |
3 | 1, 2 | eqsstri 3776 | . . 3 ⊢ 𝑅 ⊆ (𝐶 × 𝐷) |
4 | 3 | brel 5325 | . 2 ⊢ (𝐴𝑅𝐵 → (𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷)) |
5 | df-br 4805 | . . . 4 ⊢ (𝐴𝑅𝐵 ↔ 〈𝐴, 𝐵〉 ∈ 𝑅) | |
6 | 1 | eleq2i 2831 | . . . 4 ⊢ (〈𝐴, 𝐵〉 ∈ 𝑅 ↔ 〈𝐴, 𝐵〉 ∈ {〈𝑥, 𝑦〉 ∣ ((𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷) ∧ 𝜑)}) |
7 | 5, 6 | bitri 264 | . . 3 ⊢ (𝐴𝑅𝐵 ↔ 〈𝐴, 𝐵〉 ∈ {〈𝑥, 𝑦〉 ∣ ((𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷) ∧ 𝜑)}) |
8 | brab2a.1 | . . . 4 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (𝜑 ↔ 𝜓)) | |
9 | 8 | opelopab2a 5140 | . . 3 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → (〈𝐴, 𝐵〉 ∈ {〈𝑥, 𝑦〉 ∣ ((𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷) ∧ 𝜑)} ↔ 𝜓)) |
10 | 7, 9 | syl5bb 272 | . 2 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → (𝐴𝑅𝐵 ↔ 𝜓)) |
11 | 4, 10 | biadan2 677 | 1 ⊢ (𝐴𝑅𝐵 ↔ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) ∧ 𝜓)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 ∧ wa 383 = wceq 1632 ∈ wcel 2139 〈cop 4327 class class class wbr 4804 {copab 4864 × cxp 5264 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1871 ax-4 1886 ax-5 1988 ax-6 2054 ax-7 2090 ax-9 2148 ax-10 2168 ax-11 2183 ax-12 2196 ax-13 2391 ax-ext 2740 ax-sep 4933 ax-nul 4941 ax-pr 5055 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3an 1074 df-tru 1635 df-ex 1854 df-nf 1859 df-sb 2047 df-eu 2611 df-mo 2612 df-clab 2747 df-cleq 2753 df-clel 2756 df-nfc 2891 df-ral 3055 df-rex 3056 df-rab 3059 df-v 3342 df-dif 3718 df-un 3720 df-in 3722 df-ss 3729 df-nul 4059 df-if 4231 df-sn 4322 df-pr 4324 df-op 4328 df-br 4805 df-opab 4865 df-xp 5272 |
This theorem is referenced by: fnse 7462 ltxrlt 10300 ltxr 12142 issect 16614 gaorb 17940 ispgp 18207 efgcpbllema 18367 lmbr 21264 isphtpc 22994 vitalilem1 23576 vitalilem2 23577 vitalilem3 23578 iscgrg 25606 ishlg 25696 iscgra 25900 isinag 25928 isleag 25932 filnetlem1 32679 |
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