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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 7730 | . . 3 ⊢ ((Rel 𝑅 ∧ 𝐴 ∈ 𝑅) → 𝐴 = 〈(1st ‘𝐴), (2nd ‘𝐴)〉) | |
2 | simpr 487 | . . 3 ⊢ ((Rel 𝑅 ∧ 𝐴 ∈ 𝑅) → 𝐴 ∈ 𝑅) | |
3 | 1, 2 | eqeltrrd 2912 | . 2 ⊢ ((Rel 𝑅 ∧ 𝐴 ∈ 𝑅) → 〈(1st ‘𝐴), (2nd ‘𝐴)〉 ∈ 𝑅) |
4 | fvex 6676 | . . 3 ⊢ (1st ‘𝐴) ∈ V | |
5 | fvex 6676 | . . 3 ⊢ (2nd ‘𝐴) ∈ V | |
6 | 4, 5 | opelrn 5806 | . 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 398 ∈ wcel 2108 〈cop 4565 ran crn 5549 Rel wrel 5553 ‘cfv 6348 1st c1st 7679 2nd c2nd 7680 |
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 1905 ax-6 1964 ax-7 2009 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2154 ax-12 2170 ax-ext 2791 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7453 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1084 df-tru 1534 df-ex 1775 df-nf 1779 df-sb 2064 df-mo 2616 df-eu 2648 df-clab 2798 df-cleq 2812 df-clel 2891 df-nfc 2961 df-ral 3141 df-rex 3142 df-rab 3145 df-v 3495 df-sbc 3771 df-dif 3937 df-un 3939 df-in 3941 df-ss 3950 df-nul 4290 df-if 4466 df-sn 4560 df-pr 4562 df-op 4566 df-uni 4831 df-br 5058 df-opab 5120 df-mpt 5138 df-id 5453 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-iota 6307 df-fun 6350 df-fv 6356 df-1st 7681 df-2nd 7682 |
This theorem is referenced by: heicant 34919 mblfinlem1 34921 |
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