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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 8039 | . . . 4 ⊢ dom 𝐹 ⊆ ∪ 𝑥 ∈ V ({𝑥} × (1st ‘𝑥)) |
3 | elfvdm 6918 | . . . . 5 ⊢ (𝑁 ∈ (𝐹‘〈〈𝑉, 𝑊〉, 𝐾〉) → 〈〈𝑉, 𝑊〉, 𝐾〉 ∈ dom 𝐹) | |
4 | df-ov 7399 | . . . . 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 | sselid 3978 | . . 3 ⊢ (𝑁 ∈ (〈𝑉, 𝑊〉𝐹𝐾) → 〈〈𝑉, 𝑊〉, 𝐾〉 ∈ ∪ 𝑥 ∈ V ({𝑥} × (1st ‘𝑥))) |
7 | fveq2 6881 | . . . . 5 ⊢ (𝑥 = 〈𝑉, 𝑊〉 → (1st ‘𝑥) = (1st ‘〈𝑉, 𝑊〉)) | |
8 | 7 | opeliunxp2 5833 | . . . 4 ⊢ (〈〈𝑉, 𝑊〉, 𝐾〉 ∈ ∪ 𝑥 ∈ V ({𝑥} × (1st ‘𝑥)) ↔ (〈𝑉, 𝑊〉 ∈ V ∧ 𝐾 ∈ (1st ‘〈𝑉, 𝑊〉))) |
9 | 8 | simprbi 498 | . . 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 7974 | . . 3 ⊢ ((𝑉 ∈ 𝑋 ∧ 𝑊 ∈ 𝑌) → (1st ‘〈𝑉, 𝑊〉) = 𝑉) | |
12 | 11 | eleq2d 2820 | . 2 ⊢ ((𝑉 ∈ 𝑋 ∧ 𝑊 ∈ 𝑌) → (𝐾 ∈ (1st ‘〈𝑉, 𝑊〉) ↔ 𝐾 ∈ 𝑉)) |
13 | 10, 12 | imbitrid 243 | 1 ⊢ ((𝑉 ∈ 𝑋 ∧ 𝑊 ∈ 𝑌) → (𝑁 ∈ (〈𝑉, 𝑊〉𝐹𝐾) → 𝐾 ∈ 𝑉)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 397 = wceq 1542 ∈ wcel 2107 Vcvv 3475 {csn 4624 〈cop 4630 ∪ ciun 4993 × cxp 5670 dom cdm 5672 ‘cfv 6535 (class class class)co 7396 ∈ cmpo 7398 1st c1st 7960 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5295 ax-nul 5302 ax-pr 5423 ax-un 7712 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ral 3063 df-rex 3072 df-rab 3434 df-v 3477 df-sbc 3776 df-csb 3892 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-nul 4321 df-if 4525 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4905 df-iun 4995 df-br 5145 df-opab 5207 df-mpt 5228 df-id 5570 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-iota 6487 df-fun 6537 df-fv 6543 df-ov 7399 df-oprab 7400 df-mpo 7401 df-1st 7962 df-2nd 7963 |
This theorem is referenced by: mpoxopynvov0g 8186 mpoxopovel 8192 |
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