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Mirrors > Home > ILE Home > Th. List > ennnfonelemfun | GIF version |
Description: Lemma for ennnfone 12429. 𝐿 is a function. (Contributed by Jim Kingdon, 16-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) |
ennnfone.l | ⊢ 𝐿 = ∪ 𝑖 ∈ ℕ0 (𝐻‘𝑖) |
Ref | Expression |
---|---|
ennnfonelemfun | ⊢ (𝜑 → Fun 𝐿) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ennnfonelemh.dceq | . . . . . . . . 9 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦) | |
2 | ennnfonelemh.f | . . . . . . . . 9 ⊢ (𝜑 → 𝐹:ω–onto→𝐴) | |
3 | ennnfonelemh.ne | . . . . . . . . 9 ⊢ (𝜑 → ∀𝑛 ∈ ω ∃𝑘 ∈ ω ∀𝑗 ∈ suc 𝑛(𝐹‘𝑘) ≠ (𝐹‘𝑗)) | |
4 | ennnfonelemh.g | . . . . . . . . 9 ⊢ 𝐺 = (𝑥 ∈ (𝐴 ↑pm ω), 𝑦 ∈ ω ↦ if((𝐹‘𝑦) ∈ (𝐹 “ 𝑦), 𝑥, (𝑥 ∪ {⟨dom 𝑥, (𝐹‘𝑦)⟩}))) | |
5 | ennnfonelemh.n | . . . . . . . . 9 ⊢ 𝑁 = frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 0) | |
6 | ennnfonelemh.j | . . . . . . . . 9 ⊢ 𝐽 = (𝑥 ∈ ℕ0 ↦ if(𝑥 = 0, ∅, (◡𝑁‘(𝑥 − 1)))) | |
7 | ennnfonelemh.h | . . . . . . . . 9 ⊢ 𝐻 = seq0(𝐺, 𝐽) | |
8 | 1, 2, 3, 4, 5, 6, 7 | ennnfonelemh 12408 | . . . . . . . 8 ⊢ (𝜑 → 𝐻:ℕ0⟶(𝐴 ↑pm ω)) |
9 | 8 | frnd 5377 | . . . . . . 7 ⊢ (𝜑 → ran 𝐻 ⊆ (𝐴 ↑pm ω)) |
10 | 9 | sselda 3157 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑠 ∈ ran 𝐻) → 𝑠 ∈ (𝐴 ↑pm ω)) |
11 | pmfun 6671 | . . . . . 6 ⊢ (𝑠 ∈ (𝐴 ↑pm ω) → Fun 𝑠) | |
12 | 10, 11 | syl 14 | . . . . 5 ⊢ ((𝜑 ∧ 𝑠 ∈ ran 𝐻) → Fun 𝑠) |
13 | 1 | ad2antrr 488 | . . . . . . 7 ⊢ (((𝜑 ∧ 𝑠 ∈ ran 𝐻) ∧ 𝑡 ∈ ran 𝐻) → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦) |
14 | 2 | ad2antrr 488 | . . . . . . 7 ⊢ (((𝜑 ∧ 𝑠 ∈ ran 𝐻) ∧ 𝑡 ∈ ran 𝐻) → 𝐹:ω–onto→𝐴) |
15 | 3 | ad2antrr 488 | . . . . . . 7 ⊢ (((𝜑 ∧ 𝑠 ∈ ran 𝐻) ∧ 𝑡 ∈ ran 𝐻) → ∀𝑛 ∈ ω ∃𝑘 ∈ ω ∀𝑗 ∈ suc 𝑛(𝐹‘𝑘) ≠ (𝐹‘𝑗)) |
16 | simplr 528 | . . . . . . 7 ⊢ (((𝜑 ∧ 𝑠 ∈ ran 𝐻) ∧ 𝑡 ∈ ran 𝐻) → 𝑠 ∈ ran 𝐻) | |
17 | simpr 110 | . . . . . . 7 ⊢ (((𝜑 ∧ 𝑠 ∈ ran 𝐻) ∧ 𝑡 ∈ ran 𝐻) → 𝑡 ∈ ran 𝐻) | |
18 | 13, 14, 15, 4, 5, 6, 7, 16, 17 | ennnfonelemrnh 12420 | . . . . . 6 ⊢ (((𝜑 ∧ 𝑠 ∈ ran 𝐻) ∧ 𝑡 ∈ ran 𝐻) → (𝑠 ⊆ 𝑡 ∨ 𝑡 ⊆ 𝑠)) |
19 | 18 | ralrimiva 2550 | . . . . 5 ⊢ ((𝜑 ∧ 𝑠 ∈ ran 𝐻) → ∀𝑡 ∈ ran 𝐻(𝑠 ⊆ 𝑡 ∨ 𝑡 ⊆ 𝑠)) |
20 | 12, 19 | jca 306 | . . . 4 ⊢ ((𝜑 ∧ 𝑠 ∈ ran 𝐻) → (Fun 𝑠 ∧ ∀𝑡 ∈ ran 𝐻(𝑠 ⊆ 𝑡 ∨ 𝑡 ⊆ 𝑠))) |
21 | 20 | ralrimiva 2550 | . . 3 ⊢ (𝜑 → ∀𝑠 ∈ ran 𝐻(Fun 𝑠 ∧ ∀𝑡 ∈ ran 𝐻(𝑠 ⊆ 𝑡 ∨ 𝑡 ⊆ 𝑠))) |
22 | fununi 5286 | . . 3 ⊢ (∀𝑠 ∈ ran 𝐻(Fun 𝑠 ∧ ∀𝑡 ∈ ran 𝐻(𝑠 ⊆ 𝑡 ∨ 𝑡 ⊆ 𝑠)) → Fun ∪ ran 𝐻) | |
23 | 21, 22 | syl 14 | . 2 ⊢ (𝜑 → Fun ∪ ran 𝐻) |
24 | ennnfone.l | . . . 4 ⊢ 𝐿 = ∪ 𝑖 ∈ ℕ0 (𝐻‘𝑖) | |
25 | 8 | ffnd 5368 | . . . . 5 ⊢ (𝜑 → 𝐻 Fn ℕ0) |
26 | fniunfv 5766 | . . . . 5 ⊢ (𝐻 Fn ℕ0 → ∪ 𝑖 ∈ ℕ0 (𝐻‘𝑖) = ∪ ran 𝐻) | |
27 | 25, 26 | syl 14 | . . . 4 ⊢ (𝜑 → ∪ 𝑖 ∈ ℕ0 (𝐻‘𝑖) = ∪ ran 𝐻) |
28 | 24, 27 | eqtrid 2222 | . . 3 ⊢ (𝜑 → 𝐿 = ∪ ran 𝐻) |
29 | 28 | funeqd 5240 | . 2 ⊢ (𝜑 → (Fun 𝐿 ↔ Fun ∪ ran 𝐻)) |
30 | 23, 29 | mpbird 167 | 1 ⊢ (𝜑 → Fun 𝐿) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 ∨ wo 708 DECID wdc 834 = wceq 1353 ∈ wcel 2148 ≠ wne 2347 ∀wral 2455 ∃wrex 2456 ∪ cun 3129 ⊆ wss 3131 ∅c0 3424 ifcif 3536 {csn 3594 ⟨cop 3597 ∪ cuni 3811 ∪ ciun 3888 ↦ cmpt 4066 suc csuc 4367 ωcom 4591 ◡ccnv 4627 dom cdm 4628 ran crn 4629 “ cima 4631 Fun wfun 5212 Fn wfn 5213 –onto→wfo 5216 ‘cfv 5218 (class class class)co 5878 ∈ cmpo 5880 freccfrec 6394 ↑pm cpm 6652 0cc0 7814 1c1 7815 + caddc 7817 − cmin 8131 ℕ0cn0 9179 ℤcz 9256 seqcseq 10448 |
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 4120 ax-sep 4123 ax-nul 4131 ax-pow 4176 ax-pr 4211 ax-un 4435 ax-setind 4538 ax-iinf 4589 ax-cnex 7905 ax-resscn 7906 ax-1cn 7907 ax-1re 7908 ax-icn 7909 ax-addcl 7910 ax-addrcl 7911 ax-mulcl 7912 ax-addcom 7914 ax-addass 7916 ax-distr 7918 ax-i2m1 7919 ax-0lt1 7920 ax-0id 7922 ax-rnegex 7923 ax-cnre 7925 ax-pre-ltirr 7926 ax-pre-ltwlin 7927 ax-pre-lttrn 7928 ax-pre-ltadd 7930 |
This theorem depends on definitions: df-bi 117 df-dc 835 df-3or 979 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-nel 2443 df-ral 2460 df-rex 2461 df-reu 2462 df-rab 2464 df-v 2741 df-sbc 2965 df-csb 3060 df-dif 3133 df-un 3135 df-in 3137 df-ss 3144 df-nul 3425 df-if 3537 df-pw 3579 df-sn 3600 df-pr 3601 df-op 3603 df-uni 3812 df-int 3847 df-iun 3890 df-br 4006 df-opab 4067 df-mpt 4068 df-tr 4104 df-id 4295 df-iord 4368 df-on 4370 df-ilim 4371 df-suc 4373 df-iom 4592 df-xp 4634 df-rel 4635 df-cnv 4636 df-co 4637 df-dm 4638 df-rn 4639 df-res 4640 df-ima 4641 df-iota 5180 df-fun 5220 df-fn 5221 df-f 5222 df-f1 5223 df-fo 5224 df-f1o 5225 df-fv 5226 df-riota 5834 df-ov 5881 df-oprab 5882 df-mpo 5883 df-1st 6144 df-2nd 6145 df-recs 6309 df-frec 6395 df-pm 6654 df-pnf 7997 df-mnf 7998 df-xr 7999 df-ltxr 8000 df-le 8001 df-sub 8133 df-neg 8134 df-inn 8923 df-n0 9180 df-z 9257 df-uz 9532 df-seqfrec 10449 |
This theorem is referenced by: ennnfonelemf1 12422 |
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