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Mirrors > Home > ILE Home > Th. List > ennnfonelemj0 | GIF version |
Description: Lemma for ennnfone 12396. 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 9167 | . . . 4 ⊢ 0 ∈ ℕ0 | |
2 | eqid 2177 | . . . . . 6 ⊢ 0 = 0 | |
3 | 2 | iftruei 3540 | . . . . 5 ⊢ if(0 = 0, ∅, (◡𝑁‘(0 − 1))) = ∅ |
4 | 0ex 4127 | . . . . 5 ⊢ ∅ ∈ V | |
5 | 3, 4 | eqeltri 2250 | . . . 4 ⊢ if(0 = 0, ∅, (◡𝑁‘(0 − 1))) ∈ V |
6 | eqeq1 2184 | . . . . . 6 ⊢ (𝑥 = 0 → (𝑥 = 0 ↔ 0 = 0)) | |
7 | fvoveq1 5891 | . . . . . 6 ⊢ (𝑥 = 0 → (◡𝑁‘(𝑥 − 1)) = (◡𝑁‘(0 − 1))) | |
8 | 6, 7 | ifbieq2d 3558 | . . . . 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 5587 | . . . 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 2198 | . 2 ⊢ (𝐽‘0) = ∅ |
13 | dmeq 4822 | . . . 4 ⊢ (𝑔 = ∅ → dom 𝑔 = dom ∅) | |
14 | 13 | eleq1d 2246 | . . 3 ⊢ (𝑔 = ∅ → (dom 𝑔 ∈ ω ↔ dom ∅ ∈ ω)) |
15 | fun0 5269 | . . . . 5 ⊢ Fun ∅ | |
16 | 0ss 3461 | . . . . 5 ⊢ ∅ ⊆ (ω × 𝐴) | |
17 | 15, 16 | pm3.2i 272 | . . . 4 ⊢ (Fun ∅ ∧ ∅ ⊆ (ω × 𝐴)) |
18 | omex 4588 | . . . . . 6 ⊢ ω ∈ V | |
19 | ennnfonelemh.f | . . . . . 6 ⊢ (𝜑 → 𝐹:ω–onto→𝐴) | |
20 | focdmex 6109 | . . . . . 6 ⊢ (ω ∈ V → (𝐹:ω–onto→𝐴 → 𝐴 ∈ V)) | |
21 | 18, 19, 20 | mpsyl 65 | . . . . 5 ⊢ (𝜑 → 𝐴 ∈ V) |
22 | elpmg 6657 | . . . . 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 4836 | . . . . 5 ⊢ dom ∅ = ∅ | |
26 | peano1 4589 | . . . . 5 ⊢ ∅ ∈ ω | |
27 | 25, 26 | eqeltri 2250 | . . . 4 ⊢ dom ∅ ∈ ω |
28 | 27 | a1i 9 | . . 3 ⊢ (𝜑 → dom ∅ ∈ ω) |
29 | 14, 24, 28 | elrabd 2895 | . 2 ⊢ (𝜑 → ∅ ∈ {𝑔 ∈ (𝐴 ↑pm ω) ∣ dom 𝑔 ∈ ω}) |
30 | 12, 29 | eqeltrid 2264 | 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 834 = wceq 1353 ∈ wcel 2148 ≠ wne 2347 ∀wral 2455 ∃wrex 2456 {crab 2459 Vcvv 2737 ∪ cun 3127 ⊆ wss 3129 ∅c0 3422 ifcif 3534 {csn 3591 〈cop 3594 ↦ cmpt 4061 suc csuc 4361 ωcom 4585 × cxp 4620 ◡ccnv 4621 dom cdm 4622 “ cima 4625 Fun wfun 5205 –onto→wfo 5209 ‘cfv 5211 (class class class)co 5868 ∈ cmpo 5870 freccfrec 6384 ↑pm cpm 6642 0cc0 7789 1c1 7790 + caddc 7792 − cmin 8105 ℕ0cn0 9152 ℤcz 9229 seqcseq 10418 |
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 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-coll 4115 ax-sep 4118 ax-nul 4126 ax-pow 4171 ax-pr 4205 ax-un 4429 ax-setind 4532 ax-iinf 4583 ax-1cn 7882 ax-icn 7884 ax-addcl 7885 ax-mulcl 7887 ax-i2m1 7894 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ne 2348 df-ral 2460 df-rex 2461 df-reu 2462 df-rab 2464 df-v 2739 df-sbc 2963 df-csb 3058 df-dif 3131 df-un 3133 df-in 3135 df-ss 3142 df-nul 3423 df-if 3535 df-pw 3576 df-sn 3597 df-pr 3598 df-op 3600 df-uni 3808 df-int 3843 df-iun 3886 df-br 4001 df-opab 4062 df-mpt 4063 df-id 4289 df-iom 4586 df-xp 4628 df-rel 4629 df-cnv 4630 df-co 4631 df-dm 4632 df-rn 4633 df-res 4634 df-ima 4635 df-iota 5173 df-fun 5213 df-fn 5214 df-f 5215 df-f1 5216 df-fo 5217 df-f1o 5218 df-fv 5219 df-ov 5871 df-oprab 5872 df-mpo 5873 df-pm 6644 df-n0 9153 |
This theorem is referenced by: ennnfonelemh 12375 ennnfonelem0 12376 ennnfonelemp1 12377 ennnfonelemom 12379 |
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