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Mirrors > Home > ILE Home > Th. List > ennnfonelemj0 | GIF version |
Description: Lemma for ennnfone 12295. Initial state for 𝐽. (Contributed by Jim Kingdon, 20-Jul-2023.) |
Ref | Expression |
---|---|
ennnfonelemh.dceq | ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦) |
ennnfonelemh.f | ⊢ (𝜑 → 𝐹:ω–onto→𝐴) |
ennnfonelemh.ne | ⊢ (𝜑 → ∀𝑛 ∈ ω ∃𝑘 ∈ ω ∀𝑗 ∈ suc 𝑛(𝐹‘𝑘) ≠ (𝐹‘𝑗)) |
ennnfonelemh.g | ⊢ 𝐺 = (𝑥 ∈ (𝐴 ↑pm ω), 𝑦 ∈ ω ↦ if((𝐹‘𝑦) ∈ (𝐹 “ 𝑦), 𝑥, (𝑥 ∪ {〈dom 𝑥, (𝐹‘𝑦)〉}))) |
ennnfonelemh.n | ⊢ 𝑁 = frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 0) |
ennnfonelemh.j | ⊢ 𝐽 = (𝑥 ∈ ℕ0 ↦ if(𝑥 = 0, ∅, (◡𝑁‘(𝑥 − 1)))) |
ennnfonelemh.h | ⊢ 𝐻 = seq0(𝐺, 𝐽) |
Ref | Expression |
---|---|
ennnfonelemj0 | ⊢ (𝜑 → (𝐽‘0) ∈ {𝑔 ∈ (𝐴 ↑pm ω) ∣ dom 𝑔 ∈ ω}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0nn0 9120 | . . . 4 ⊢ 0 ∈ ℕ0 | |
2 | eqid 2164 | . . . . . 6 ⊢ 0 = 0 | |
3 | 2 | iftruei 3521 | . . . . 5 ⊢ if(0 = 0, ∅, (◡𝑁‘(0 − 1))) = ∅ |
4 | 0ex 4103 | . . . . 5 ⊢ ∅ ∈ V | |
5 | 3, 4 | eqeltri 2237 | . . . 4 ⊢ if(0 = 0, ∅, (◡𝑁‘(0 − 1))) ∈ V |
6 | eqeq1 2171 | . . . . . 6 ⊢ (𝑥 = 0 → (𝑥 = 0 ↔ 0 = 0)) | |
7 | fvoveq1 5859 | . . . . . 6 ⊢ (𝑥 = 0 → (◡𝑁‘(𝑥 − 1)) = (◡𝑁‘(0 − 1))) | |
8 | 6, 7 | ifbieq2d 3539 | . . . . 5 ⊢ (𝑥 = 0 → if(𝑥 = 0, ∅, (◡𝑁‘(𝑥 − 1))) = if(0 = 0, ∅, (◡𝑁‘(0 − 1)))) |
9 | ennnfonelemh.j | . . . . 5 ⊢ 𝐽 = (𝑥 ∈ ℕ0 ↦ if(𝑥 = 0, ∅, (◡𝑁‘(𝑥 − 1)))) | |
10 | 8, 9 | fvmptg 5556 | . . . 4 ⊢ ((0 ∈ ℕ0 ∧ if(0 = 0, ∅, (◡𝑁‘(0 − 1))) ∈ V) → (𝐽‘0) = if(0 = 0, ∅, (◡𝑁‘(0 − 1)))) |
11 | 1, 5, 10 | mp2an 423 | . . 3 ⊢ (𝐽‘0) = if(0 = 0, ∅, (◡𝑁‘(0 − 1))) |
12 | 11, 3 | eqtri 2185 | . 2 ⊢ (𝐽‘0) = ∅ |
13 | dmeq 4798 | . . . 4 ⊢ (𝑔 = ∅ → dom 𝑔 = dom ∅) | |
14 | 13 | eleq1d 2233 | . . 3 ⊢ (𝑔 = ∅ → (dom 𝑔 ∈ ω ↔ dom ∅ ∈ ω)) |
15 | fun0 5240 | . . . . 5 ⊢ Fun ∅ | |
16 | 0ss 3442 | . . . . 5 ⊢ ∅ ⊆ (ω × 𝐴) | |
17 | 15, 16 | pm3.2i 270 | . . . 4 ⊢ (Fun ∅ ∧ ∅ ⊆ (ω × 𝐴)) |
18 | omex 4564 | . . . . . 6 ⊢ ω ∈ V | |
19 | ennnfonelemh.f | . . . . . 6 ⊢ (𝜑 → 𝐹:ω–onto→𝐴) | |
20 | focdmex 10689 | . . . . . 6 ⊢ ((ω ∈ V ∧ 𝐹:ω–onto→𝐴) → 𝐴 ∈ V) | |
21 | 18, 19, 20 | sylancr 411 | . . . . 5 ⊢ (𝜑 → 𝐴 ∈ V) |
22 | elpmg 6621 | . . . . 5 ⊢ ((𝐴 ∈ V ∧ ω ∈ V) → (∅ ∈ (𝐴 ↑pm ω) ↔ (Fun ∅ ∧ ∅ ⊆ (ω × 𝐴)))) | |
23 | 21, 18, 22 | sylancl 410 | . . . 4 ⊢ (𝜑 → (∅ ∈ (𝐴 ↑pm ω) ↔ (Fun ∅ ∧ ∅ ⊆ (ω × 𝐴)))) |
24 | 17, 23 | mpbiri 167 | . . 3 ⊢ (𝜑 → ∅ ∈ (𝐴 ↑pm ω)) |
25 | dm0 4812 | . . . . 5 ⊢ dom ∅ = ∅ | |
26 | peano1 4565 | . . . . 5 ⊢ ∅ ∈ ω | |
27 | 25, 26 | eqeltri 2237 | . . . 4 ⊢ dom ∅ ∈ ω |
28 | 27 | a1i 9 | . . 3 ⊢ (𝜑 → dom ∅ ∈ ω) |
29 | 14, 24, 28 | elrabd 2879 | . 2 ⊢ (𝜑 → ∅ ∈ {𝑔 ∈ (𝐴 ↑pm ω) ∣ dom 𝑔 ∈ ω}) |
30 | 12, 29 | eqeltrid 2251 | 1 ⊢ (𝜑 → (𝐽‘0) ∈ {𝑔 ∈ (𝐴 ↑pm ω) ∣ dom 𝑔 ∈ ω}) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ↔ wb 104 DECID wdc 824 = wceq 1342 ∈ wcel 2135 ≠ wne 2334 ∀wral 2442 ∃wrex 2443 {crab 2446 Vcvv 2721 ∪ cun 3109 ⊆ wss 3111 ∅c0 3404 ifcif 3515 {csn 3570 〈cop 3573 ↦ cmpt 4037 suc csuc 4337 ωcom 4561 × cxp 4596 ◡ccnv 4597 dom cdm 4598 “ cima 4601 Fun wfun 5176 –onto→wfo 5180 ‘cfv 5182 (class class class)co 5836 ∈ cmpo 5838 freccfrec 6349 ↑pm cpm 6606 0cc0 7744 1c1 7745 + caddc 7747 − cmin 8060 ℕ0cn0 9105 ℤcz 9182 seqcseq 10370 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 604 ax-in2 605 ax-io 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-13 2137 ax-14 2138 ax-ext 2146 ax-coll 4091 ax-sep 4094 ax-nul 4102 ax-pow 4147 ax-pr 4181 ax-un 4405 ax-setind 4508 ax-iinf 4559 ax-1cn 7837 ax-icn 7839 ax-addcl 7840 ax-mulcl 7842 ax-i2m1 7849 |
This theorem depends on definitions: df-bi 116 df-3an 969 df-tru 1345 df-fal 1348 df-nf 1448 df-sb 1750 df-eu 2016 df-mo 2017 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ne 2335 df-ral 2447 df-rex 2448 df-reu 2449 df-rab 2451 df-v 2723 df-sbc 2947 df-csb 3041 df-dif 3113 df-un 3115 df-in 3117 df-ss 3124 df-nul 3405 df-if 3516 df-pw 3555 df-sn 3576 df-pr 3577 df-op 3579 df-uni 3784 df-int 3819 df-iun 3862 df-br 3977 df-opab 4038 df-mpt 4039 df-id 4265 df-iom 4562 df-xp 4604 df-rel 4605 df-cnv 4606 df-co 4607 df-dm 4608 df-rn 4609 df-res 4610 df-ima 4611 df-iota 5147 df-fun 5184 df-fn 5185 df-f 5186 df-f1 5187 df-fo 5188 df-f1o 5189 df-fv 5190 df-ov 5839 df-oprab 5840 df-mpo 5841 df-pm 6608 df-n0 9106 |
This theorem is referenced by: ennnfonelemh 12274 ennnfonelem0 12275 ennnfonelemp1 12276 ennnfonelemom 12278 |
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