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Mirrors > Home > ILE Home > Th. List > ennnfonelem0 | GIF version |
Description: Lemma for ennnfone 12380. 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 5497 | . . 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 12356 | . . . 4 ⊢ (𝜑 → (𝐽‘0) ∈ {𝑔 ∈ (𝐴 ↑pm ω) ∣ dom 𝑔 ∈ ω}) |
10 | 3, 4, 5, 6, 7, 8, 1 | ennnfonelemg 12358 | . . . 4 ⊢ ((𝜑 ∧ (𝑓 ∈ {𝑔 ∈ (𝐴 ↑pm ω) ∣ dom 𝑔 ∈ ω} ∧ 𝑗 ∈ ω)) → (𝑓𝐺𝑗) ∈ {𝑔 ∈ (𝐴 ↑pm ω) ∣ dom 𝑔 ∈ ω}) |
11 | 0zd 9224 | . . . 4 ⊢ (𝜑 → 0 ∈ ℤ) | |
12 | 3, 4, 5, 6, 7, 8, 1 | ennnfonelemjn 12357 | . . . 4 ⊢ ((𝜑 ∧ 𝑓 ∈ (ℤ≥‘(0 + 1))) → (𝐽‘𝑓) ∈ ω) |
13 | 9, 10, 11, 12 | seq1cd 10421 | . . 3 ⊢ (𝜑 → (seq0(𝐺, 𝐽)‘0) = (𝐽‘0)) |
14 | 2, 13 | eqtrid 2215 | . 2 ⊢ (𝜑 → (𝐻‘0) = (𝐽‘0)) |
15 | 0nn0 9150 | . . . 4 ⊢ 0 ∈ ℕ0 | |
16 | eqid 2170 | . . . . . 6 ⊢ 0 = 0 | |
17 | 16 | iftruei 3532 | . . . . 5 ⊢ if(0 = 0, ∅, (◡𝑁‘(0 − 1))) = ∅ |
18 | 0ex 4116 | . . . . 5 ⊢ ∅ ∈ V | |
19 | 17, 18 | eqeltri 2243 | . . . 4 ⊢ if(0 = 0, ∅, (◡𝑁‘(0 − 1))) ∈ V |
20 | eqeq1 2177 | . . . . . 6 ⊢ (𝑥 = 0 → (𝑥 = 0 ↔ 0 = 0)) | |
21 | fvoveq1 5876 | . . . . . 6 ⊢ (𝑥 = 0 → (◡𝑁‘(𝑥 − 1)) = (◡𝑁‘(0 − 1))) | |
22 | 20, 21 | ifbieq2d 3550 | . . . . 5 ⊢ (𝑥 = 0 → if(𝑥 = 0, ∅, (◡𝑁‘(𝑥 − 1))) = if(0 = 0, ∅, (◡𝑁‘(0 − 1)))) |
23 | 22, 8 | fvmptg 5572 | . . . 4 ⊢ ((0 ∈ ℕ0 ∧ if(0 = 0, ∅, (◡𝑁‘(0 − 1))) ∈ V) → (𝐽‘0) = if(0 = 0, ∅, (◡𝑁‘(0 − 1)))) |
24 | 15, 19, 23 | mp2an 424 | . . 3 ⊢ (𝐽‘0) = if(0 = 0, ∅, (◡𝑁‘(0 − 1))) |
25 | 24, 17 | eqtri 2191 | . 2 ⊢ (𝐽‘0) = ∅ |
26 | 14, 25 | eqtrdi 2219 | 1 ⊢ (𝜑 → (𝐻‘0) = ∅) |
Colors of variables: wff set class |
Syntax hints: → wi 4 DECID wdc 829 = wceq 1348 ∈ wcel 2141 ≠ wne 2340 ∀wral 2448 ∃wrex 2449 {crab 2452 Vcvv 2730 ∪ cun 3119 ∅c0 3414 ifcif 3526 {csn 3583 〈cop 3586 ↦ cmpt 4050 suc csuc 4350 ωcom 4574 ◡ccnv 4610 dom cdm 4611 “ cima 4614 –onto→wfo 5196 ‘cfv 5198 (class class class)co 5853 ∈ cmpo 5855 freccfrec 6369 ↑pm cpm 6627 0cc0 7774 1c1 7775 + caddc 7777 − cmin 8090 ℕ0cn0 9135 ℤcz 9212 seqcseq 10401 |
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 609 ax-in2 610 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-13 2143 ax-14 2144 ax-ext 2152 ax-coll 4104 ax-sep 4107 ax-nul 4115 ax-pow 4160 ax-pr 4194 ax-un 4418 ax-setind 4521 ax-iinf 4572 ax-cnex 7865 ax-resscn 7866 ax-1cn 7867 ax-1re 7868 ax-icn 7869 ax-addcl 7870 ax-addrcl 7871 ax-mulcl 7872 ax-addcom 7874 ax-addass 7876 ax-distr 7878 ax-i2m1 7879 ax-0lt1 7880 ax-0id 7882 ax-rnegex 7883 ax-cnre 7885 ax-pre-ltirr 7886 ax-pre-ltwlin 7887 ax-pre-lttrn 7888 ax-pre-ltadd 7890 |
This theorem depends on definitions: df-bi 116 df-dc 830 df-3or 974 df-3an 975 df-tru 1351 df-fal 1354 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ne 2341 df-nel 2436 df-ral 2453 df-rex 2454 df-reu 2455 df-rab 2457 df-v 2732 df-sbc 2956 df-csb 3050 df-dif 3123 df-un 3125 df-in 3127 df-ss 3134 df-nul 3415 df-if 3527 df-pw 3568 df-sn 3589 df-pr 3590 df-op 3592 df-uni 3797 df-int 3832 df-iun 3875 df-br 3990 df-opab 4051 df-mpt 4052 df-tr 4088 df-id 4278 df-iord 4351 df-on 4353 df-ilim 4354 df-suc 4356 df-iom 4575 df-xp 4617 df-rel 4618 df-cnv 4619 df-co 4620 df-dm 4621 df-rn 4622 df-res 4623 df-ima 4624 df-iota 5160 df-fun 5200 df-fn 5201 df-f 5202 df-f1 5203 df-fo 5204 df-f1o 5205 df-fv 5206 df-riota 5809 df-ov 5856 df-oprab 5857 df-mpo 5858 df-1st 6119 df-2nd 6120 df-recs 6284 df-frec 6370 df-pm 6629 df-pnf 7956 df-mnf 7957 df-xr 7958 df-ltxr 7959 df-le 7960 df-sub 8092 df-neg 8093 df-inn 8879 df-n0 9136 df-z 9213 df-uz 9488 df-seqfrec 10402 |
This theorem is referenced by: ennnfonelem1 12362 ennnfonelemkh 12367 ennnfonelemhf1o 12368 |
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