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Mirrors > Home > ILE Home > Th. List > 2ndrn | 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 | simpr 109 | . 2 ⊢ ((Rel 𝑅 ∧ 𝐴 ∈ 𝑅) → 𝐴 ∈ 𝑅) | |
2 | 1st2nd 6033 | . . 3 ⊢ ((Rel 𝑅 ∧ 𝐴 ∈ 𝑅) → 𝐴 = 〈(1st ‘𝐴), (2nd ‘𝐴)〉) | |
3 | 2, 1 | eqeltrrd 2192 | . 2 ⊢ ((Rel 𝑅 ∧ 𝐴 ∈ 𝑅) → 〈(1st ‘𝐴), (2nd ‘𝐴)〉 ∈ 𝑅) |
4 | 1stexg 6019 | . . . 4 ⊢ (𝐴 ∈ 𝑅 → (1st ‘𝐴) ∈ V) | |
5 | 2ndexg 6020 | . . . 4 ⊢ (𝐴 ∈ 𝑅 → (2nd ‘𝐴) ∈ V) | |
6 | 4, 5 | jca 302 | . . 3 ⊢ (𝐴 ∈ 𝑅 → ((1st ‘𝐴) ∈ V ∧ (2nd ‘𝐴) ∈ V)) |
7 | opelrng 4731 | . . . 4 ⊢ (((1st ‘𝐴) ∈ V ∧ (2nd ‘𝐴) ∈ V ∧ 〈(1st ‘𝐴), (2nd ‘𝐴)〉 ∈ 𝑅) → (2nd ‘𝐴) ∈ ran 𝑅) | |
8 | 7 | 3expa 1164 | . . 3 ⊢ ((((1st ‘𝐴) ∈ V ∧ (2nd ‘𝐴) ∈ V) ∧ 〈(1st ‘𝐴), (2nd ‘𝐴)〉 ∈ 𝑅) → (2nd ‘𝐴) ∈ ran 𝑅) |
9 | 6, 8 | sylan 279 | . 2 ⊢ ((𝐴 ∈ 𝑅 ∧ 〈(1st ‘𝐴), (2nd ‘𝐴)〉 ∈ 𝑅) → (2nd ‘𝐴) ∈ ran 𝑅) |
10 | 1, 3, 9 | syl2anc 406 | 1 ⊢ ((Rel 𝑅 ∧ 𝐴 ∈ 𝑅) → (2nd ‘𝐴) ∈ ran 𝑅) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ∈ wcel 1463 Vcvv 2657 〈cop 3496 ran crn 4500 Rel wrel 4504 ‘cfv 5081 1st c1st 5990 2nd c2nd 5991 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 681 ax-5 1406 ax-7 1407 ax-gen 1408 ax-ie1 1452 ax-ie2 1453 ax-8 1465 ax-10 1466 ax-11 1467 ax-i12 1468 ax-bndl 1469 ax-4 1470 ax-13 1474 ax-14 1475 ax-17 1489 ax-i9 1493 ax-ial 1497 ax-i5r 1498 ax-ext 2097 ax-sep 4006 ax-pow 4058 ax-pr 4091 ax-un 4315 |
This theorem depends on definitions: df-bi 116 df-3an 947 df-tru 1317 df-nf 1420 df-sb 1719 df-eu 1978 df-mo 1979 df-clab 2102 df-cleq 2108 df-clel 2111 df-nfc 2244 df-ral 2395 df-rex 2396 df-v 2659 df-sbc 2879 df-un 3041 df-in 3043 df-ss 3050 df-pw 3478 df-sn 3499 df-pr 3500 df-op 3502 df-uni 3703 df-br 3896 df-opab 3950 df-mpt 3951 df-id 4175 df-xp 4505 df-rel 4506 df-cnv 4507 df-co 4508 df-dm 4509 df-rn 4510 df-iota 5046 df-fun 5083 df-fn 5084 df-f 5085 df-fo 5087 df-fv 5089 df-1st 5992 df-2nd 5993 |
This theorem is referenced by: (None) |
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