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Mirrors > Home > MPE Home > Th. List > brab | Structured version Visualization version GIF version |
Description: The law of concretion for a binary relation. (Contributed by NM, 16-Aug-1999.) |
Ref | Expression |
---|---|
opelopab.1 | ⊢ 𝐴 ∈ V |
opelopab.2 | ⊢ 𝐵 ∈ V |
opelopab.3 | ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) |
opelopab.4 | ⊢ (𝑦 = 𝐵 → (𝜓 ↔ 𝜒)) |
brab.5 | ⊢ 𝑅 = {〈𝑥, 𝑦〉 ∣ 𝜑} |
Ref | Expression |
---|---|
brab | ⊢ (𝐴𝑅𝐵 ↔ 𝜒) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | opelopab.1 | . 2 ⊢ 𝐴 ∈ V | |
2 | opelopab.2 | . 2 ⊢ 𝐵 ∈ V | |
3 | opelopab.3 | . . 3 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) | |
4 | opelopab.4 | . . 3 ⊢ (𝑦 = 𝐵 → (𝜓 ↔ 𝜒)) | |
5 | brab.5 | . . 3 ⊢ 𝑅 = {〈𝑥, 𝑦〉 ∣ 𝜑} | |
6 | 3, 4, 5 | brabg 5494 | . 2 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐴𝑅𝐵 ↔ 𝜒)) |
7 | 1, 2, 6 | mp2an 690 | 1 ⊢ (𝐴𝑅𝐵 ↔ 𝜒) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 = wceq 1541 ∈ wcel 2106 Vcvv 3443 class class class wbr 5103 {copab 5165 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-ext 2708 ax-sep 5254 ax-nul 5261 ax-pr 5382 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-sb 2068 df-clab 2715 df-cleq 2729 df-clel 2815 df-rab 3406 df-v 3445 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-nul 4281 df-if 4485 df-sn 4585 df-pr 4587 df-op 4591 df-br 5104 df-opab 5166 |
This theorem is referenced by: opbrop 5727 f1oweALT 7897 frxp 8050 fnwelem 8055 xpord2lem 8066 xpord3lem 8073 poseq 8082 dftpos4 8168 dfac3 10015 axdc2lem 10342 brdom7disj 10425 brdom6disj 10426 ordpipq 10836 ltresr 11034 shftfn 14917 2shfti 14924 ishpg 27529 brcgr 27677 ex-opab 29204 br8d 31356 br8 34138 br6 34139 br4 34140 dfbigcup2 34415 brsegle 34624 heiborlem2 36202 |
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