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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 4874 | . . 3 ⊢ (𝐴𝑅𝐵 ↔ 〈𝐴, 𝐵〉 ∈ 𝑅) | |
2 | brabga.2 | . . . 4 ⊢ 𝑅 = {〈𝑥, 𝑦〉 ∣ 𝜑} | |
3 | 2 | eleq2i 2898 | . . 3 ⊢ (〈𝐴, 𝐵〉 ∈ 𝑅 ↔ 〈𝐴, 𝐵〉 ∈ {〈𝑥, 𝑦〉 ∣ 𝜑}) |
4 | 1, 3 | bitri 267 | . 2 ⊢ (𝐴𝑅𝐵 ↔ 〈𝐴, 𝐵〉 ∈ {〈𝑥, 𝑦〉 ∣ 𝜑}) |
5 | opelopabga.1 | . . 3 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (𝜑 ↔ 𝜓)) | |
6 | 5 | opelopabga 5214 | . 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (〈𝐴, 𝐵〉 ∈ {〈𝑥, 𝑦〉 ∣ 𝜑} ↔ 𝜓)) |
7 | 4, 6 | syl5bb 275 | 1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴𝑅𝐵 ↔ 𝜓)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 ∧ wa 386 = wceq 1658 ∈ wcel 2166 〈cop 4403 class class class wbr 4873 {copab 4935 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1896 ax-4 1910 ax-5 2011 ax-6 2077 ax-7 2114 ax-9 2175 ax-10 2194 ax-11 2209 ax-12 2222 ax-13 2391 ax-ext 2803 ax-sep 5005 ax-nul 5013 ax-pr 5127 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 881 df-3an 1115 df-tru 1662 df-ex 1881 df-nf 1885 df-sb 2070 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-rab 3126 df-v 3416 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-nul 4145 df-if 4307 df-sn 4398 df-pr 4400 df-op 4404 df-br 4874 df-opab 4936 |
This theorem is referenced by: braba 5218 brabg 5220 epelg 5256 brcog 5521 fmptco 6646 ofrfval 7165 isfsupp 8548 wemaplem1 8720 oemapval 8857 wemapwe 8871 fpwwe2lem2 9769 fpwwelem 9782 clim 14602 rlim 14603 vdwmc 16053 isstruct2 16232 brssc 16826 isfunc 16876 isfull 16922 isfth 16926 ipole 17511 eqgval 17994 frgpuplem 18538 dvdsr 19000 islindf 20518 ulmval 24533 hpgbr 26069 isausgr 26463 issubgr 26568 isrgr 26857 isrusgr 26859 istrlson 27009 upgrwlkdvspth 27041 ispthson 27044 isspthson 27045 erclwwlkeq 27356 erclwwlkneq 27420 hlimi 28600 isinftm 30280 metidv 30480 ismntoplly 30614 brae 30849 braew 30850 brfae 30856 brcoss 34734 brcoels 34738 climf 40649 climf2 40693 nelbr 42176 isomgr 42541 iscllaw 42672 iscomlaw 42673 isasslaw 42675 islininds 43082 lindepsnlininds 43088 |
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