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Mirrors > Home > ILE Home > Th. List > ennnfonelem0 | GIF version |
Description: Lemma for ennnfone 12196. Initial value. (Contributed by Jim Kingdon, 15-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 |
---|---|
ennnfonelem0 | ⊢ (𝜑 → (𝐻‘0) = ∅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ennnfonelemh.h | . . . 4 ⊢ 𝐻 = seq0(𝐺, 𝐽) | |
2 | 1 | fveq1i 5470 | . . 3 ⊢ (𝐻‘0) = (seq0(𝐺, 𝐽)‘0) |
3 | ennnfonelemh.dceq | . . . . 5 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦) | |
4 | ennnfonelemh.f | . . . . 5 ⊢ (𝜑 → 𝐹:ω–onto→𝐴) | |
5 | ennnfonelemh.ne | . . . . 5 ⊢ (𝜑 → ∀𝑛 ∈ ω ∃𝑘 ∈ ω ∀𝑗 ∈ suc 𝑛(𝐹‘𝑘) ≠ (𝐹‘𝑗)) | |
6 | ennnfonelemh.g | . . . . 5 ⊢ 𝐺 = (𝑥 ∈ (𝐴 ↑pm ω), 𝑦 ∈ ω ↦ if((𝐹‘𝑦) ∈ (𝐹 “ 𝑦), 𝑥, (𝑥 ∪ {〈dom 𝑥, (𝐹‘𝑦)〉}))) | |
7 | ennnfonelemh.n | . . . . 5 ⊢ 𝑁 = frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 0) | |
8 | ennnfonelemh.j | . . . . 5 ⊢ 𝐽 = (𝑥 ∈ ℕ0 ↦ if(𝑥 = 0, ∅, (◡𝑁‘(𝑥 − 1)))) | |
9 | 3, 4, 5, 6, 7, 8, 1 | ennnfonelemj0 12172 | . . . 4 ⊢ (𝜑 → (𝐽‘0) ∈ {𝑔 ∈ (𝐴 ↑pm ω) ∣ dom 𝑔 ∈ ω}) |
10 | 3, 4, 5, 6, 7, 8, 1 | ennnfonelemg 12174 | . . . 4 ⊢ ((𝜑 ∧ (𝑓 ∈ {𝑔 ∈ (𝐴 ↑pm ω) ∣ dom 𝑔 ∈ ω} ∧ 𝑗 ∈ ω)) → (𝑓𝐺𝑗) ∈ {𝑔 ∈ (𝐴 ↑pm ω) ∣ dom 𝑔 ∈ ω}) |
11 | 0zd 9180 | . . . 4 ⊢ (𝜑 → 0 ∈ ℤ) | |
12 | 3, 4, 5, 6, 7, 8, 1 | ennnfonelemjn 12173 | . . . 4 ⊢ ((𝜑 ∧ 𝑓 ∈ (ℤ≥‘(0 + 1))) → (𝐽‘𝑓) ∈ ω) |
13 | 9, 10, 11, 12 | seq1cd 10368 | . . 3 ⊢ (𝜑 → (seq0(𝐺, 𝐽)‘0) = (𝐽‘0)) |
14 | 2, 13 | syl5eq 2202 | . 2 ⊢ (𝜑 → (𝐻‘0) = (𝐽‘0)) |
15 | 0nn0 9106 | . . . 4 ⊢ 0 ∈ ℕ0 | |
16 | eqid 2157 | . . . . . 6 ⊢ 0 = 0 | |
17 | 16 | iftruei 3511 | . . . . 5 ⊢ if(0 = 0, ∅, (◡𝑁‘(0 − 1))) = ∅ |
18 | 0ex 4092 | . . . . 5 ⊢ ∅ ∈ V | |
19 | 17, 18 | eqeltri 2230 | . . . 4 ⊢ if(0 = 0, ∅, (◡𝑁‘(0 − 1))) ∈ V |
20 | eqeq1 2164 | . . . . . 6 ⊢ (𝑥 = 0 → (𝑥 = 0 ↔ 0 = 0)) | |
21 | fvoveq1 5848 | . . . . . 6 ⊢ (𝑥 = 0 → (◡𝑁‘(𝑥 − 1)) = (◡𝑁‘(0 − 1))) | |
22 | 20, 21 | ifbieq2d 3529 | . . . . 5 ⊢ (𝑥 = 0 → if(𝑥 = 0, ∅, (◡𝑁‘(𝑥 − 1))) = if(0 = 0, ∅, (◡𝑁‘(0 − 1)))) |
23 | 22, 8 | fvmptg 5545 | . . . 4 ⊢ ((0 ∈ ℕ0 ∧ if(0 = 0, ∅, (◡𝑁‘(0 − 1))) ∈ V) → (𝐽‘0) = if(0 = 0, ∅, (◡𝑁‘(0 − 1)))) |
24 | 15, 19, 23 | mp2an 423 | . . 3 ⊢ (𝐽‘0) = if(0 = 0, ∅, (◡𝑁‘(0 − 1))) |
25 | 24, 17 | eqtri 2178 | . 2 ⊢ (𝐽‘0) = ∅ |
26 | 14, 25 | eqtrdi 2206 | 1 ⊢ (𝜑 → (𝐻‘0) = ∅) |
Colors of variables: wff set class |
Syntax hints: → wi 4 DECID wdc 820 = wceq 1335 ∈ wcel 2128 ≠ wne 2327 ∀wral 2435 ∃wrex 2436 {crab 2439 Vcvv 2712 ∪ cun 3100 ∅c0 3394 ifcif 3505 {csn 3560 〈cop 3563 ↦ cmpt 4026 suc csuc 4326 ωcom 4550 ◡ccnv 4586 dom cdm 4587 “ cima 4590 –onto→wfo 5169 ‘cfv 5171 (class class class)co 5825 ∈ cmpo 5827 freccfrec 6338 ↑pm cpm 6595 0cc0 7733 1c1 7734 + caddc 7736 − cmin 8047 ℕ0cn0 9091 ℤcz 9168 seqcseq 10348 |
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 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-10 1485 ax-11 1486 ax-i12 1487 ax-bndl 1489 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-13 2130 ax-14 2131 ax-ext 2139 ax-coll 4080 ax-sep 4083 ax-nul 4091 ax-pow 4136 ax-pr 4170 ax-un 4394 ax-setind 4497 ax-iinf 4548 ax-cnex 7824 ax-resscn 7825 ax-1cn 7826 ax-1re 7827 ax-icn 7828 ax-addcl 7829 ax-addrcl 7830 ax-mulcl 7831 ax-addcom 7833 ax-addass 7835 ax-distr 7837 ax-i2m1 7838 ax-0lt1 7839 ax-0id 7841 ax-rnegex 7842 ax-cnre 7844 ax-pre-ltirr 7845 ax-pre-ltwlin 7846 ax-pre-lttrn 7847 ax-pre-ltadd 7849 |
This theorem depends on definitions: df-bi 116 df-dc 821 df-3or 964 df-3an 965 df-tru 1338 df-fal 1341 df-nf 1441 df-sb 1743 df-eu 2009 df-mo 2010 df-clab 2144 df-cleq 2150 df-clel 2153 df-nfc 2288 df-ne 2328 df-nel 2423 df-ral 2440 df-rex 2441 df-reu 2442 df-rab 2444 df-v 2714 df-sbc 2938 df-csb 3032 df-dif 3104 df-un 3106 df-in 3108 df-ss 3115 df-nul 3395 df-if 3506 df-pw 3545 df-sn 3566 df-pr 3567 df-op 3569 df-uni 3774 df-int 3809 df-iun 3852 df-br 3967 df-opab 4027 df-mpt 4028 df-tr 4064 df-id 4254 df-iord 4327 df-on 4329 df-ilim 4330 df-suc 4332 df-iom 4551 df-xp 4593 df-rel 4594 df-cnv 4595 df-co 4596 df-dm 4597 df-rn 4598 df-res 4599 df-ima 4600 df-iota 5136 df-fun 5173 df-fn 5174 df-f 5175 df-f1 5176 df-fo 5177 df-f1o 5178 df-fv 5179 df-riota 5781 df-ov 5828 df-oprab 5829 df-mpo 5830 df-1st 6089 df-2nd 6090 df-recs 6253 df-frec 6339 df-pm 6597 df-pnf 7915 df-mnf 7916 df-xr 7917 df-ltxr 7918 df-le 7919 df-sub 8049 df-neg 8050 df-inn 8835 df-n0 9092 df-z 9169 df-uz 9441 df-seqfrec 10349 |
This theorem is referenced by: ennnfonelem1 12178 ennnfonelemkh 12183 ennnfonelemhf1o 12184 |
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