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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 4584 | . . 3 ⊢ (𝐴𝑅𝐵 ↔ 〈𝐴, 𝐵〉 ∈ 𝑅) | |
2 | brabga.2 | . . . 4 ⊢ 𝑅 = {〈𝑥, 𝑦〉 ∣ 𝜑} | |
3 | 2 | eleq2i 2680 | . . 3 ⊢ (〈𝐴, 𝐵〉 ∈ 𝑅 ↔ 〈𝐴, 𝐵〉 ∈ {〈𝑥, 𝑦〉 ∣ 𝜑}) |
4 | 1, 3 | bitri 263 | . 2 ⊢ (𝐴𝑅𝐵 ↔ 〈𝐴, 𝐵〉 ∈ {〈𝑥, 𝑦〉 ∣ 𝜑}) |
5 | opelopabga.1 | . . 3 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (𝜑 ↔ 𝜓)) | |
6 | 5 | opelopabga 4913 | . 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (〈𝐴, 𝐵〉 ∈ {〈𝑥, 𝑦〉 ∣ 𝜑} ↔ 𝜓)) |
7 | 4, 6 | syl5bb 271 | 1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴𝑅𝐵 ↔ 𝜓)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∧ wa 383 = wceq 1475 ∈ wcel 1977 〈cop 4131 class class class wbr 4583 {copab 4642 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pr 4833 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-rab 2905 df-v 3175 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-br 4584 df-opab 4644 |
This theorem is referenced by: braba 4917 brabg 4919 epelg 4950 brcog 5210 fmptco 6303 ofrfval 6803 isfsupp 8162 wemaplem1 8334 oemapval 8463 wemapwe 8477 fpwwe2lem2 9333 fpwwelem 9346 clim 14073 rlim 14074 vdwmc 15520 isstruct2 15704 brssc 16297 isfunc 16347 isfull 16393 isfth 16397 ipole 16981 eqgval 17466 frgpuplem 18008 dvdsr 18469 islindf 19970 ulmval 23938 hpgbr 25452 isuhgra 25827 isushgra 25830 isumgra 25844 isuslgra 25872 isusgra 25873 isausgra 25883 iscusgra 25985 iswlkon 26062 istrlon 26071 ispthon 26106 isspthon 26113 isconngra 26200 isconngra1 26201 erclwwlkeq 26339 erclwwlkneq 26351 iseupa 26492 hlimi 27429 isinftm 29066 metidv 29263 ismntoplly 29397 brae 29631 braew 29632 brfae 29638 climf 38689 climf2 38733 isausgr 40394 issubgr 40495 isrgr 40759 isrusgr 40761 istrlson 40914 upgrwlkdvspth 40945 ispthson 40948 isspthson 40949 erclwwlkseq 41239 erclwwlksneq 41251 iscllaw 41615 iscomlaw 41616 isasslaw 41618 islininds 42029 lindepsnlininds 42035 |
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