Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > mpt2xopn0yelv | 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 |
---|---|
mpt2xopn0yelv.f | ⊢ 𝐹 = (𝑥 ∈ V, 𝑦 ∈ (1st ‘𝑥) ↦ 𝐶) |
Ref | Expression |
---|---|
mpt2xopn0yelv | ⊢ ((𝑉 ∈ 𝑋 ∧ 𝑊 ∈ 𝑌) → (𝑁 ∈ (〈𝑉, 𝑊〉𝐹𝐾) → 𝐾 ∈ 𝑉)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mpt2xopn0yelv.f | . . . . 5 ⊢ 𝐹 = (𝑥 ∈ V, 𝑦 ∈ (1st ‘𝑥) ↦ 𝐶) | |
2 | 1 | dmmpt2ssx 7124 | . . . 4 ⊢ dom 𝐹 ⊆ ∪ 𝑥 ∈ V ({𝑥} × (1st ‘𝑥)) |
3 | elfvdm 6130 | . . . . 5 ⊢ (𝑁 ∈ (𝐹‘〈〈𝑉, 𝑊〉, 𝐾〉) → 〈〈𝑉, 𝑊〉, 𝐾〉 ∈ dom 𝐹) | |
4 | df-ov 6552 | . . . . 5 ⊢ (〈𝑉, 𝑊〉𝐹𝐾) = (𝐹‘〈〈𝑉, 𝑊〉, 𝐾〉) | |
5 | 3, 4 | eleq2s 2706 | . . . 4 ⊢ (𝑁 ∈ (〈𝑉, 𝑊〉𝐹𝐾) → 〈〈𝑉, 𝑊〉, 𝐾〉 ∈ dom 𝐹) |
6 | 2, 5 | sseldi 3566 | . . 3 ⊢ (𝑁 ∈ (〈𝑉, 𝑊〉𝐹𝐾) → 〈〈𝑉, 𝑊〉, 𝐾〉 ∈ ∪ 𝑥 ∈ V ({𝑥} × (1st ‘𝑥))) |
7 | fveq2 6103 | . . . . 5 ⊢ (𝑥 = 〈𝑉, 𝑊〉 → (1st ‘𝑥) = (1st ‘〈𝑉, 𝑊〉)) | |
8 | 7 | opeliunxp2 5182 | . . . 4 ⊢ (〈〈𝑉, 𝑊〉, 𝐾〉 ∈ ∪ 𝑥 ∈ V ({𝑥} × (1st ‘𝑥)) ↔ (〈𝑉, 𝑊〉 ∈ V ∧ 𝐾 ∈ (1st ‘〈𝑉, 𝑊〉))) |
9 | 8 | simprbi 479 | . . 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 7071 | . . 3 ⊢ ((𝑉 ∈ 𝑋 ∧ 𝑊 ∈ 𝑌) → (1st ‘〈𝑉, 𝑊〉) = 𝑉) | |
12 | 11 | eleq2d 2673 | . 2 ⊢ ((𝑉 ∈ 𝑋 ∧ 𝑊 ∈ 𝑌) → (𝐾 ∈ (1st ‘〈𝑉, 𝑊〉) ↔ 𝐾 ∈ 𝑉)) |
13 | 10, 12 | syl5ib 233 | 1 ⊢ ((𝑉 ∈ 𝑋 ∧ 𝑊 ∈ 𝑌) → (𝑁 ∈ (〈𝑉, 𝑊〉𝐹𝐾) → 𝐾 ∈ 𝑉)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 = wceq 1475 ∈ wcel 1977 Vcvv 3173 {csn 4125 〈cop 4131 ∪ ciun 4455 × cxp 5036 dom cdm 5038 ‘cfv 5804 (class class class)co 6549 ↦ cmpt2 6551 1st c1st 7057 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-iota 5768 df-fun 5806 df-fv 5812 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-1st 7059 df-2nd 7060 |
This theorem is referenced by: mpt2xopynvov0g 7227 mpt2xopovel 7233 nbgrael 25955 |
Copyright terms: Public domain | W3C validator |