![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 5150 | . . 3 ⊢ (𝐴𝑅𝐵 ↔ 〈𝐴, 𝐵〉 ∈ 𝑅) | |
2 | brabga.2 | . . . 4 ⊢ 𝑅 = {〈𝑥, 𝑦〉 ∣ 𝜑} | |
3 | 2 | eleq2i 2817 | . . 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 5535 | . 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 394 = wceq 1533 ∈ wcel 2098 〈cop 4636 class class class wbr 5149 {copab 5211 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-ext 2696 ax-sep 5300 ax-nul 5307 ax-pr 5429 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-sb 2060 df-clab 2703 df-cleq 2717 df-clel 2802 df-rab 3419 df-v 3463 df-dif 3947 df-un 3949 df-ss 3961 df-nul 4323 df-if 4531 df-sn 4631 df-pr 4633 df-op 4637 df-br 5150 df-opab 5212 |
This theorem is referenced by: braba 5539 brabg 5541 epelg 5583 brcog 5869 fmptco 7138 ofrfvalg 7693 isfsupp 9391 wemaplem1 9571 oemapval 9708 wemapwe 9722 fpwwe2lem2 10657 fpwwelem 10670 clim 15474 rlim 15475 vdwmc 16950 isstruct2 17121 brssc 17800 isfunc 17853 isfull 17902 isfth 17906 ipole 18529 eqgval 19140 frgpuplem 19739 dvdsr 20313 islindf 21763 ulmval 26361 hpgbr 28636 isausgr 29049 issubgr 29156 isrgr 29445 isrusgr 29447 istrlson 29593 upgrwlkdvspth 29625 ispthson 29628 isspthson 29629 erclwwlkeq 29900 erclwwlkneq 29949 hlimi 31070 isinftm 32981 brfldext 33470 brfinext 33476 metidv 33624 ismntoplly 33757 brae 33991 braew 33992 brfae 33998 satfbrsuc 35107 prv 35169 bj-epelg 36678 bj-ideqgALT 36768 bj-idreseq 36772 bj-idreseqb 36773 bj-ideqg1ALT 36775 brcoss 38033 brcoels 38037 brdmqss 38248 aks6d1c1p1 41710 climf 45148 climf2 45192 nelbr 46792 iscllaw 47437 iscomlaw 47438 isasslaw 47440 islininds 47700 lindepsnlininds 47706 |
Copyright terms: Public domain | W3C validator |