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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 5144 | . . 3 ⊢ (𝐴𝑅𝐵 ↔ 〈𝐴, 𝐵〉 ∈ 𝑅) | |
| 2 | brabga.2 | . . . 4 ⊢ 𝑅 = {〈𝑥, 𝑦〉 ∣ 𝜑} | |
| 3 | 2 | eleq2i 2833 | . . 3 ⊢ (〈𝐴, 𝐵〉 ∈ 𝑅 ↔ 〈𝐴, 𝐵〉 ∈ {〈𝑥, 𝑦〉 ∣ 𝜑}) |
| 4 | 1, 3 | bitri 275 | . 2 ⊢ (𝐴𝑅𝐵 ↔ 〈𝐴, 𝐵〉 ∈ {〈𝑥, 𝑦〉 ∣ 𝜑}) |
| 5 | opelopabga.1 | . . 3 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (𝜑 ↔ 𝜓)) | |
| 6 | 5 | opelopabga 5538 | . 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (〈𝐴, 𝐵〉 ∈ {〈𝑥, 𝑦〉 ∣ 𝜑} ↔ 𝜓)) |
| 7 | 4, 6 | bitrid 283 | 1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴𝑅𝐵 ↔ 𝜓)) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1540 ∈ wcel 2108 〈cop 4632 class class class wbr 5143 {copab 5205 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-br 5144 df-opab 5206 |
| This theorem is referenced by: braba 5542 brabg 5544 epelg 5585 brcog 5877 fmptco 7149 ofrfvalg 7705 isfsupp 9405 wemaplem1 9586 oemapval 9723 wemapwe 9737 fpwwe2lem2 10672 fpwwelem 10685 clim 15530 rlim 15531 vdwmc 17016 isstruct2 17186 brssc 17858 isfunc 17909 isfull 17957 isfth 17961 ipole 18579 eqgval 19195 frgpuplem 19790 dvdsr 20362 islindf 21832 ulmval 26423 hpgbr 28768 isausgr 29181 issubgr 29288 isrgr 29577 isrusgr 29579 istrlson 29725 upgrwlkdvspth 29759 ispthson 29762 isspthson 29763 erclwwlkeq 30037 erclwwlkneq 30086 hlimi 31207 isinftm 33188 brfldext 33698 brfinext 33704 fldext2chn 33769 constrextdg2lem 33789 metidv 33891 ismntoplly 34026 brae 34242 braew 34243 brfae 34249 satfbrsuc 35371 prv 35433 bj-epelg 37069 bj-ideqgALT 37159 bj-idreseq 37163 bj-idreseqb 37164 bj-ideqg1ALT 37166 brcoss 38432 brcoels 38436 brdmqss 38647 aks6d1c1p1 42108 climf 45637 climf2 45681 nelbr 47286 iscllaw 48105 iscomlaw 48106 isasslaw 48108 islininds 48363 lindepsnlininds 48369 |
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