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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 6141 | . . 3 ⊢ ((Rel 𝑅 ∧ 𝐴 ∈ 𝑅) → 𝐴 = 〈(1st ‘𝐴), (2nd ‘𝐴)〉) | |
3 | 2, 1 | eqeltrrd 2242 | . 2 ⊢ ((Rel 𝑅 ∧ 𝐴 ∈ 𝑅) → 〈(1st ‘𝐴), (2nd ‘𝐴)〉 ∈ 𝑅) |
4 | 1stexg 6127 | . . . 4 ⊢ (𝐴 ∈ 𝑅 → (1st ‘𝐴) ∈ V) | |
5 | 2ndexg 6128 | . . . 4 ⊢ (𝐴 ∈ 𝑅 → (2nd ‘𝐴) ∈ V) | |
6 | 4, 5 | jca 304 | . . 3 ⊢ (𝐴 ∈ 𝑅 → ((1st ‘𝐴) ∈ V ∧ (2nd ‘𝐴) ∈ V)) |
7 | opelrng 4830 | . . . 4 ⊢ (((1st ‘𝐴) ∈ V ∧ (2nd ‘𝐴) ∈ V ∧ 〈(1st ‘𝐴), (2nd ‘𝐴)〉 ∈ 𝑅) → (2nd ‘𝐴) ∈ ran 𝑅) | |
8 | 7 | 3expa 1192 | . . 3 ⊢ ((((1st ‘𝐴) ∈ V ∧ (2nd ‘𝐴) ∈ V) ∧ 〈(1st ‘𝐴), (2nd ‘𝐴)〉 ∈ 𝑅) → (2nd ‘𝐴) ∈ ran 𝑅) |
9 | 6, 8 | sylan 281 | . 2 ⊢ ((𝐴 ∈ 𝑅 ∧ 〈(1st ‘𝐴), (2nd ‘𝐴)〉 ∈ 𝑅) → (2nd ‘𝐴) ∈ ran 𝑅) |
10 | 1, 3, 9 | syl2anc 409 | 1 ⊢ ((Rel 𝑅 ∧ 𝐴 ∈ 𝑅) → (2nd ‘𝐴) ∈ ran 𝑅) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ∈ wcel 2135 Vcvv 2721 〈cop 3573 ran crn 4599 Rel wrel 4603 ‘cfv 5182 1st c1st 6098 2nd c2nd 6099 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-13 2137 ax-14 2138 ax-ext 2146 ax-sep 4094 ax-pow 4147 ax-pr 4181 ax-un 4405 |
This theorem depends on definitions: df-bi 116 df-3an 969 df-tru 1345 df-nf 1448 df-sb 1750 df-eu 2016 df-mo 2017 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ral 2447 df-rex 2448 df-v 2723 df-sbc 2947 df-un 3115 df-in 3117 df-ss 3124 df-pw 3555 df-sn 3576 df-pr 3577 df-op 3579 df-uni 3784 df-br 3977 df-opab 4038 df-mpt 4039 df-id 4265 df-xp 4604 df-rel 4605 df-cnv 4606 df-co 4607 df-dm 4608 df-rn 4609 df-iota 5147 df-fun 5184 df-fn 5185 df-f 5186 df-fo 5188 df-fv 5190 df-1st 6100 df-2nd 6101 |
This theorem is referenced by: (None) |
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