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Mirrors > Home > ILE Home > Th. List > ennnfonelemj0 | GIF version |
Description: Lemma for ennnfone 12451. 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 9211 | . . . 4 ⊢ 0 ∈ ℕ0 | |
2 | eqid 2189 | . . . . . 6 ⊢ 0 = 0 | |
3 | 2 | iftruei 3555 | . . . . 5 ⊢ if(0 = 0, ∅, (◡𝑁‘(0 − 1))) = ∅ |
4 | 0ex 4145 | . . . . 5 ⊢ ∅ ∈ V | |
5 | 3, 4 | eqeltri 2262 | . . . 4 ⊢ if(0 = 0, ∅, (◡𝑁‘(0 − 1))) ∈ V |
6 | eqeq1 2196 | . . . . . 6 ⊢ (𝑥 = 0 → (𝑥 = 0 ↔ 0 = 0)) | |
7 | fvoveq1 5915 | . . . . . 6 ⊢ (𝑥 = 0 → (◡𝑁‘(𝑥 − 1)) = (◡𝑁‘(0 − 1))) | |
8 | 6, 7 | ifbieq2d 3573 | . . . . 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 5609 | . . . 4 ⊢ ((0 ∈ ℕ0 ∧ if(0 = 0, ∅, (◡𝑁‘(0 − 1))) ∈ V) → (𝐽‘0) = if(0 = 0, ∅, (◡𝑁‘(0 − 1)))) |
11 | 1, 5, 10 | mp2an 426 | . . 3 ⊢ (𝐽‘0) = if(0 = 0, ∅, (◡𝑁‘(0 − 1))) |
12 | 11, 3 | eqtri 2210 | . 2 ⊢ (𝐽‘0) = ∅ |
13 | dmeq 4842 | . . . 4 ⊢ (𝑔 = ∅ → dom 𝑔 = dom ∅) | |
14 | 13 | eleq1d 2258 | . . 3 ⊢ (𝑔 = ∅ → (dom 𝑔 ∈ ω ↔ dom ∅ ∈ ω)) |
15 | fun0 5290 | . . . . 5 ⊢ Fun ∅ | |
16 | 0ss 3476 | . . . . 5 ⊢ ∅ ⊆ (ω × 𝐴) | |
17 | 15, 16 | pm3.2i 272 | . . . 4 ⊢ (Fun ∅ ∧ ∅ ⊆ (ω × 𝐴)) |
18 | omex 4607 | . . . . . 6 ⊢ ω ∈ V | |
19 | ennnfonelemh.f | . . . . . 6 ⊢ (𝜑 → 𝐹:ω–onto→𝐴) | |
20 | focdmex 6135 | . . . . . 6 ⊢ (ω ∈ V → (𝐹:ω–onto→𝐴 → 𝐴 ∈ V)) | |
21 | 18, 19, 20 | mpsyl 65 | . . . . 5 ⊢ (𝜑 → 𝐴 ∈ V) |
22 | elpmg 6683 | . . . . 5 ⊢ ((𝐴 ∈ V ∧ ω ∈ V) → (∅ ∈ (𝐴 ↑pm ω) ↔ (Fun ∅ ∧ ∅ ⊆ (ω × 𝐴)))) | |
23 | 21, 18, 22 | sylancl 413 | . . . 4 ⊢ (𝜑 → (∅ ∈ (𝐴 ↑pm ω) ↔ (Fun ∅ ∧ ∅ ⊆ (ω × 𝐴)))) |
24 | 17, 23 | mpbiri 168 | . . 3 ⊢ (𝜑 → ∅ ∈ (𝐴 ↑pm ω)) |
25 | dm0 4856 | . . . . 5 ⊢ dom ∅ = ∅ | |
26 | peano1 4608 | . . . . 5 ⊢ ∅ ∈ ω | |
27 | 25, 26 | eqeltri 2262 | . . . 4 ⊢ dom ∅ ∈ ω |
28 | 27 | a1i 9 | . . 3 ⊢ (𝜑 → dom ∅ ∈ ω) |
29 | 14, 24, 28 | elrabd 2910 | . 2 ⊢ (𝜑 → ∅ ∈ {𝑔 ∈ (𝐴 ↑pm ω) ∣ dom 𝑔 ∈ ω}) |
30 | 12, 29 | eqeltrid 2276 | 1 ⊢ (𝜑 → (𝐽‘0) ∈ {𝑔 ∈ (𝐴 ↑pm ω) ∣ dom 𝑔 ∈ ω}) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 ↔ wb 105 DECID wdc 835 = wceq 1364 ∈ wcel 2160 ≠ wne 2360 ∀wral 2468 ∃wrex 2469 {crab 2472 Vcvv 2752 ∪ cun 3142 ⊆ wss 3144 ∅c0 3437 ifcif 3549 {csn 3607 ⟨cop 3610 ↦ cmpt 4079 suc csuc 4380 ωcom 4604 × cxp 4639 ◡ccnv 4640 dom cdm 4641 “ cima 4644 Fun wfun 5226 –onto→wfo 5230 ‘cfv 5232 (class class class)co 5892 ∈ cmpo 5894 freccfrec 6410 ↑pm cpm 6668 0cc0 7831 1c1 7832 + caddc 7834 − cmin 8148 ℕ0cn0 9196 ℤcz 9273 seqcseq 10465 |
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 615 ax-in2 616 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-13 2162 ax-14 2163 ax-ext 2171 ax-coll 4133 ax-sep 4136 ax-nul 4144 ax-pow 4189 ax-pr 4224 ax-un 4448 ax-setind 4551 ax-iinf 4602 ax-1cn 7924 ax-icn 7926 ax-addcl 7927 ax-mulcl 7929 ax-i2m1 7936 |
This theorem depends on definitions: df-bi 117 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1472 df-sb 1774 df-eu 2041 df-mo 2042 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ne 2361 df-ral 2473 df-rex 2474 df-reu 2475 df-rab 2477 df-v 2754 df-sbc 2978 df-csb 3073 df-dif 3146 df-un 3148 df-in 3150 df-ss 3157 df-nul 3438 df-if 3550 df-pw 3592 df-sn 3613 df-pr 3614 df-op 3616 df-uni 3825 df-int 3860 df-iun 3903 df-br 4019 df-opab 4080 df-mpt 4081 df-id 4308 df-iom 4605 df-xp 4647 df-rel 4648 df-cnv 4649 df-co 4650 df-dm 4651 df-rn 4652 df-res 4653 df-ima 4654 df-iota 5193 df-fun 5234 df-fn 5235 df-f 5236 df-f1 5237 df-fo 5238 df-f1o 5239 df-fv 5240 df-ov 5895 df-oprab 5896 df-mpo 5897 df-pm 6670 df-n0 9197 |
This theorem is referenced by: ennnfonelemh 12430 ennnfonelem0 12431 ennnfonelemp1 12432 ennnfonelemom 12434 |
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