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Mirrors > Home > MPE Home > Th. List > mpoxopn0yelv | Structured version Visualization version 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 7792 | . . . 4 ⊢ dom 𝐹 ⊆ ∪ 𝑥 ∈ V ({𝑥} × (1st ‘𝑥)) |
3 | elfvdm 6709 | . . . . 5 ⊢ (𝑁 ∈ (𝐹‘〈〈𝑉, 𝑊〉, 𝐾〉) → 〈〈𝑉, 𝑊〉, 𝐾〉 ∈ dom 𝐹) | |
4 | df-ov 7176 | . . . . 5 ⊢ (〈𝑉, 𝑊〉𝐹𝐾) = (𝐹‘〈〈𝑉, 𝑊〉, 𝐾〉) | |
5 | 3, 4 | eleq2s 2852 | . . . 4 ⊢ (𝑁 ∈ (〈𝑉, 𝑊〉𝐹𝐾) → 〈〈𝑉, 𝑊〉, 𝐾〉 ∈ dom 𝐹) |
6 | 2, 5 | sseldi 3876 | . . 3 ⊢ (𝑁 ∈ (〈𝑉, 𝑊〉𝐹𝐾) → 〈〈𝑉, 𝑊〉, 𝐾〉 ∈ ∪ 𝑥 ∈ V ({𝑥} × (1st ‘𝑥))) |
7 | fveq2 6677 | . . . . 5 ⊢ (𝑥 = 〈𝑉, 𝑊〉 → (1st ‘𝑥) = (1st ‘〈𝑉, 𝑊〉)) | |
8 | 7 | opeliunxp2 5682 | . . . 4 ⊢ (〈〈𝑉, 𝑊〉, 𝐾〉 ∈ ∪ 𝑥 ∈ V ({𝑥} × (1st ‘𝑥)) ↔ (〈𝑉, 𝑊〉 ∈ V ∧ 𝐾 ∈ (1st ‘〈𝑉, 𝑊〉))) |
9 | 8 | simprbi 500 | . . 3 ⊢ (〈〈𝑉, 𝑊〉, 𝐾〉 ∈ ∪ 𝑥 ∈ V ({𝑥} × (1st ‘𝑥)) → 𝐾 ∈ (1st ‘〈𝑉, 𝑊〉)) |
10 | 6, 9 | syl 17 | . 2 ⊢ (𝑁 ∈ (〈𝑉, 𝑊〉𝐹𝐾) → 𝐾 ∈ (1st ‘〈𝑉, 𝑊〉)) |
11 | op1stg 7729 | . . 3 ⊢ ((𝑉 ∈ 𝑋 ∧ 𝑊 ∈ 𝑌) → (1st ‘〈𝑉, 𝑊〉) = 𝑉) | |
12 | 11 | eleq2d 2819 | . 2 ⊢ ((𝑉 ∈ 𝑋 ∧ 𝑊 ∈ 𝑌) → (𝐾 ∈ (1st ‘〈𝑉, 𝑊〉) ↔ 𝐾 ∈ 𝑉)) |
13 | 10, 12 | syl5ib 247 | 1 ⊢ ((𝑉 ∈ 𝑋 ∧ 𝑊 ∈ 𝑌) → (𝑁 ∈ (〈𝑉, 𝑊〉𝐹𝐾) → 𝐾 ∈ 𝑉)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1542 ∈ wcel 2114 Vcvv 3399 {csn 4517 〈cop 4523 ∪ ciun 4882 × cxp 5524 dom cdm 5526 ‘cfv 6340 (class class class)co 7173 ∈ cmpo 7175 1st c1st 7715 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-ext 2711 ax-sep 5168 ax-nul 5175 ax-pr 5297 ax-un 7482 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2075 df-mo 2541 df-eu 2571 df-clab 2718 df-cleq 2731 df-clel 2812 df-nfc 2882 df-ne 2936 df-ral 3059 df-rex 3060 df-rab 3063 df-v 3401 df-sbc 3682 df-csb 3792 df-dif 3847 df-un 3849 df-in 3851 df-ss 3861 df-nul 4213 df-if 4416 df-sn 4518 df-pr 4520 df-op 4524 df-uni 4798 df-iun 4884 df-br 5032 df-opab 5094 df-mpt 5112 df-id 5430 df-xp 5532 df-rel 5533 df-cnv 5534 df-co 5535 df-dm 5536 df-rn 5537 df-res 5538 df-ima 5539 df-iota 6298 df-fun 6342 df-fv 6348 df-ov 7176 df-oprab 7177 df-mpo 7178 df-1st 7717 df-2nd 7718 |
This theorem is referenced by: mpoxopynvov0g 7912 mpoxopovel 7918 |
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