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Mirrors > Home > ILE Home > Th. List > mpoxopn0yelv | GIF version |
Description: If there is an element of the value of an operation given by a maps-to rule, where the first argument is a pair and the base set of the second argument is the first component of the first argument, then the second argument is an element of the first component of the first argument. (Contributed by Alexander van der Vekens, 10-Oct-2017.) |
Ref | Expression |
---|---|
mpoxopn0yelv.f | ⊢ 𝐹 = (𝑥 ∈ V, 𝑦 ∈ (1st ‘𝑥) ↦ 𝐶) |
Ref | Expression |
---|---|
mpoxopn0yelv | ⊢ ((𝑉 ∈ 𝑋 ∧ 𝑊 ∈ 𝑌) → (𝑁 ∈ (〈𝑉, 𝑊〉𝐹𝐾) → 𝐾 ∈ 𝑉)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mpoxopn0yelv.f | . . . . 5 ⊢ 𝐹 = (𝑥 ∈ V, 𝑦 ∈ (1st ‘𝑥) ↦ 𝐶) | |
2 | 1 | dmmpossx 6159 | . . . 4 ⊢ dom 𝐹 ⊆ ∪ 𝑥 ∈ V ({𝑥} × (1st ‘𝑥)) |
3 | 1 | mpofun 5935 | . . . . . . 7 ⊢ Fun 𝐹 |
4 | funrel 5199 | . . . . . . 7 ⊢ (Fun 𝐹 → Rel 𝐹) | |
5 | 3, 4 | ax-mp 5 | . . . . . 6 ⊢ Rel 𝐹 |
6 | relelfvdm 5512 | . . . . . 6 ⊢ ((Rel 𝐹 ∧ 𝑁 ∈ (𝐹‘〈〈𝑉, 𝑊〉, 𝐾〉)) → 〈〈𝑉, 𝑊〉, 𝐾〉 ∈ dom 𝐹) | |
7 | 5, 6 | mpan 421 | . . . . 5 ⊢ (𝑁 ∈ (𝐹‘〈〈𝑉, 𝑊〉, 𝐾〉) → 〈〈𝑉, 𝑊〉, 𝐾〉 ∈ dom 𝐹) |
8 | df-ov 5839 | . . . . 5 ⊢ (〈𝑉, 𝑊〉𝐹𝐾) = (𝐹‘〈〈𝑉, 𝑊〉, 𝐾〉) | |
9 | 7, 8 | eleq2s 2259 | . . . 4 ⊢ (𝑁 ∈ (〈𝑉, 𝑊〉𝐹𝐾) → 〈〈𝑉, 𝑊〉, 𝐾〉 ∈ dom 𝐹) |
10 | 2, 9 | sseldi 3135 | . . 3 ⊢ (𝑁 ∈ (〈𝑉, 𝑊〉𝐹𝐾) → 〈〈𝑉, 𝑊〉, 𝐾〉 ∈ ∪ 𝑥 ∈ V ({𝑥} × (1st ‘𝑥))) |
11 | fveq2 5480 | . . . . 5 ⊢ (𝑥 = 〈𝑉, 𝑊〉 → (1st ‘𝑥) = (1st ‘〈𝑉, 𝑊〉)) | |
12 | 11 | opeliunxp2 4738 | . . . 4 ⊢ (〈〈𝑉, 𝑊〉, 𝐾〉 ∈ ∪ 𝑥 ∈ V ({𝑥} × (1st ‘𝑥)) ↔ (〈𝑉, 𝑊〉 ∈ V ∧ 𝐾 ∈ (1st ‘〈𝑉, 𝑊〉))) |
13 | 12 | simprbi 273 | . . 3 ⊢ (〈〈𝑉, 𝑊〉, 𝐾〉 ∈ ∪ 𝑥 ∈ V ({𝑥} × (1st ‘𝑥)) → 𝐾 ∈ (1st ‘〈𝑉, 𝑊〉)) |
14 | 10, 13 | syl 14 | . 2 ⊢ (𝑁 ∈ (〈𝑉, 𝑊〉𝐹𝐾) → 𝐾 ∈ (1st ‘〈𝑉, 𝑊〉)) |
15 | op1stg 6110 | . . 3 ⊢ ((𝑉 ∈ 𝑋 ∧ 𝑊 ∈ 𝑌) → (1st ‘〈𝑉, 𝑊〉) = 𝑉) | |
16 | 15 | eleq2d 2234 | . 2 ⊢ ((𝑉 ∈ 𝑋 ∧ 𝑊 ∈ 𝑌) → (𝐾 ∈ (1st ‘〈𝑉, 𝑊〉) ↔ 𝐾 ∈ 𝑉)) |
17 | 14, 16 | syl5ib 153 | 1 ⊢ ((𝑉 ∈ 𝑋 ∧ 𝑊 ∈ 𝑌) → (𝑁 ∈ (〈𝑉, 𝑊〉𝐹𝐾) → 𝐾 ∈ 𝑉)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 = wceq 1342 ∈ wcel 2135 Vcvv 2721 {csn 3570 〈cop 3573 ∪ ciun 3860 × cxp 4596 dom cdm 4598 Rel wrel 4603 Fun wfun 5176 ‘cfv 5182 (class class class)co 5836 ∈ cmpo 5838 1st c1st 6098 |
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-rab 2451 df-v 2723 df-sbc 2947 df-csb 3041 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-iun 3862 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-res 4610 df-ima 4611 df-iota 5147 df-fun 5184 df-fv 5190 df-ov 5839 df-oprab 5840 df-mpo 5841 df-1st 6100 df-2nd 6101 |
This theorem is referenced by: mpoxopovel 6200 |
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