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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 8043 | . . 3 ⊢ ((Rel 𝑅 ∧ 𝐴 ∈ 𝑅) → 𝐴 = ⟨(1st ‘𝐴), (2nd ‘𝐴)⟩) | |
2 | simpr 484 | . . 3 ⊢ ((Rel 𝑅 ∧ 𝐴 ∈ 𝑅) → 𝐴 ∈ 𝑅) | |
3 | 1, 2 | eqeltrrd 2830 | . 2 ⊢ ((Rel 𝑅 ∧ 𝐴 ∈ 𝑅) → ⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ ∈ 𝑅) |
4 | fvex 6910 | . . 3 ⊢ (1st ‘𝐴) ∈ V | |
5 | fvex 6910 | . . 3 ⊢ (2nd ‘𝐴) ∈ V | |
6 | 4, 5 | opelrn 5945 | . 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 2099 ⟨cop 4635 ran crn 5679 Rel wrel 5683 ‘cfv 6548 1st c1st 7991 2nd c2nd 7992 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-sep 5299 ax-nul 5306 ax-pr 5429 ax-un 7740 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-ral 3059 df-rex 3068 df-rab 3430 df-v 3473 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4324 df-if 4530 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4909 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5576 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-iota 6500 df-fun 6550 df-fv 6556 df-1st 7993 df-2nd 7994 |
This theorem is referenced by: gsumhashmul 32783 heicant 37128 mblfinlem1 37130 |
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