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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 6199 | . . . 4 ⊢ dom 𝐹 ⊆ ∪ 𝑥 ∈ V ({𝑥} × (1st ‘𝑥)) |
3 | 1 | mpofun 5976 | . . . . . . 7 ⊢ Fun 𝐹 |
4 | funrel 5233 | . . . . . . 7 ⊢ (Fun 𝐹 → Rel 𝐹) | |
5 | 3, 4 | ax-mp 5 | . . . . . 6 ⊢ Rel 𝐹 |
6 | relelfvdm 5547 | . . . . . 6 ⊢ ((Rel 𝐹 ∧ 𝑁 ∈ (𝐹‘〈〈𝑉, 𝑊〉, 𝐾〉)) → 〈〈𝑉, 𝑊〉, 𝐾〉 ∈ dom 𝐹) | |
7 | 5, 6 | mpan 424 | . . . . 5 ⊢ (𝑁 ∈ (𝐹‘〈〈𝑉, 𝑊〉, 𝐾〉) → 〈〈𝑉, 𝑊〉, 𝐾〉 ∈ dom 𝐹) |
8 | df-ov 5877 | . . . . 5 ⊢ (〈𝑉, 𝑊〉𝐹𝐾) = (𝐹‘〈〈𝑉, 𝑊〉, 𝐾〉) | |
9 | 7, 8 | eleq2s 2272 | . . . 4 ⊢ (𝑁 ∈ (〈𝑉, 𝑊〉𝐹𝐾) → 〈〈𝑉, 𝑊〉, 𝐾〉 ∈ dom 𝐹) |
10 | 2, 9 | sselid 3153 | . . 3 ⊢ (𝑁 ∈ (〈𝑉, 𝑊〉𝐹𝐾) → 〈〈𝑉, 𝑊〉, 𝐾〉 ∈ ∪ 𝑥 ∈ V ({𝑥} × (1st ‘𝑥))) |
11 | fveq2 5515 | . . . . 5 ⊢ (𝑥 = 〈𝑉, 𝑊〉 → (1st ‘𝑥) = (1st ‘〈𝑉, 𝑊〉)) | |
12 | 11 | opeliunxp2 4767 | . . . 4 ⊢ (〈〈𝑉, 𝑊〉, 𝐾〉 ∈ ∪ 𝑥 ∈ V ({𝑥} × (1st ‘𝑥)) ↔ (〈𝑉, 𝑊〉 ∈ V ∧ 𝐾 ∈ (1st ‘〈𝑉, 𝑊〉))) |
13 | 12 | simprbi 275 | . . 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 6150 | . . 3 ⊢ ((𝑉 ∈ 𝑋 ∧ 𝑊 ∈ 𝑌) → (1st ‘〈𝑉, 𝑊〉) = 𝑉) | |
16 | 15 | eleq2d 2247 | . 2 ⊢ ((𝑉 ∈ 𝑋 ∧ 𝑊 ∈ 𝑌) → (𝐾 ∈ (1st ‘〈𝑉, 𝑊〉) ↔ 𝐾 ∈ 𝑉)) |
17 | 14, 16 | imbitrid 154 | 1 ⊢ ((𝑉 ∈ 𝑋 ∧ 𝑊 ∈ 𝑌) → (𝑁 ∈ (〈𝑉, 𝑊〉𝐹𝐾) → 𝐾 ∈ 𝑉)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 = wceq 1353 ∈ wcel 2148 Vcvv 2737 {csn 3592 〈cop 3595 ∪ ciun 3886 × cxp 4624 dom cdm 4626 Rel wrel 4631 Fun wfun 5210 ‘cfv 5216 (class class class)co 5874 ∈ cmpo 5876 1st c1st 6138 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-sep 4121 ax-pow 4174 ax-pr 4209 ax-un 4433 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ral 2460 df-rex 2461 df-rab 2464 df-v 2739 df-sbc 2963 df-csb 3058 df-un 3133 df-in 3135 df-ss 3142 df-pw 3577 df-sn 3598 df-pr 3599 df-op 3601 df-uni 3810 df-iun 3888 df-br 4004 df-opab 4065 df-mpt 4066 df-id 4293 df-xp 4632 df-rel 4633 df-cnv 4634 df-co 4635 df-dm 4636 df-rn 4637 df-res 4638 df-ima 4639 df-iota 5178 df-fun 5218 df-fv 5224 df-ov 5877 df-oprab 5878 df-mpo 5879 df-1st 6140 df-2nd 6141 |
This theorem is referenced by: mpoxopovel 6241 |
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