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Mirrors > Home > ILE Home > Th. List > ennnfonelemh | GIF version |
Description: Lemma for ennnfone 11777. (Contributed by Jim Kingdon, 8-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 |
---|---|
ennnfonelemh | ⊢ (𝜑 → 𝐻:ℕ0⟶(𝐴 ↑pm ω)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ennnfonelemh.dceq | . . . . 5 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦) | |
2 | ennnfonelemh.f | . . . . 5 ⊢ (𝜑 → 𝐹:ω–onto→𝐴) | |
3 | ennnfonelemh.ne | . . . . 5 ⊢ (𝜑 → ∀𝑛 ∈ ω ∃𝑘 ∈ ω ∀𝑗 ∈ suc 𝑛(𝐹‘𝑘) ≠ (𝐹‘𝑗)) | |
4 | ennnfonelemh.g | . . . . 5 ⊢ 𝐺 = (𝑥 ∈ (𝐴 ↑pm ω), 𝑦 ∈ ω ↦ if((𝐹‘𝑦) ∈ (𝐹 “ 𝑦), 𝑥, (𝑥 ∪ {〈dom 𝑥, (𝐹‘𝑦)〉}))) | |
5 | ennnfonelemh.n | . . . . 5 ⊢ 𝑁 = frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 0) | |
6 | ennnfonelemh.j | . . . . 5 ⊢ 𝐽 = (𝑥 ∈ ℕ0 ↦ if(𝑥 = 0, ∅, (◡𝑁‘(𝑥 − 1)))) | |
7 | ennnfonelemh.h | . . . . 5 ⊢ 𝐻 = seq0(𝐺, 𝐽) | |
8 | 1, 2, 3, 4, 5, 6, 7 | ennnfonelemj0 11753 | . . . 4 ⊢ (𝜑 → (𝐽‘0) ∈ {𝑔 ∈ (𝐴 ↑pm ω) ∣ dom 𝑔 ∈ ω}) |
9 | 1, 2, 3, 4, 5, 6, 7 | ennnfonelemg 11755 | . . . 4 ⊢ ((𝜑 ∧ (𝑓 ∈ {𝑔 ∈ (𝐴 ↑pm ω) ∣ dom 𝑔 ∈ ω} ∧ 𝑗 ∈ ω)) → (𝑓𝐺𝑗) ∈ {𝑔 ∈ (𝐴 ↑pm ω) ∣ dom 𝑔 ∈ ω}) |
10 | nn0uz 9256 | . . . 4 ⊢ ℕ0 = (ℤ≥‘0) | |
11 | 0zd 8964 | . . . 4 ⊢ (𝜑 → 0 ∈ ℤ) | |
12 | 1, 2, 3, 4, 5, 6, 7 | ennnfonelemjn 11754 | . . . 4 ⊢ ((𝜑 ∧ 𝑓 ∈ (ℤ≥‘(0 + 1))) → (𝐽‘𝑓) ∈ ω) |
13 | 8, 9, 10, 11, 12 | seqf2 10124 | . . 3 ⊢ (𝜑 → seq0(𝐺, 𝐽):ℕ0⟶{𝑔 ∈ (𝐴 ↑pm ω) ∣ dom 𝑔 ∈ ω}) |
14 | ssrab2 3146 | . . . 4 ⊢ {𝑔 ∈ (𝐴 ↑pm ω) ∣ dom 𝑔 ∈ ω} ⊆ (𝐴 ↑pm ω) | |
15 | 14 | a1i 9 | . . 3 ⊢ (𝜑 → {𝑔 ∈ (𝐴 ↑pm ω) ∣ dom 𝑔 ∈ ω} ⊆ (𝐴 ↑pm ω)) |
16 | 13, 15 | fssd 5241 | . 2 ⊢ (𝜑 → seq0(𝐺, 𝐽):ℕ0⟶(𝐴 ↑pm ω)) |
17 | 7 | feq1i 5221 | . 2 ⊢ (𝐻:ℕ0⟶(𝐴 ↑pm ω) ↔ seq0(𝐺, 𝐽):ℕ0⟶(𝐴 ↑pm ω)) |
18 | 16, 17 | sylibr 133 | 1 ⊢ (𝜑 → 𝐻:ℕ0⟶(𝐴 ↑pm ω)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 DECID wdc 802 = wceq 1312 ∈ wcel 1461 ≠ wne 2280 ∀wral 2388 ∃wrex 2389 {crab 2392 ∪ cun 3033 ⊆ wss 3035 ∅c0 3327 ifcif 3438 {csn 3491 〈cop 3494 ↦ cmpt 3947 suc csuc 4245 ωcom 4462 ◡ccnv 4496 dom cdm 4497 “ cima 4500 ⟶wf 5075 –onto→wfo 5077 ‘cfv 5079 (class class class)co 5726 ∈ cmpo 5728 freccfrec 6239 ↑pm cpm 6495 0cc0 7541 1c1 7542 + caddc 7544 − cmin 7850 ℕ0cn0 8875 ℤcz 8952 seqcseq 10105 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 586 ax-in2 587 ax-io 681 ax-5 1404 ax-7 1405 ax-gen 1406 ax-ie1 1450 ax-ie2 1451 ax-8 1463 ax-10 1464 ax-11 1465 ax-i12 1466 ax-bndl 1467 ax-4 1468 ax-13 1472 ax-14 1473 ax-17 1487 ax-i9 1491 ax-ial 1495 ax-i5r 1496 ax-ext 2095 ax-coll 4001 ax-sep 4004 ax-nul 4012 ax-pow 4056 ax-pr 4089 ax-un 4313 ax-setind 4410 ax-iinf 4460 ax-cnex 7630 ax-resscn 7631 ax-1cn 7632 ax-1re 7633 ax-icn 7634 ax-addcl 7635 ax-addrcl 7636 ax-mulcl 7637 ax-addcom 7639 ax-addass 7641 ax-distr 7643 ax-i2m1 7644 ax-0lt1 7645 ax-0id 7647 ax-rnegex 7648 ax-cnre 7650 ax-pre-ltirr 7651 ax-pre-ltwlin 7652 ax-pre-lttrn 7653 ax-pre-ltadd 7655 |
This theorem depends on definitions: df-bi 116 df-dc 803 df-3or 944 df-3an 945 df-tru 1315 df-fal 1318 df-nf 1418 df-sb 1717 df-eu 1976 df-mo 1977 df-clab 2100 df-cleq 2106 df-clel 2109 df-nfc 2242 df-ne 2281 df-nel 2376 df-ral 2393 df-rex 2394 df-reu 2395 df-rab 2397 df-v 2657 df-sbc 2877 df-csb 2970 df-dif 3037 df-un 3039 df-in 3041 df-ss 3048 df-nul 3328 df-if 3439 df-pw 3476 df-sn 3497 df-pr 3498 df-op 3500 df-uni 3701 df-int 3736 df-iun 3779 df-br 3894 df-opab 3948 df-mpt 3949 df-tr 3985 df-id 4173 df-iord 4246 df-on 4248 df-ilim 4249 df-suc 4251 df-iom 4463 df-xp 4503 df-rel 4504 df-cnv 4505 df-co 4506 df-dm 4507 df-rn 4508 df-res 4509 df-ima 4510 df-iota 5044 df-fun 5081 df-fn 5082 df-f 5083 df-f1 5084 df-fo 5085 df-f1o 5086 df-fv 5087 df-riota 5682 df-ov 5729 df-oprab 5730 df-mpo 5731 df-1st 5990 df-2nd 5991 df-recs 6154 df-frec 6240 df-pm 6497 df-pnf 7720 df-mnf 7721 df-xr 7722 df-ltxr 7723 df-le 7724 df-sub 7852 df-neg 7853 df-inn 8625 df-n0 8876 df-z 8953 df-uz 9223 df-seqfrec 10106 |
This theorem is referenced by: ennnfonelemp1 11758 ennnfonelemrnh 11768 ennnfonelemfun 11769 ennnfonelemf1 11770 |
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