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Mirrors > Home > MPE Home > Th. List > Mathboxes > finxpreclem1 | Structured version Visualization version GIF version |
Description: Lemma for ↑↑ recursion theorems. (Contributed by ML, 17-Oct-2020.) |
Ref | Expression |
---|---|
finxpreclem1 | ⊢ (𝑋 ∈ 𝑈 → ∅ = ((𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1o ∧ 𝑥 ∈ 𝑈), ∅, if(𝑥 ∈ (V × 𝑈), 〈∪ 𝑛, (1st ‘𝑥)〉, 〈𝑛, 𝑥〉)))‘〈1o, 𝑋〉)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-ov 7158 | . 2 ⊢ (1o(𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1o ∧ 𝑥 ∈ 𝑈), ∅, if(𝑥 ∈ (V × 𝑈), 〈∪ 𝑛, (1st ‘𝑥)〉, 〈𝑛, 𝑥〉)))𝑋) = ((𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1o ∧ 𝑥 ∈ 𝑈), ∅, if(𝑥 ∈ (V × 𝑈), 〈∪ 𝑛, (1st ‘𝑥)〉, 〈𝑛, 𝑥〉)))‘〈1o, 𝑋〉) | |
2 | eqidd 2822 | . . 3 ⊢ (𝑋 ∈ 𝑈 → (𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1o ∧ 𝑥 ∈ 𝑈), ∅, if(𝑥 ∈ (V × 𝑈), 〈∪ 𝑛, (1st ‘𝑥)〉, 〈𝑛, 𝑥〉))) = (𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1o ∧ 𝑥 ∈ 𝑈), ∅, if(𝑥 ∈ (V × 𝑈), 〈∪ 𝑛, (1st ‘𝑥)〉, 〈𝑛, 𝑥〉)))) | |
3 | eleq1a 2908 | . . . . . 6 ⊢ (𝑋 ∈ 𝑈 → (𝑥 = 𝑋 → 𝑥 ∈ 𝑈)) | |
4 | 3 | anim2d 613 | . . . . 5 ⊢ (𝑋 ∈ 𝑈 → ((𝑛 = 1o ∧ 𝑥 = 𝑋) → (𝑛 = 1o ∧ 𝑥 ∈ 𝑈))) |
5 | iftrue 4472 | . . . . 5 ⊢ ((𝑛 = 1o ∧ 𝑥 ∈ 𝑈) → if((𝑛 = 1o ∧ 𝑥 ∈ 𝑈), ∅, if(𝑥 ∈ (V × 𝑈), 〈∪ 𝑛, (1st ‘𝑥)〉, 〈𝑛, 𝑥〉)) = ∅) | |
6 | 4, 5 | syl6 35 | . . . 4 ⊢ (𝑋 ∈ 𝑈 → ((𝑛 = 1o ∧ 𝑥 = 𝑋) → if((𝑛 = 1o ∧ 𝑥 ∈ 𝑈), ∅, if(𝑥 ∈ (V × 𝑈), 〈∪ 𝑛, (1st ‘𝑥)〉, 〈𝑛, 𝑥〉)) = ∅)) |
7 | 6 | imp 409 | . . 3 ⊢ ((𝑋 ∈ 𝑈 ∧ (𝑛 = 1o ∧ 𝑥 = 𝑋)) → if((𝑛 = 1o ∧ 𝑥 ∈ 𝑈), ∅, if(𝑥 ∈ (V × 𝑈), 〈∪ 𝑛, (1st ‘𝑥)〉, 〈𝑛, 𝑥〉)) = ∅) |
8 | 1onn 8264 | . . . 4 ⊢ 1o ∈ ω | |
9 | 8 | a1i 11 | . . 3 ⊢ (𝑋 ∈ 𝑈 → 1o ∈ ω) |
10 | elex 3512 | . . 3 ⊢ (𝑋 ∈ 𝑈 → 𝑋 ∈ V) | |
11 | 0ex 5210 | . . . 4 ⊢ ∅ ∈ V | |
12 | 11 | a1i 11 | . . 3 ⊢ (𝑋 ∈ 𝑈 → ∅ ∈ V) |
13 | 2, 7, 9, 10, 12 | ovmpod 7301 | . 2 ⊢ (𝑋 ∈ 𝑈 → (1o(𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1o ∧ 𝑥 ∈ 𝑈), ∅, if(𝑥 ∈ (V × 𝑈), 〈∪ 𝑛, (1st ‘𝑥)〉, 〈𝑛, 𝑥〉)))𝑋) = ∅) |
14 | 1, 13 | syl5reqr 2871 | 1 ⊢ (𝑋 ∈ 𝑈 → ∅ = ((𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1o ∧ 𝑥 ∈ 𝑈), ∅, if(𝑥 ∈ (V × 𝑈), 〈∪ 𝑛, (1st ‘𝑥)〉, 〈𝑛, 𝑥〉)))‘〈1o, 𝑋〉)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1533 ∈ wcel 2110 Vcvv 3494 ∅c0 4290 ifcif 4466 〈cop 4572 ∪ cuni 4837 × cxp 5552 ‘cfv 6354 (class class class)co 7155 ∈ cmpo 7157 ωcom 7579 1st c1st 7686 1oc1o 8094 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5202 ax-nul 5209 ax-pr 5329 ax-un 7460 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3772 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4567 df-pr 4569 df-tp 4571 df-op 4573 df-uni 4838 df-br 5066 df-opab 5128 df-tr 5172 df-id 5459 df-eprel 5464 df-po 5473 df-so 5474 df-fr 5513 df-we 5515 df-xp 5560 df-rel 5561 df-cnv 5562 df-co 5563 df-dm 5564 df-ord 6193 df-on 6194 df-lim 6195 df-suc 6196 df-iota 6313 df-fun 6356 df-fv 6362 df-ov 7158 df-oprab 7159 df-mpo 7160 df-om 7580 df-1o 8101 |
This theorem is referenced by: finxp1o 34672 |
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