ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  ennnfonelemf1 GIF version

Theorem ennnfonelemf1 12384
Description: Lemma for ennnfone 12391. 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 12383 . . . 4 (𝜑 → Fun 𝐿)
109funfnd 5239 . . 3 (𝜑𝐿 Fn dom 𝐿)
111, 2, 3, 4, 5, 6, 7ennnfonelemh 12370 . . . . . . . . 9 (𝜑𝐻:ℕ0⟶(𝐴pm ω))
1211ffnd 5358 . . . . . . . 8 (𝜑𝐻 Fn ℕ0)
13 fniunfv 5753 . . . . . . . 8 (𝐻 Fn ℕ0 𝑖 ∈ ℕ0 (𝐻𝑖) = ran 𝐻)
1412, 13syl 14 . . . . . . 7 (𝜑 𝑖 ∈ ℕ0 (𝐻𝑖) = ran 𝐻)
158, 14eqtrid 2220 . . . . . 6 (𝜑𝐿 = ran 𝐻)
1615rneqd 4849 . . . . 5 (𝜑 → ran 𝐿 = ran ran 𝐻)
17 rnuni 5032 . . . . 5 ran ran 𝐻 = 𝑥 ∈ ran 𝐻ran 𝑥
1816, 17eqtrdi 2224 . . . 4 (𝜑 → ran 𝐿 = 𝑥 ∈ ran 𝐻ran 𝑥)
1911frnd 5367 . . . . . . . . . 10 (𝜑 → ran 𝐻 ⊆ (𝐴pm ω))
2019sselda 3153 . . . . . . . . 9 ((𝜑𝑥 ∈ ran 𝐻) → 𝑥 ∈ (𝐴pm ω))
21 elpmi 6657 . . . . . . . . 9 (𝑥 ∈ (𝐴pm ω) → (𝑥:dom 𝑥𝐴 ∧ dom 𝑥 ⊆ ω))
2220, 21syl 14 . . . . . . . 8 ((𝜑𝑥 ∈ ran 𝐻) → (𝑥:dom 𝑥𝐴 ∧ dom 𝑥 ⊆ ω))
2322simpld 112 . . . . . . 7 ((𝜑𝑥 ∈ ran 𝐻) → 𝑥:dom 𝑥𝐴)
2423frnd 5367 . . . . . 6 ((𝜑𝑥 ∈ ran 𝐻) → ran 𝑥𝐴)
2524ralrimiva 2548 . . . . 5 (𝜑 → ∀𝑥 ∈ ran 𝐻ran 𝑥𝐴)
26 iunss 3923 . . . . 5 ( 𝑥 ∈ ran 𝐻ran 𝑥𝐴 ↔ ∀𝑥 ∈ ran 𝐻ran 𝑥𝐴)
2725, 26sylibr 134 . . . 4 (𝜑 𝑥 ∈ ran 𝐻ran 𝑥𝐴)
2818, 27eqsstrd 3189 . . 3 (𝜑 → ran 𝐿𝐴)
29 df-f 5212 . . 3 (𝐿:dom 𝐿𝐴 ↔ (𝐿 Fn dom 𝐿 ∧ ran 𝐿𝐴))
3010, 28, 29sylanbrc 417 . 2 (𝜑𝐿:dom 𝐿𝐴)
3119sselda 3153 . . . . . . . 8 ((𝜑𝑠 ∈ ran 𝐻) → 𝑠 ∈ (𝐴pm ω))
32 pmfun 6658 . . . . . . . 8 (𝑠 ∈ (𝐴pm ω) → Fun 𝑠)
3331, 32syl 14 . . . . . . 7 ((𝜑𝑠 ∈ ran 𝐻) → Fun 𝑠)
3411ffund 5361 . . . . . . . . . 10 (𝜑 → Fun 𝐻)
3534adantr 276 . . . . . . . . 9 ((𝜑𝑠 ∈ ran 𝐻) → Fun 𝐻)
36 simpr 110 . . . . . . . . 9 ((𝜑𝑠 ∈ ran 𝐻) → 𝑠 ∈ ran 𝐻)
37 elrnrexdm 5647 . . . . . . . . 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 276 . . . . . . . . . . . 12 ((𝜑𝑞 ∈ dom 𝐻) → ∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦)
402adantr 276 . . . . . . . . . . . 12 ((𝜑𝑞 ∈ dom 𝐻) → 𝐹:ω–onto𝐴)
413adantr 276 . . . . . . . . . . . 12 ((𝜑𝑞 ∈ dom 𝐻) → ∀𝑛 ∈ ω ∃𝑘 ∈ ω ∀𝑗 ∈ suc 𝑛(𝐹𝑘) ≠ (𝐹𝑗))
4211fdmd 5364 . . . . . . . . . . . . . 14 (𝜑 → dom 𝐻 = ℕ0)
4342eleq2d 2245 . . . . . . . . . . . . 13 (𝜑 → (𝑞 ∈ dom 𝐻𝑞 ∈ ℕ0))
4443biimpa 296 . . . . . . . . . . . 12 ((𝜑𝑞 ∈ dom 𝐻) → 𝑞 ∈ ℕ0)
4539, 40, 41, 4, 5, 6, 7, 44ennnfonelemhf1o 12379 . . . . . . . . . . 11 ((𝜑𝑞 ∈ dom 𝐻) → (𝐻𝑞):dom (𝐻𝑞)–1-1-onto→(𝐹 “ (𝑁𝑞)))
46 f1ocnv 5466 . . . . . . . . . . 11 ((𝐻𝑞):dom (𝐻𝑞)–1-1-onto→(𝐹 “ (𝑁𝑞)) → (𝐻𝑞):(𝐹 “ (𝑁𝑞))–1-1-onto→dom (𝐻𝑞))
47 f1ofun 5455 . . . . . . . . . . 11 ((𝐻𝑞):(𝐹 “ (𝑁𝑞))–1-1-onto→dom (𝐻𝑞) → Fun (𝐻𝑞))
4845, 46, 473syl 17 . . . . . . . . . 10 ((𝜑𝑞 ∈ dom 𝐻) → Fun (𝐻𝑞))
4948ad2ant2r 509 . . . . . . . . 9 (((𝜑𝑠 ∈ ran 𝐻) ∧ (𝑞 ∈ dom 𝐻𝑠 = (𝐻𝑞))) → Fun (𝐻𝑞))
50 simprr 531 . . . . . . . . . . 11 (((𝜑𝑠 ∈ ran 𝐻) ∧ (𝑞 ∈ dom 𝐻𝑠 = (𝐻𝑞))) → 𝑠 = (𝐻𝑞))
5150cnveqd 4796 . . . . . . . . . 10 (((𝜑𝑠 ∈ ran 𝐻) ∧ (𝑞 ∈ dom 𝐻𝑠 = (𝐻𝑞))) → 𝑠 = (𝐻𝑞))
5251funeqd 5230 . . . . . . . . 9 (((𝜑𝑠 ∈ ran 𝐻) ∧ (𝑞 ∈ dom 𝐻𝑠 = (𝐻𝑞))) → (Fun 𝑠 ↔ Fun (𝐻𝑞)))
5349, 52mpbird 167 . . . . . . . 8 (((𝜑𝑠 ∈ ran 𝐻) ∧ (𝑞 ∈ dom 𝐻𝑠 = (𝐻𝑞))) → Fun 𝑠)
5438, 53rexlimddv 2597 . . . . . . 7 ((𝜑𝑠 ∈ ran 𝐻) → Fun 𝑠)
551ad2antrr 488 . . . . . . . . 9 (((𝜑𝑠 ∈ ran 𝐻) ∧ 𝑡 ∈ ran 𝐻) → ∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦)
562ad2antrr 488 . . . . . . . . 9 (((𝜑𝑠 ∈ ran 𝐻) ∧ 𝑡 ∈ ran 𝐻) → 𝐹:ω–onto𝐴)
573ad2antrr 488 . . . . . . . . 9 (((𝜑𝑠 ∈ ran 𝐻) ∧ 𝑡 ∈ ran 𝐻) → ∀𝑛 ∈ ω ∃𝑘 ∈ ω ∀𝑗 ∈ suc 𝑛(𝐹𝑘) ≠ (𝐹𝑗))
58 simplr 528 . . . . . . . . 9 (((𝜑𝑠 ∈ ran 𝐻) ∧ 𝑡 ∈ ran 𝐻) → 𝑠 ∈ ran 𝐻)
59 simpr 110 . . . . . . . . 9 (((𝜑𝑠 ∈ ran 𝐻) ∧ 𝑡 ∈ ran 𝐻) → 𝑡 ∈ ran 𝐻)
6055, 56, 57, 4, 5, 6, 7, 58, 59ennnfonelemrnh 12382 . . . . . . . 8 (((𝜑𝑠 ∈ ran 𝐻) ∧ 𝑡 ∈ ran 𝐻) → (𝑠𝑡𝑡𝑠))
6160ralrimiva 2548 . . . . . . 7 ((𝜑𝑠 ∈ ran 𝐻) → ∀𝑡 ∈ ran 𝐻(𝑠𝑡𝑡𝑠))
6233, 54, 61jca31 309 . . . . . 6 ((𝜑𝑠 ∈ ran 𝐻) → ((Fun 𝑠 ∧ Fun 𝑠) ∧ ∀𝑡 ∈ ran 𝐻(𝑠𝑡𝑡𝑠)))
6362ralrimiva 2548 . . . . 5 (𝜑 → ∀𝑠 ∈ ran 𝐻((Fun 𝑠 ∧ Fun 𝑠) ∧ ∀𝑡 ∈ ran 𝐻(𝑠𝑡𝑡𝑠)))
64 fun11uni 5278 . . . . 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 114 . . 3 (𝜑 → Fun ran 𝐻)
6715cnveqd 4796 . . . 4 (𝜑𝐿 = ran 𝐻)
6867funeqd 5230 . . 3 (𝜑 → (Fun 𝐿 ↔ Fun ran 𝐻))
6966, 68mpbird 167 . 2 (𝜑 → Fun 𝐿)
70 df-f1 5213 . 2 (𝐿:dom 𝐿1-1𝐴 ↔ (𝐿:dom 𝐿𝐴 ∧ Fun 𝐿))
7130, 69, 70sylanbrc 417 1 (𝜑𝐿:dom 𝐿1-1𝐴)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  wo 708  DECID wdc 834   = wceq 1353  wcel 2146  wne 2345  wral 2453  wrex 2454  cun 3125  wss 3127  c0 3420  ifcif 3532  {csn 3589  cop 3592   cuni 3805   ciun 3882  cmpt 4059  suc csuc 4359  ωcom 4583  ccnv 4619  dom cdm 4620  ran crn 4621  cima 4623  Fun wfun 5202   Fn wfn 5203  wf 5204  1-1wf1 5205  ontowfo 5206  1-1-ontowf1o 5207  cfv 5208  (class class class)co 5865  cmpo 5867  freccfrec 6381  pm cpm 6639  0cc0 7786  1c1 7787   + caddc 7789  cmin 8102  0cn0 9147  cz 9224  seqcseq 10413
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 1445  ax-7 1446  ax-gen 1447  ax-ie1 1491  ax-ie2 1492  ax-8 1502  ax-10 1503  ax-11 1504  ax-i12 1505  ax-bndl 1507  ax-4 1508  ax-17 1524  ax-i9 1528  ax-ial 1532  ax-i5r 1533  ax-13 2148  ax-14 2149  ax-ext 2157  ax-coll 4113  ax-sep 4116  ax-nul 4124  ax-pow 4169  ax-pr 4203  ax-un 4427  ax-setind 4530  ax-iinf 4581  ax-cnex 7877  ax-resscn 7878  ax-1cn 7879  ax-1re 7880  ax-icn 7881  ax-addcl 7882  ax-addrcl 7883  ax-mulcl 7884  ax-addcom 7886  ax-addass 7888  ax-distr 7890  ax-i2m1 7891  ax-0lt1 7892  ax-0id 7894  ax-rnegex 7895  ax-cnre 7897  ax-pre-ltirr 7898  ax-pre-ltwlin 7899  ax-pre-lttrn 7900  ax-pre-ltadd 7902
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 1459  df-sb 1761  df-eu 2027  df-mo 2028  df-clab 2162  df-cleq 2168  df-clel 2171  df-nfc 2306  df-ne 2346  df-nel 2441  df-ral 2458  df-rex 2459  df-reu 2460  df-rab 2462  df-v 2737  df-sbc 2961  df-csb 3056  df-dif 3129  df-un 3131  df-in 3133  df-ss 3140  df-nul 3421  df-if 3533  df-pw 3574  df-sn 3595  df-pr 3596  df-op 3598  df-uni 3806  df-int 3841  df-iun 3884  df-br 3999  df-opab 4060  df-mpt 4061  df-tr 4097  df-id 4287  df-iord 4360  df-on 4362  df-ilim 4363  df-suc 4365  df-iom 4584  df-xp 4626  df-rel 4627  df-cnv 4628  df-co 4629  df-dm 4630  df-rn 4631  df-res 4632  df-ima 4633  df-iota 5170  df-fun 5210  df-fn 5211  df-f 5212  df-f1 5213  df-fo 5214  df-f1o 5215  df-fv 5216  df-riota 5821  df-ov 5868  df-oprab 5869  df-mpo 5870  df-1st 6131  df-2nd 6132  df-recs 6296  df-frec 6382  df-pm 6641  df-pnf 7968  df-mnf 7969  df-xr 7970  df-ltxr 7971  df-le 7972  df-sub 8104  df-neg 8105  df-inn 8891  df-n0 9148  df-z 9225  df-uz 9500  df-seqfrec 10414
This theorem is referenced by:  ennnfonelemrn  12385  ennnfonelemen  12387
  Copyright terms: Public domain W3C validator