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Mirrors > Home > MPE Home > Th. List > 1st2ndbr | Structured version Visualization version GIF version |
Description: Express an element of a relation as a relationship between first and second components. (Contributed by Mario Carneiro, 22-Jun-2016.) |
Ref | Expression |
---|---|
1st2ndbr | ⊢ ((Rel 𝐵 ∧ 𝐴 ∈ 𝐵) → (1st ‘𝐴)𝐵(2nd ‘𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 1st2nd 7972 | . . 3 ⊢ ((Rel 𝐵 ∧ 𝐴 ∈ 𝐵) → 𝐴 = ⟨(1st ‘𝐴), (2nd ‘𝐴)⟩) | |
2 | simpr 486 | . . 3 ⊢ ((Rel 𝐵 ∧ 𝐴 ∈ 𝐵) → 𝐴 ∈ 𝐵) | |
3 | 1, 2 | eqeltrrd 2839 | . 2 ⊢ ((Rel 𝐵 ∧ 𝐴 ∈ 𝐵) → ⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ ∈ 𝐵) |
4 | df-br 5107 | . 2 ⊢ ((1st ‘𝐴)𝐵(2nd ‘𝐴) ↔ ⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ ∈ 𝐵) | |
5 | 3, 4 | sylibr 233 | 1 ⊢ ((Rel 𝐵 ∧ 𝐴 ∈ 𝐵) → (1st ‘𝐴)𝐵(2nd ‘𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 397 ∈ wcel 2107 ⟨cop 4593 class class class wbr 5106 Rel wrel 5639 ‘cfv 6497 1st c1st 7920 2nd c2nd 7921 |
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 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2708 ax-sep 5257 ax-nul 5264 ax-pr 5385 ax-un 7673 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ral 3066 df-rex 3075 df-rab 3409 df-v 3448 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-nul 4284 df-if 4488 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-br 5107 df-opab 5169 df-mpt 5190 df-id 5532 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-iota 6449 df-fun 6499 df-fv 6505 df-1st 7922 df-2nd 7923 |
This theorem is referenced by: cofuval 17769 cofu1 17771 cofu2 17773 cofucl 17775 cofuass 17776 cofulid 17777 cofurid 17778 funcres 17783 cofull 17822 cofth 17823 isnat2 17836 fuccocl 17854 fucidcl 17855 fuclid 17856 fucrid 17857 fucass 17858 fucsect 17862 fucinv 17863 invfuc 17864 fuciso 17865 natpropd 17866 fucpropd 17867 homahom 17926 homadm 17927 homacd 17928 homadmcd 17929 catciso 17998 prfval 18088 prfcl 18092 prf1st 18093 prf2nd 18094 1st2ndprf 18095 evlfcllem 18111 evlfcl 18112 curf1cl 18118 curf2cl 18121 curfcl 18122 uncf1 18126 uncf2 18127 curfuncf 18128 uncfcurf 18129 diag1cl 18132 diag2cl 18136 curf2ndf 18137 yon1cl 18153 oyon1cl 18161 yonedalem1 18162 yonedalem21 18163 yonedalem3a 18164 yonedalem4c 18167 yonedalem22 18168 yonedalem3b 18169 yonedalem3 18170 yonedainv 18171 yonffthlem 18172 yoniso 18175 utop2nei 23605 utop3cls 23606 thincciso 47076 |
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