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Mirrors > Home > ILE Home > Th. List > ennnfonelemh | GIF version |
Description: Lemma for ennnfone 12471. (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 12447 | . . . 4 ⊢ (𝜑 → (𝐽‘0) ∈ {𝑔 ∈ (𝐴 ↑pm ω) ∣ dom 𝑔 ∈ ω}) |
9 | 1, 2, 3, 4, 5, 6, 7 | ennnfonelemg 12449 | . . . 4 ⊢ ((𝜑 ∧ (𝑓 ∈ {𝑔 ∈ (𝐴 ↑pm ω) ∣ dom 𝑔 ∈ ω} ∧ 𝑗 ∈ ω)) → (𝑓𝐺𝑗) ∈ {𝑔 ∈ (𝐴 ↑pm ω) ∣ dom 𝑔 ∈ ω}) |
10 | nn0uz 9587 | . . . 4 ⊢ ℕ0 = (ℤ≥‘0) | |
11 | 0zd 9290 | . . . 4 ⊢ (𝜑 → 0 ∈ ℤ) | |
12 | 1, 2, 3, 4, 5, 6, 7 | ennnfonelemjn 12448 | . . . 4 ⊢ ((𝜑 ∧ 𝑓 ∈ (ℤ≥‘(0 + 1))) → (𝐽‘𝑓) ∈ ω) |
13 | 8, 9, 10, 11, 12 | seqf2 10490 | . . 3 ⊢ (𝜑 → seq0(𝐺, 𝐽):ℕ0⟶{𝑔 ∈ (𝐴 ↑pm ω) ∣ dom 𝑔 ∈ ω}) |
14 | ssrab2 3255 | . . . 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 5394 | . 2 ⊢ (𝜑 → seq0(𝐺, 𝐽):ℕ0⟶(𝐴 ↑pm ω)) |
17 | 7 | feq1i 5374 | . 2 ⊢ (𝐻:ℕ0⟶(𝐴 ↑pm ω) ↔ seq0(𝐺, 𝐽):ℕ0⟶(𝐴 ↑pm ω)) |
18 | 16, 17 | sylibr 134 | 1 ⊢ (𝜑 → 𝐻:ℕ0⟶(𝐴 ↑pm ω)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 DECID wdc 835 = wceq 1364 ∈ wcel 2160 ≠ wne 2360 ∀wral 2468 ∃wrex 2469 {crab 2472 ∪ cun 3142 ⊆ wss 3144 ∅c0 3437 ifcif 3549 {csn 3607 〈cop 3610 ↦ cmpt 4079 suc csuc 4380 ωcom 4604 ◡ccnv 4640 dom cdm 4641 “ cima 4644 ⟶wf 5228 –onto→wfo 5230 ‘cfv 5232 (class class class)co 5892 ∈ cmpo 5894 freccfrec 6410 ↑pm cpm 6670 0cc0 7836 1c1 7837 + caddc 7839 − cmin 8153 ℕ0cn0 9201 ℤcz 9278 seqcseq 10471 |
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-cnex 7927 ax-resscn 7928 ax-1cn 7929 ax-1re 7930 ax-icn 7931 ax-addcl 7932 ax-addrcl 7933 ax-mulcl 7934 ax-addcom 7936 ax-addass 7938 ax-distr 7940 ax-i2m1 7941 ax-0lt1 7942 ax-0id 7944 ax-rnegex 7945 ax-cnre 7947 ax-pre-ltirr 7948 ax-pre-ltwlin 7949 ax-pre-lttrn 7950 ax-pre-ltadd 7952 |
This theorem depends on definitions: df-bi 117 df-dc 836 df-3or 981 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-nel 2456 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-tr 4117 df-id 4308 df-iord 4381 df-on 4383 df-ilim 4384 df-suc 4386 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-riota 5848 df-ov 5895 df-oprab 5896 df-mpo 5897 df-1st 6160 df-2nd 6161 df-recs 6325 df-frec 6411 df-pm 6672 df-pnf 8019 df-mnf 8020 df-xr 8021 df-ltxr 8022 df-le 8023 df-sub 8155 df-neg 8156 df-inn 8945 df-n0 9202 df-z 9279 df-uz 9554 df-seqfrec 10472 |
This theorem is referenced by: ennnfonelemp1 12452 ennnfonelemrnh 12462 ennnfonelemfun 12463 ennnfonelemf1 12464 |
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