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Mirrors > Home > ILE Home > Th. List > ennnfonelemj0 | GIF version |
Description: Lemma for ennnfone 11783. 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 8896 | . . . 4 ⊢ 0 ∈ ℕ0 | |
2 | eqid 2115 | . . . . . 6 ⊢ 0 = 0 | |
3 | 2 | iftruei 3446 | . . . . 5 ⊢ if(0 = 0, ∅, (◡𝑁‘(0 − 1))) = ∅ |
4 | 0ex 4015 | . . . . 5 ⊢ ∅ ∈ V | |
5 | 3, 4 | eqeltri 2187 | . . . 4 ⊢ if(0 = 0, ∅, (◡𝑁‘(0 − 1))) ∈ V |
6 | eqeq1 2121 | . . . . . 6 ⊢ (𝑥 = 0 → (𝑥 = 0 ↔ 0 = 0)) | |
7 | fvoveq1 5751 | . . . . . 6 ⊢ (𝑥 = 0 → (◡𝑁‘(𝑥 − 1)) = (◡𝑁‘(0 − 1))) | |
8 | 6, 7 | ifbieq2d 3462 | . . . . 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 5451 | . . . 4 ⊢ ((0 ∈ ℕ0 ∧ if(0 = 0, ∅, (◡𝑁‘(0 − 1))) ∈ V) → (𝐽‘0) = if(0 = 0, ∅, (◡𝑁‘(0 − 1)))) |
11 | 1, 5, 10 | mp2an 420 | . . 3 ⊢ (𝐽‘0) = if(0 = 0, ∅, (◡𝑁‘(0 − 1))) |
12 | 11, 3 | eqtri 2135 | . 2 ⊢ (𝐽‘0) = ∅ |
13 | dmeq 4699 | . . . 4 ⊢ (𝑔 = ∅ → dom 𝑔 = dom ∅) | |
14 | 13 | eleq1d 2183 | . . 3 ⊢ (𝑔 = ∅ → (dom 𝑔 ∈ ω ↔ dom ∅ ∈ ω)) |
15 | fun0 5139 | . . . . 5 ⊢ Fun ∅ | |
16 | 0ss 3367 | . . . . 5 ⊢ ∅ ⊆ (ω × 𝐴) | |
17 | 15, 16 | pm3.2i 268 | . . . 4 ⊢ (Fun ∅ ∧ ∅ ⊆ (ω × 𝐴)) |
18 | omex 4467 | . . . . . 6 ⊢ ω ∈ V | |
19 | ennnfonelemh.f | . . . . . 6 ⊢ (𝜑 → 𝐹:ω–onto→𝐴) | |
20 | focdmex 10426 | . . . . . 6 ⊢ ((ω ∈ V ∧ 𝐹:ω–onto→𝐴) → 𝐴 ∈ V) | |
21 | 18, 19, 20 | sylancr 408 | . . . . 5 ⊢ (𝜑 → 𝐴 ∈ V) |
22 | elpmg 6512 | . . . . 5 ⊢ ((𝐴 ∈ V ∧ ω ∈ V) → (∅ ∈ (𝐴 ↑pm ω) ↔ (Fun ∅ ∧ ∅ ⊆ (ω × 𝐴)))) | |
23 | 21, 18, 22 | sylancl 407 | . . . 4 ⊢ (𝜑 → (∅ ∈ (𝐴 ↑pm ω) ↔ (Fun ∅ ∧ ∅ ⊆ (ω × 𝐴)))) |
24 | 17, 23 | mpbiri 167 | . . 3 ⊢ (𝜑 → ∅ ∈ (𝐴 ↑pm ω)) |
25 | dm0 4713 | . . . . 5 ⊢ dom ∅ = ∅ | |
26 | peano1 4468 | . . . . 5 ⊢ ∅ ∈ ω | |
27 | 25, 26 | eqeltri 2187 | . . . 4 ⊢ dom ∅ ∈ ω |
28 | 27 | a1i 9 | . . 3 ⊢ (𝜑 → dom ∅ ∈ ω) |
29 | 14, 24, 28 | elrabd 2811 | . 2 ⊢ (𝜑 → ∅ ∈ {𝑔 ∈ (𝐴 ↑pm ω) ∣ dom 𝑔 ∈ ω}) |
30 | 12, 29 | syl5eqel 2201 | 1 ⊢ (𝜑 → (𝐽‘0) ∈ {𝑔 ∈ (𝐴 ↑pm ω) ∣ dom 𝑔 ∈ ω}) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ↔ wb 104 DECID wdc 802 = wceq 1314 ∈ wcel 1463 ≠ wne 2282 ∀wral 2390 ∃wrex 2391 {crab 2394 Vcvv 2657 ∪ cun 3035 ⊆ wss 3037 ∅c0 3329 ifcif 3440 {csn 3493 〈cop 3496 ↦ cmpt 3949 suc csuc 4247 ωcom 4464 × cxp 4497 ◡ccnv 4498 dom cdm 4499 “ cima 4502 Fun wfun 5075 –onto→wfo 5079 ‘cfv 5081 (class class class)co 5728 ∈ cmpo 5730 freccfrec 6241 ↑pm cpm 6497 0cc0 7547 1c1 7548 + caddc 7550 − cmin 7856 ℕ0cn0 8881 ℤcz 8958 seqcseq 10111 |
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 1406 ax-7 1407 ax-gen 1408 ax-ie1 1452 ax-ie2 1453 ax-8 1465 ax-10 1466 ax-11 1467 ax-i12 1468 ax-bndl 1469 ax-4 1470 ax-13 1474 ax-14 1475 ax-17 1489 ax-i9 1493 ax-ial 1497 ax-i5r 1498 ax-ext 2097 ax-coll 4003 ax-sep 4006 ax-nul 4014 ax-pow 4058 ax-pr 4091 ax-un 4315 ax-setind 4412 ax-iinf 4462 ax-1cn 7638 ax-icn 7640 ax-addcl 7641 ax-mulcl 7643 ax-i2m1 7650 |
This theorem depends on definitions: df-bi 116 df-3an 947 df-tru 1317 df-fal 1320 df-nf 1420 df-sb 1719 df-eu 1978 df-mo 1979 df-clab 2102 df-cleq 2108 df-clel 2111 df-nfc 2244 df-ne 2283 df-ral 2395 df-rex 2396 df-reu 2397 df-rab 2399 df-v 2659 df-sbc 2879 df-csb 2972 df-dif 3039 df-un 3041 df-in 3043 df-ss 3050 df-nul 3330 df-if 3441 df-pw 3478 df-sn 3499 df-pr 3500 df-op 3502 df-uni 3703 df-int 3738 df-iun 3781 df-br 3896 df-opab 3950 df-mpt 3951 df-id 4175 df-iom 4465 df-xp 4505 df-rel 4506 df-cnv 4507 df-co 4508 df-dm 4509 df-rn 4510 df-res 4511 df-ima 4512 df-iota 5046 df-fun 5083 df-fn 5084 df-f 5085 df-f1 5086 df-fo 5087 df-f1o 5088 df-fv 5089 df-ov 5731 df-oprab 5732 df-mpo 5733 df-pm 6499 df-n0 8882 |
This theorem is referenced by: ennnfonelemh 11762 ennnfonelem0 11763 ennnfonelemp1 11764 ennnfonelemom 11766 |
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