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Mirrors > Home > ILE Home > Th. List > nninfwlpoimlemdc | GIF version |
Description: Lemma for nninfwlpoim 7178. (Contributed by Jim Kingdon, 8-Dec-2024.) |
Ref | Expression |
---|---|
nninfwlpoimlemg.f | ⊢ (𝜑 → 𝐹:ω⟶2o) |
nninfwlpoimlemg.g | ⊢ 𝐺 = (𝑖 ∈ ω ↦ if(∃𝑥 ∈ suc 𝑖(𝐹‘𝑥) = ∅, ∅, 1o)) |
nninfwlpoilemdc.eq | ⊢ (𝜑 → ∀𝑥 ∈ ℕ∞ ∀𝑦 ∈ ℕ∞ DECID 𝑥 = 𝑦) |
Ref | Expression |
---|---|
nninfwlpoimlemdc | ⊢ (𝜑 → DECID ∀𝑛 ∈ ω (𝐹‘𝑛) = 1o) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqeq2 2187 | . . . 4 ⊢ (𝑦 = (𝑖 ∈ ω ↦ 1o) → (𝐺 = 𝑦 ↔ 𝐺 = (𝑖 ∈ ω ↦ 1o))) | |
2 | 1 | dcbid 838 | . . 3 ⊢ (𝑦 = (𝑖 ∈ ω ↦ 1o) → (DECID 𝐺 = 𝑦 ↔ DECID 𝐺 = (𝑖 ∈ ω ↦ 1o))) |
3 | eqeq1 2184 | . . . . . 6 ⊢ (𝑥 = 𝐺 → (𝑥 = 𝑦 ↔ 𝐺 = 𝑦)) | |
4 | 3 | dcbid 838 | . . . . 5 ⊢ (𝑥 = 𝐺 → (DECID 𝑥 = 𝑦 ↔ DECID 𝐺 = 𝑦)) |
5 | 4 | ralbidv 2477 | . . . 4 ⊢ (𝑥 = 𝐺 → (∀𝑦 ∈ ℕ∞ DECID 𝑥 = 𝑦 ↔ ∀𝑦 ∈ ℕ∞ DECID 𝐺 = 𝑦)) |
6 | nninfwlpoilemdc.eq | . . . 4 ⊢ (𝜑 → ∀𝑥 ∈ ℕ∞ ∀𝑦 ∈ ℕ∞ DECID 𝑥 = 𝑦) | |
7 | nninfwlpoimlemg.f | . . . . 5 ⊢ (𝜑 → 𝐹:ω⟶2o) | |
8 | nninfwlpoimlemg.g | . . . . 5 ⊢ 𝐺 = (𝑖 ∈ ω ↦ if(∃𝑥 ∈ suc 𝑖(𝐹‘𝑥) = ∅, ∅, 1o)) | |
9 | 7, 8 | nninfwlpoimlemg 7175 | . . . 4 ⊢ (𝜑 → 𝐺 ∈ ℕ∞) |
10 | 5, 6, 9 | rspcdva 2848 | . . 3 ⊢ (𝜑 → ∀𝑦 ∈ ℕ∞ DECID 𝐺 = 𝑦) |
11 | infnninf 7124 | . . . 4 ⊢ (𝑖 ∈ ω ↦ 1o) ∈ ℕ∞ | |
12 | 11 | a1i 9 | . . 3 ⊢ (𝜑 → (𝑖 ∈ ω ↦ 1o) ∈ ℕ∞) |
13 | 2, 10, 12 | rspcdva 2848 | . 2 ⊢ (𝜑 → DECID 𝐺 = (𝑖 ∈ ω ↦ 1o)) |
14 | 7, 8 | nninfwlpoimlemginf 7176 | . . 3 ⊢ (𝜑 → (𝐺 = (𝑖 ∈ ω ↦ 1o) ↔ ∀𝑛 ∈ ω (𝐹‘𝑛) = 1o)) |
15 | 14 | dcbid 838 | . 2 ⊢ (𝜑 → (DECID 𝐺 = (𝑖 ∈ ω ↦ 1o) ↔ DECID ∀𝑛 ∈ ω (𝐹‘𝑛) = 1o)) |
16 | 13, 15 | mpbid 147 | 1 ⊢ (𝜑 → DECID ∀𝑛 ∈ ω (𝐹‘𝑛) = 1o) |
Colors of variables: wff set class |
Syntax hints: → wi 4 DECID wdc 834 = wceq 1353 ∈ wcel 2148 ∀wral 2455 ∃wrex 2456 ∅c0 3424 ifcif 3536 ↦ cmpt 4066 suc csuc 4367 ωcom 4591 ⟶wf 5214 ‘cfv 5218 1oc1o 6412 2oc2o 6413 ℕ∞xnninf 7120 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-coll 4120 ax-sep 4123 ax-nul 4131 ax-pow 4176 ax-pr 4211 ax-un 4435 ax-setind 4538 ax-iinf 4589 |
This theorem depends on definitions: df-bi 117 df-dc 835 df-3or 979 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ne 2348 df-ral 2460 df-rex 2461 df-reu 2462 df-rab 2464 df-v 2741 df-sbc 2965 df-csb 3060 df-dif 3133 df-un 3135 df-in 3137 df-ss 3144 df-nul 3425 df-if 3537 df-pw 3579 df-sn 3600 df-pr 3601 df-op 3603 df-uni 3812 df-int 3847 df-iun 3890 df-br 4006 df-opab 4067 df-mpt 4068 df-tr 4104 df-id 4295 df-iord 4368 df-on 4370 df-suc 4373 df-iom 4592 df-xp 4634 df-rel 4635 df-cnv 4636 df-co 4637 df-dm 4638 df-rn 4639 df-res 4640 df-ima 4641 df-iota 5180 df-fun 5220 df-fn 5221 df-f 5222 df-f1 5223 df-fo 5224 df-f1o 5225 df-fv 5226 df-ov 5880 df-oprab 5881 df-mpo 5882 df-1o 6419 df-2o 6420 df-er 6537 df-map 6652 df-en 6743 df-fin 6745 df-nninf 7121 |
This theorem is referenced by: nninfwlpoim 7178 |
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