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Theorem ennnfonelemf1 11942
Description: Lemma for ennnfone 11949. 𝐿 is one-to-one. (Contributed by Jim Kingdon, 16-Jul-2023.)
Hypotheses
Ref Expression
ennnfonelemh.dceq (𝜑 → ∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦)
ennnfonelemh.f (𝜑𝐹:ω–onto𝐴)
ennnfonelemh.ne (𝜑 → ∀𝑛 ∈ ω ∃𝑘 ∈ ω ∀𝑗 ∈ suc 𝑛(𝐹𝑘) ≠ (𝐹𝑗))
ennnfonelemh.g 𝐺 = (𝑥 ∈ (𝐴pm ω), 𝑦 ∈ ω ↦ if((𝐹𝑦) ∈ (𝐹𝑦), 𝑥, (𝑥 ∪ {⟨dom 𝑥, (𝐹𝑦)⟩})))
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 (𝐻𝑖)
Assertion
Ref Expression
ennnfonelemf1 (𝜑𝐿:dom 𝐿1-1𝐴)
Distinct variable groups:   𝐴,𝑗,𝑥,𝑦   𝑥,𝐹,𝑦,𝑗,𝑘   𝑛,𝐹   𝑗,𝐺   𝑖,𝐻   𝑗,𝐻,𝑥,𝑦,𝑘   𝑗,𝐽   𝑥,𝑁,𝑦,𝑘,𝑗   𝜑,𝑗,𝑥,𝑦,𝑘   𝑘,𝑛,𝑗
Allowed substitution hints:   𝜑(𝑖,𝑛)   𝐴(𝑖,𝑘,𝑛)   𝐹(𝑖)   𝐺(𝑥,𝑦,𝑖,𝑘,𝑛)   𝐻(𝑛)   𝐽(𝑥,𝑦,𝑖,𝑘,𝑛)   𝐿(𝑥,𝑦,𝑖,𝑗,𝑘,𝑛)   𝑁(𝑖,𝑛)

Proof of Theorem ennnfonelemf1
Dummy variables 𝑞 𝑠 𝑡 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ennnfonelemh.dceq . . . . 5 (𝜑 → ∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦)
2 ennnfonelemh.f . . . . 5 (𝜑𝐹:ω–onto𝐴)
3 ennnfonelemh.ne . . . . 5 (𝜑 → ∀𝑛 ∈ ω ∃𝑘 ∈ ω ∀𝑗 ∈ suc 𝑛(𝐹𝑘) ≠ (𝐹𝑗))
4 ennnfonelemh.g . . . . 5 𝐺 = (𝑥 ∈ (𝐴pm ω), 𝑦 ∈ ω ↦ if((𝐹𝑦) ∈ (𝐹𝑦), 𝑥, (𝑥 ∪ {⟨dom 𝑥, (𝐹𝑦)⟩})))
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 ennnfone.l . . . . 5 𝐿 = 𝑖 ∈ ℕ0 (𝐻𝑖)
91, 2, 3, 4, 5, 6, 7, 8ennnfonelemfun 11941 . . . 4 (𝜑 → Fun 𝐿)
109funfnd 5154 . . 3 (𝜑𝐿 Fn dom 𝐿)
111, 2, 3, 4, 5, 6, 7ennnfonelemh 11928 . . . . . . . . 9 (𝜑𝐻:ℕ0⟶(𝐴pm ω))
1211ffnd 5273 . . . . . . . 8 (𝜑𝐻 Fn ℕ0)
13 fniunfv 5663 . . . . . . . 8 (𝐻 Fn ℕ0 𝑖 ∈ ℕ0 (𝐻𝑖) = ran 𝐻)
1412, 13syl 14 . . . . . . 7 (𝜑 𝑖 ∈ ℕ0 (𝐻𝑖) = ran 𝐻)
158, 14syl5eq 2184 . . . . . 6 (𝜑𝐿 = ran 𝐻)
1615rneqd 4768 . . . . 5 (𝜑 → ran 𝐿 = ran ran 𝐻)
17 rnuni 4950 . . . . 5 ran ran 𝐻 = 𝑥 ∈ ran 𝐻ran 𝑥
1816, 17syl6eq 2188 . . . 4 (𝜑 → ran 𝐿 = 𝑥 ∈ ran 𝐻ran 𝑥)
1911frnd 5282 . . . . . . . . . 10 (𝜑 → ran 𝐻 ⊆ (𝐴pm ω))
2019sselda 3097 . . . . . . . . 9 ((𝜑𝑥 ∈ ran 𝐻) → 𝑥 ∈ (𝐴pm ω))
21 elpmi 6561 . . . . . . . . 9 (𝑥 ∈ (𝐴pm ω) → (𝑥:dom 𝑥𝐴 ∧ dom 𝑥 ⊆ ω))
2220, 21syl 14 . . . . . . . 8 ((𝜑𝑥 ∈ ran 𝐻) → (𝑥:dom 𝑥𝐴 ∧ dom 𝑥 ⊆ ω))
2322simpld 111 . . . . . . 7 ((𝜑𝑥 ∈ ran 𝐻) → 𝑥:dom 𝑥𝐴)
2423frnd 5282 . . . . . 6 ((𝜑𝑥 ∈ ran 𝐻) → ran 𝑥𝐴)
2524ralrimiva 2505 . . . . 5 (𝜑 → ∀𝑥 ∈ ran 𝐻ran 𝑥𝐴)
26 iunss 3854 . . . . 5 ( 𝑥 ∈ ran 𝐻ran 𝑥𝐴 ↔ ∀𝑥 ∈ ran 𝐻ran 𝑥𝐴)
2725, 26sylibr 133 . . . 4 (𝜑 𝑥 ∈ ran 𝐻ran 𝑥𝐴)
2818, 27eqsstrd 3133 . . 3 (𝜑 → ran 𝐿𝐴)
29 df-f 5127 . . 3 (𝐿:dom 𝐿𝐴 ↔ (𝐿 Fn dom 𝐿 ∧ ran 𝐿𝐴))
3010, 28, 29sylanbrc 413 . 2 (𝜑𝐿:dom 𝐿𝐴)
3119sselda 3097 . . . . . . . 8 ((𝜑𝑠 ∈ ran 𝐻) → 𝑠 ∈ (𝐴pm ω))
32 pmfun 6562 . . . . . . . 8 (𝑠 ∈ (𝐴pm ω) → Fun 𝑠)
3331, 32syl 14 . . . . . . 7 ((𝜑𝑠 ∈ ran 𝐻) → Fun 𝑠)
3411ffund 5276 . . . . . . . . . 10 (𝜑 → Fun 𝐻)
3534adantr 274 . . . . . . . . 9 ((𝜑𝑠 ∈ ran 𝐻) → Fun 𝐻)
36 simpr 109 . . . . . . . . 9 ((𝜑𝑠 ∈ ran 𝐻) → 𝑠 ∈ ran 𝐻)
37 elrnrexdm 5559 . . . . . . . . 9 (Fun 𝐻 → (𝑠 ∈ ran 𝐻 → ∃𝑞 ∈ dom 𝐻 𝑠 = (𝐻𝑞)))
3835, 36, 37sylc 62 . . . . . . . 8 ((𝜑𝑠 ∈ ran 𝐻) → ∃𝑞 ∈ dom 𝐻 𝑠 = (𝐻𝑞))
391adantr 274 . . . . . . . . . . . 12 ((𝜑𝑞 ∈ dom 𝐻) → ∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦)
402adantr 274 . . . . . . . . . . . 12 ((𝜑𝑞 ∈ dom 𝐻) → 𝐹:ω–onto𝐴)
413adantr 274 . . . . . . . . . . . 12 ((𝜑𝑞 ∈ dom 𝐻) → ∀𝑛 ∈ ω ∃𝑘 ∈ ω ∀𝑗 ∈ suc 𝑛(𝐹𝑘) ≠ (𝐹𝑗))
4211fdmd 5279 . . . . . . . . . . . . . 14 (𝜑 → dom 𝐻 = ℕ0)
4342eleq2d 2209 . . . . . . . . . . . . 13 (𝜑 → (𝑞 ∈ dom 𝐻𝑞 ∈ ℕ0))
4443biimpa 294 . . . . . . . . . . . 12 ((𝜑𝑞 ∈ dom 𝐻) → 𝑞 ∈ ℕ0)
4539, 40, 41, 4, 5, 6, 7, 44ennnfonelemhf1o 11937 . . . . . . . . . . 11 ((𝜑𝑞 ∈ dom 𝐻) → (𝐻𝑞):dom (𝐻𝑞)–1-1-onto→(𝐹 “ (𝑁𝑞)))
46 f1ocnv 5380 . . . . . . . . . . 11 ((𝐻𝑞):dom (𝐻𝑞)–1-1-onto→(𝐹 “ (𝑁𝑞)) → (𝐻𝑞):(𝐹 “ (𝑁𝑞))–1-1-onto→dom (𝐻𝑞))
47 f1ofun 5369 . . . . . . . . . . 11 ((𝐻𝑞):(𝐹 “ (𝑁𝑞))–1-1-onto→dom (𝐻𝑞) → Fun (𝐻𝑞))
4845, 46, 473syl 17 . . . . . . . . . 10 ((𝜑𝑞 ∈ dom 𝐻) → Fun (𝐻𝑞))
4948ad2ant2r 500 . . . . . . . . 9 (((𝜑𝑠 ∈ ran 𝐻) ∧ (𝑞 ∈ dom 𝐻𝑠 = (𝐻𝑞))) → Fun (𝐻𝑞))
50 simprr 521 . . . . . . . . . . 11 (((𝜑𝑠 ∈ ran 𝐻) ∧ (𝑞 ∈ dom 𝐻𝑠 = (𝐻𝑞))) → 𝑠 = (𝐻𝑞))
5150cnveqd 4715 . . . . . . . . . 10 (((𝜑𝑠 ∈ ran 𝐻) ∧ (𝑞 ∈ dom 𝐻𝑠 = (𝐻𝑞))) → 𝑠 = (𝐻𝑞))
5251funeqd 5145 . . . . . . . . 9 (((𝜑𝑠 ∈ ran 𝐻) ∧ (𝑞 ∈ dom 𝐻𝑠 = (𝐻𝑞))) → (Fun 𝑠 ↔ Fun (𝐻𝑞)))
5349, 52mpbird 166 . . . . . . . 8 (((𝜑𝑠 ∈ ran 𝐻) ∧ (𝑞 ∈ dom 𝐻𝑠 = (𝐻𝑞))) → Fun 𝑠)
5438, 53rexlimddv 2554 . . . . . . 7 ((𝜑𝑠 ∈ ran 𝐻) → Fun 𝑠)
551ad2antrr 479 . . . . . . . . 9 (((𝜑𝑠 ∈ ran 𝐻) ∧ 𝑡 ∈ ran 𝐻) → ∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦)
562ad2antrr 479 . . . . . . . . 9 (((𝜑𝑠 ∈ ran 𝐻) ∧ 𝑡 ∈ ran 𝐻) → 𝐹:ω–onto𝐴)
573ad2antrr 479 . . . . . . . . 9 (((𝜑𝑠 ∈ ran 𝐻) ∧ 𝑡 ∈ ran 𝐻) → ∀𝑛 ∈ ω ∃𝑘 ∈ ω ∀𝑗 ∈ suc 𝑛(𝐹𝑘) ≠ (𝐹𝑗))
58 simplr 519 . . . . . . . . 9 (((𝜑𝑠 ∈ ran 𝐻) ∧ 𝑡 ∈ ran 𝐻) → 𝑠 ∈ ran 𝐻)
59 simpr 109 . . . . . . . . 9 (((𝜑𝑠 ∈ ran 𝐻) ∧ 𝑡 ∈ ran 𝐻) → 𝑡 ∈ ran 𝐻)
6055, 56, 57, 4, 5, 6, 7, 58, 59ennnfonelemrnh 11940 . . . . . . . 8 (((𝜑𝑠 ∈ ran 𝐻) ∧ 𝑡 ∈ ran 𝐻) → (𝑠𝑡𝑡𝑠))
6160ralrimiva 2505 . . . . . . 7 ((𝜑𝑠 ∈ ran 𝐻) → ∀𝑡 ∈ ran 𝐻(𝑠𝑡𝑡𝑠))
6233, 54, 61jca31 307 . . . . . 6 ((𝜑𝑠 ∈ ran 𝐻) → ((Fun 𝑠 ∧ Fun 𝑠) ∧ ∀𝑡 ∈ ran 𝐻(𝑠𝑡𝑡𝑠)))
6362ralrimiva 2505 . . . . 5 (𝜑 → ∀𝑠 ∈ ran 𝐻((Fun 𝑠 ∧ Fun 𝑠) ∧ ∀𝑡 ∈ ran 𝐻(𝑠𝑡𝑡𝑠)))
64 fun11uni 5193 . . . . 5 (∀𝑠 ∈ ran 𝐻((Fun 𝑠 ∧ Fun 𝑠) ∧ ∀𝑡 ∈ ran 𝐻(𝑠𝑡𝑡𝑠)) → (Fun ran 𝐻 ∧ Fun ran 𝐻))
6563, 64syl 14 . . . 4 (𝜑 → (Fun ran 𝐻 ∧ Fun ran 𝐻))
6665simprd 113 . . 3 (𝜑 → Fun ran 𝐻)
6715cnveqd 4715 . . . 4 (𝜑𝐿 = ran 𝐻)
6867funeqd 5145 . . 3 (𝜑 → (Fun 𝐿 ↔ Fun ran 𝐻))
6966, 68mpbird 166 . 2 (𝜑 → Fun 𝐿)
70 df-f1 5128 . 2 (𝐿:dom 𝐿1-1𝐴 ↔ (𝐿:dom 𝐿𝐴 ∧ Fun 𝐿))
7130, 69, 70sylanbrc 413 1 (𝜑𝐿:dom 𝐿1-1𝐴)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wo 697  DECID wdc 819   = wceq 1331  wcel 1480  wne 2308  wral 2416  wrex 2417  cun 3069  wss 3071  c0 3363  ifcif 3474  {csn 3527  cop 3530   cuni 3736   ciun 3813  cmpt 3989  suc csuc 4287  ωcom 4504  ccnv 4538  dom cdm 4539  ran crn 4540  cima 4542  Fun wfun 5117   Fn wfn 5118  wf 5119  1-1wf1 5120  ontowfo 5121  1-1-ontowf1o 5122  cfv 5123  (class class class)co 5774  cmpo 5776  freccfrec 6287  pm cpm 6543  0cc0 7632  1c1 7633   + caddc 7635  cmin 7945  0cn0 8989  cz 9066  seqcseq 10230
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 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-coll 4043  ax-sep 4046  ax-nul 4054  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-setind 4452  ax-iinf 4502  ax-cnex 7723  ax-resscn 7724  ax-1cn 7725  ax-1re 7726  ax-icn 7727  ax-addcl 7728  ax-addrcl 7729  ax-mulcl 7730  ax-addcom 7732  ax-addass 7734  ax-distr 7736  ax-i2m1 7737  ax-0lt1 7738  ax-0id 7740  ax-rnegex 7741  ax-cnre 7743  ax-pre-ltirr 7744  ax-pre-ltwlin 7745  ax-pre-lttrn 7746  ax-pre-ltadd 7748
This theorem depends on definitions:  df-bi 116  df-dc 820  df-3or 963  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-nel 2404  df-ral 2421  df-rex 2422  df-reu 2423  df-rab 2425  df-v 2688  df-sbc 2910  df-csb 3004  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-nul 3364  df-if 3475  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-int 3772  df-iun 3815  df-br 3930  df-opab 3990  df-mpt 3991  df-tr 4027  df-id 4215  df-iord 4288  df-on 4290  df-ilim 4291  df-suc 4293  df-iom 4505  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-f1 5128  df-fo 5129  df-f1o 5130  df-fv 5131  df-riota 5730  df-ov 5777  df-oprab 5778  df-mpo 5779  df-1st 6038  df-2nd 6039  df-recs 6202  df-frec 6288  df-pm 6545  df-pnf 7814  df-mnf 7815  df-xr 7816  df-ltxr 7817  df-le 7818  df-sub 7947  df-neg 7948  df-inn 8733  df-n0 8990  df-z 9067  df-uz 9339  df-seqfrec 10231
This theorem is referenced by:  ennnfonelemrn  11943  ennnfonelemen  11945
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