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Mirrors > Home > MPE Home > Th. List > brabg | Structured version Visualization version GIF version |
Description: The law of concretion for a binary relation. (Contributed by NM, 16-Aug-1999.) (Revised by Mario Carneiro, 19-Dec-2013.) |
Ref | Expression |
---|---|
opelopabg.1 | ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) |
opelopabg.2 | ⊢ (𝑦 = 𝐵 → (𝜓 ↔ 𝜒)) |
brabg.5 | ⊢ 𝑅 = {〈𝑥, 𝑦〉 ∣ 𝜑} |
Ref | Expression |
---|---|
brabg | ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → (𝐴𝑅𝐵 ↔ 𝜒)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | opelopabg.1 | . . 3 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) | |
2 | opelopabg.2 | . . 3 ⊢ (𝑦 = 𝐵 → (𝜓 ↔ 𝜒)) | |
3 | 1, 2 | sylan9bb 508 | . 2 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (𝜑 ↔ 𝜒)) |
4 | brabg.5 | . 2 ⊢ 𝑅 = {〈𝑥, 𝑦〉 ∣ 𝜑} | |
5 | 3, 4 | brabga 5540 | 1 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → (𝐴𝑅𝐵 ↔ 𝜒)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 394 = wceq 1534 ∈ wcel 2099 class class class wbr 5153 {copab 5215 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-ext 2697 ax-sep 5304 ax-nul 5311 ax-pr 5433 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-sb 2061 df-clab 2704 df-cleq 2718 df-clel 2803 df-rab 3420 df-v 3464 df-dif 3950 df-un 3952 df-ss 3964 df-nul 4326 df-if 4534 df-sn 4634 df-pr 4636 df-op 4640 df-br 5154 df-opab 5216 |
This theorem is referenced by: brab 5549 ideqg 5858 brcnvg 5886 f1owe 7365 brrpssg 7736 soseq 8173 breng 8983 brenOLD 8985 brdom2g 8986 brdomgOLD 8988 brwdom 9610 brttrcl 9756 ltprord 11073 shftfib 15077 efgrelexlema 19747 isref 23504 sltval 27677 brsslt 27815 lrrecval 27953 istrkgld 28386 islnopp 28666 axcontlem5 28902 cmbr 31517 leopg 32055 cvbr 32215 mdbr 32227 dmdbr 32232 isfne 36051 brabg2 37418 isriscg 37685 brssr 38199 lcvbr 38719 bropabg 42989 |
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