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Mirrors > Home > MPE Home > Th. List > mpoxopoveqd | Structured version Visualization version GIF version |
Description: 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, deduction version. (Contributed by Alexander van der Vekens, 11-Oct-2017.) |
Ref | Expression |
---|---|
mpoxopoveq.f | ⊢ 𝐹 = (𝑥 ∈ V, 𝑦 ∈ (1st ‘𝑥) ↦ {𝑛 ∈ (1st ‘𝑥) ∣ 𝜑}) |
mpoxopoveqd.1 | ⊢ (𝜓 → (𝑉 ∈ 𝑋 ∧ 𝑊 ∈ 𝑌)) |
mpoxopoveqd.2 | ⊢ ((𝜓 ∧ ¬ 𝐾 ∈ 𝑉) → {𝑛 ∈ 𝑉 ∣ [〈𝑉, 𝑊〉 / 𝑥][𝐾 / 𝑦]𝜑} = ∅) |
Ref | Expression |
---|---|
mpoxopoveqd | ⊢ (𝜓 → (〈𝑉, 𝑊〉𝐹𝐾) = {𝑛 ∈ 𝑉 ∣ [〈𝑉, 𝑊〉 / 𝑥][𝐾 / 𝑦]𝜑}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mpoxopoveq.f | . . . . 5 ⊢ 𝐹 = (𝑥 ∈ V, 𝑦 ∈ (1st ‘𝑥) ↦ {𝑛 ∈ (1st ‘𝑥) ∣ 𝜑}) | |
2 | 1 | mpoxopoveq 7915 | . . . 4 ⊢ (((𝑉 ∈ 𝑋 ∧ 𝑊 ∈ 𝑌) ∧ 𝐾 ∈ 𝑉) → (〈𝑉, 𝑊〉𝐹𝐾) = {𝑛 ∈ 𝑉 ∣ [〈𝑉, 𝑊〉 / 𝑥][𝐾 / 𝑦]𝜑}) |
3 | 2 | ex 416 | . . 3 ⊢ ((𝑉 ∈ 𝑋 ∧ 𝑊 ∈ 𝑌) → (𝐾 ∈ 𝑉 → (〈𝑉, 𝑊〉𝐹𝐾) = {𝑛 ∈ 𝑉 ∣ [〈𝑉, 𝑊〉 / 𝑥][𝐾 / 𝑦]𝜑})) |
4 | mpoxopoveqd.1 | . . 3 ⊢ (𝜓 → (𝑉 ∈ 𝑋 ∧ 𝑊 ∈ 𝑌)) | |
5 | 3, 4 | syl11 33 | . 2 ⊢ (𝐾 ∈ 𝑉 → (𝜓 → (〈𝑉, 𝑊〉𝐹𝐾) = {𝑛 ∈ 𝑉 ∣ [〈𝑉, 𝑊〉 / 𝑥][𝐾 / 𝑦]𝜑})) |
6 | df-nel 3039 | . . . . . 6 ⊢ (𝐾 ∉ 𝑉 ↔ ¬ 𝐾 ∈ 𝑉) | |
7 | 1 | mpoxopynvov0 7914 | . . . . . 6 ⊢ (𝐾 ∉ 𝑉 → (〈𝑉, 𝑊〉𝐹𝐾) = ∅) |
8 | 6, 7 | sylbir 238 | . . . . 5 ⊢ (¬ 𝐾 ∈ 𝑉 → (〈𝑉, 𝑊〉𝐹𝐾) = ∅) |
9 | 8 | adantr 484 | . . . 4 ⊢ ((¬ 𝐾 ∈ 𝑉 ∧ 𝜓) → (〈𝑉, 𝑊〉𝐹𝐾) = ∅) |
10 | mpoxopoveqd.2 | . . . . . 6 ⊢ ((𝜓 ∧ ¬ 𝐾 ∈ 𝑉) → {𝑛 ∈ 𝑉 ∣ [〈𝑉, 𝑊〉 / 𝑥][𝐾 / 𝑦]𝜑} = ∅) | |
11 | 10 | eqcomd 2744 | . . . . 5 ⊢ ((𝜓 ∧ ¬ 𝐾 ∈ 𝑉) → ∅ = {𝑛 ∈ 𝑉 ∣ [〈𝑉, 𝑊〉 / 𝑥][𝐾 / 𝑦]𝜑}) |
12 | 11 | ancoms 462 | . . . 4 ⊢ ((¬ 𝐾 ∈ 𝑉 ∧ 𝜓) → ∅ = {𝑛 ∈ 𝑉 ∣ [〈𝑉, 𝑊〉 / 𝑥][𝐾 / 𝑦]𝜑}) |
13 | 9, 12 | eqtrd 2773 | . . 3 ⊢ ((¬ 𝐾 ∈ 𝑉 ∧ 𝜓) → (〈𝑉, 𝑊〉𝐹𝐾) = {𝑛 ∈ 𝑉 ∣ [〈𝑉, 𝑊〉 / 𝑥][𝐾 / 𝑦]𝜑}) |
14 | 13 | ex 416 | . 2 ⊢ (¬ 𝐾 ∈ 𝑉 → (𝜓 → (〈𝑉, 𝑊〉𝐹𝐾) = {𝑛 ∈ 𝑉 ∣ [〈𝑉, 𝑊〉 / 𝑥][𝐾 / 𝑦]𝜑})) |
15 | 5, 14 | pm2.61i 185 | 1 ⊢ (𝜓 → (〈𝑉, 𝑊〉𝐹𝐾) = {𝑛 ∈ 𝑉 ∣ [〈𝑉, 𝑊〉 / 𝑥][𝐾 / 𝑦]𝜑}) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 399 = wceq 1542 ∈ wcel 2113 ∉ wnel 3038 {crab 3057 Vcvv 3398 [wsbc 3682 ∅c0 4212 〈cop 4523 ‘cfv 6340 (class class class)co 7171 ∈ cmpo 7173 1st c1st 7713 |
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 1916 ax-6 1974 ax-7 2019 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2161 ax-12 2178 ax-ext 2710 ax-sep 5168 ax-nul 5175 ax-pr 5297 ax-un 7480 |
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 2074 df-mo 2540 df-eu 2570 df-clab 2717 df-cleq 2730 df-clel 2811 df-nfc 2881 df-ne 2935 df-nel 3039 df-ral 3058 df-rex 3059 df-rab 3062 df-v 3400 df-sbc 3683 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 7174 df-oprab 7175 df-mpo 7176 df-1st 7715 df-2nd 7716 |
This theorem is referenced by: (None) |
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