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Mirrors > Home > MPE Home > Th. List > 1st2nd | Structured version Visualization version GIF version |
Description: Reconstruction of a member of a relation in terms of its ordered pair components. (Contributed by NM, 29-Aug-2006.) |
Ref | Expression |
---|---|
1st2nd | ⊢ ((Rel 𝐵 ∧ 𝐴 ∈ 𝐵) → 𝐴 = ⟨(1st ‘𝐴), (2nd ‘𝐴)⟩) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-rel 5641 | . . 3 ⊢ (Rel 𝐵 ↔ 𝐵 ⊆ (V × V)) | |
2 | ssel2 3940 | . . 3 ⊢ ((𝐵 ⊆ (V × V) ∧ 𝐴 ∈ 𝐵) → 𝐴 ∈ (V × V)) | |
3 | 1, 2 | sylanb 582 | . 2 ⊢ ((Rel 𝐵 ∧ 𝐴 ∈ 𝐵) → 𝐴 ∈ (V × V)) |
4 | 1st2nd2 7961 | . 2 ⊢ (𝐴 ∈ (V × V) → 𝐴 = ⟨(1st ‘𝐴), (2nd ‘𝐴)⟩) | |
5 | 3, 4 | syl 17 | 1 ⊢ ((Rel 𝐵 ∧ 𝐴 ∈ 𝐵) → 𝐴 = ⟨(1st ‘𝐴), (2nd ‘𝐴)⟩) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 397 = wceq 1542 ∈ wcel 2107 Vcvv 3446 ⊆ wss 3911 ⟨cop 4593 × cxp 5632 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: 2ndrn 7974 1st2ndbr 7975 funfv1st2nd 7979 funelss 7980 elopabi 7995 cnvf1olem 8043 ordpinq 10880 addassnq 10895 mulassnq 10896 distrnq 10898 mulidnq 10900 recmulnq 10901 ltexnq 10912 fsumcnv 15659 fprodcnv 15867 cofulid 17777 cofurid 17778 idffth 17821 cofull 17822 cofth 17823 ressffth 17826 isnat2 17836 nat1st2nd 17839 homadmcd 17929 catciso 17998 prf1st 18093 prf2nd 18094 1st2ndprf 18095 curfuncf 18128 uncfcurf 18129 curf2ndf 18137 yonffthlem 18172 yoniso 18175 dprd2dlem2 19820 dprd2dlem1 19821 dprd2da 19822 mdetunilem9 21972 2ndcctbss 22809 utop2nei 23605 utop3cls 23606 caubl 24675 wlkop 28579 nvop2 29553 nvvop 29554 nvop 29621 phop 29763 fgreu 31591 1stpreimas 31622 gsumhashmul 31901 cvmliftlem1 33882 heiborlem3 36275 rngoi 36361 drngoi 36413 isdrngo1 36418 iscrngo2 36459 |
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