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Mirrors > Home > MPE Home > Th. List > brabga | Structured version Visualization version GIF version |
Description: The law of concretion for a binary relation. (Contributed by Mario Carneiro, 19-Dec-2013.) |
Ref | Expression |
---|---|
opelopabga.1 | ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (𝜑 ↔ 𝜓)) |
brabga.2 | ⊢ 𝑅 = {〈𝑥, 𝑦〉 ∣ 𝜑} |
Ref | Expression |
---|---|
brabga | ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴𝑅𝐵 ↔ 𝜓)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-br 5075 | . . 3 ⊢ (𝐴𝑅𝐵 ↔ 〈𝐴, 𝐵〉 ∈ 𝑅) | |
2 | brabga.2 | . . . 4 ⊢ 𝑅 = {〈𝑥, 𝑦〉 ∣ 𝜑} | |
3 | 2 | eleq2i 2830 | . . 3 ⊢ (〈𝐴, 𝐵〉 ∈ 𝑅 ↔ 〈𝐴, 𝐵〉 ∈ {〈𝑥, 𝑦〉 ∣ 𝜑}) |
4 | 1, 3 | bitri 274 | . 2 ⊢ (𝐴𝑅𝐵 ↔ 〈𝐴, 𝐵〉 ∈ {〈𝑥, 𝑦〉 ∣ 𝜑}) |
5 | opelopabga.1 | . . 3 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (𝜑 ↔ 𝜓)) | |
6 | 5 | opelopabga 5446 | . 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (〈𝐴, 𝐵〉 ∈ {〈𝑥, 𝑦〉 ∣ 𝜑} ↔ 𝜓)) |
7 | 4, 6 | bitrid 282 | 1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴𝑅𝐵 ↔ 𝜓)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 = wceq 1539 ∈ wcel 2106 〈cop 4567 class class class wbr 5074 {copab 5136 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-ext 2709 ax-sep 5223 ax-nul 5230 ax-pr 5352 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-sb 2068 df-clab 2716 df-cleq 2730 df-clel 2816 df-rab 3073 df-v 3434 df-dif 3890 df-un 3892 df-nul 4257 df-if 4460 df-sn 4562 df-pr 4564 df-op 4568 df-br 5075 df-opab 5137 |
This theorem is referenced by: braba 5450 brabg 5452 epelg 5496 brcog 5775 fmptco 7001 ofrfvalg 7541 isfsupp 9132 wemaplem1 9305 oemapval 9441 wemapwe 9455 fpwwe2lem2 10388 fpwwelem 10401 clim 15203 rlim 15204 vdwmc 16679 isstruct2 16850 brssc 17526 isfunc 17579 isfull 17626 isfth 17630 ipole 18252 eqgval 18805 frgpuplem 19378 dvdsr 19888 islindf 21019 ulmval 25539 hpgbr 27121 isausgr 27534 issubgr 27638 isrgr 27926 isrusgr 27928 istrlson 28075 upgrwlkdvspth 28107 ispthson 28110 isspthson 28111 erclwwlkeq 28382 erclwwlkneq 28431 hlimi 29550 isinftm 31435 brfldext 31722 brfinext 31728 metidv 31842 ismntoplly 31975 brae 32209 braew 32210 brfae 32216 satfbrsuc 33328 prv 33390 bj-epelg 35239 bj-ideqgALT 35329 bj-idreseq 35333 bj-idreseqb 35334 bj-ideqg1ALT 35336 brcoss 36554 brcoels 36558 brdmqss 36759 climf 43163 climf2 43207 nelbr 44766 isomgr 45275 iscllaw 45383 iscomlaw 45384 isasslaw 45386 islininds 45787 lindepsnlininds 45793 |
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