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Mirrors > Home > MPE Home > Th. List > 2ndrn | Structured version Visualization version GIF version |
Description: The second ordered pair component of a member of a relation belongs to the range of the relation. (Contributed by NM, 17-Sep-2006.) |
Ref | Expression |
---|---|
2ndrn | ⊢ ((Rel 𝑅 ∧ 𝐴 ∈ 𝑅) → (2nd ‘𝐴) ∈ ran 𝑅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 1st2nd 8029 | . . 3 ⊢ ((Rel 𝑅 ∧ 𝐴 ∈ 𝑅) → 𝐴 = 〈(1st ‘𝐴), (2nd ‘𝐴)〉) | |
2 | simpr 484 | . . 3 ⊢ ((Rel 𝑅 ∧ 𝐴 ∈ 𝑅) → 𝐴 ∈ 𝑅) | |
3 | 1, 2 | eqeltrrd 2833 | . 2 ⊢ ((Rel 𝑅 ∧ 𝐴 ∈ 𝑅) → 〈(1st ‘𝐴), (2nd ‘𝐴)〉 ∈ 𝑅) |
4 | fvex 6904 | . . 3 ⊢ (1st ‘𝐴) ∈ V | |
5 | fvex 6904 | . . 3 ⊢ (2nd ‘𝐴) ∈ V | |
6 | 4, 5 | opelrn 5942 | . 2 ⊢ (〈(1st ‘𝐴), (2nd ‘𝐴)〉 ∈ 𝑅 → (2nd ‘𝐴) ∈ ran 𝑅) |
7 | 3, 6 | syl 17 | 1 ⊢ ((Rel 𝑅 ∧ 𝐴 ∈ 𝑅) → (2nd ‘𝐴) ∈ ran 𝑅) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 ∈ wcel 2105 〈cop 4634 ran crn 5677 Rel wrel 5681 ‘cfv 6543 1st c1st 7977 2nd c2nd 7978 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-sep 5299 ax-nul 5306 ax-pr 5427 ax-un 7729 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-ral 3061 df-rex 3070 df-rab 3432 df-v 3475 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5574 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-iota 6495 df-fun 6545 df-fv 6551 df-1st 7979 df-2nd 7980 |
This theorem is referenced by: gsumhashmul 32645 heicant 36989 mblfinlem1 36991 |
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