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Theorem fseqenlem2 9765
Description: Lemma for fseqen 9767. (Contributed by Mario Carneiro, 17-May-2015.)
Hypotheses
Ref Expression
fseqenlem.a (𝜑𝐴𝑉)
fseqenlem.b (𝜑𝐵𝐴)
fseqenlem.f (𝜑𝐹:(𝐴 × 𝐴)–1-1-onto𝐴)
fseqenlem.g 𝐺 = seqω((𝑛 ∈ V, 𝑓 ∈ V ↦ (𝑥 ∈ (𝐴m suc 𝑛) ↦ ((𝑓‘(𝑥𝑛))𝐹(𝑥𝑛)))), {⟨∅, 𝐵⟩})
fseqenlem.k 𝐾 = (𝑦 𝑘 ∈ ω (𝐴m 𝑘) ↦ ⟨dom 𝑦, ((𝐺‘dom 𝑦)‘𝑦)⟩)
Assertion
Ref Expression
fseqenlem2 (𝜑𝐾: 𝑘 ∈ ω (𝐴m 𝑘)–1-1→(ω × 𝐴))
Distinct variable groups:   𝑦,𝐵   𝑓,𝑛,𝑥,𝐹   𝑦,𝑘,𝐺   𝑓,𝑘,𝑦,𝐴,𝑛,𝑥   𝜑,𝑘,𝑛,𝑥,𝑦
Allowed substitution hints:   𝜑(𝑓)   𝐵(𝑥,𝑓,𝑘,𝑛)   𝐹(𝑦,𝑘)   𝐺(𝑥,𝑓,𝑛)   𝐾(𝑥,𝑦,𝑓,𝑘,𝑛)   𝑉(𝑥,𝑦,𝑓,𝑘,𝑛)

Proof of Theorem fseqenlem2
Dummy variables 𝑧 𝑤 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eliun 4933 . . . . 5 (𝑦 𝑘 ∈ ω (𝐴m 𝑘) ↔ ∃𝑘 ∈ ω 𝑦 ∈ (𝐴m 𝑘))
2 elmapi 8611 . . . . . . . . . 10 (𝑦 ∈ (𝐴m 𝑘) → 𝑦:𝑘𝐴)
32ad2antll 725 . . . . . . . . 9 ((𝜑 ∧ (𝑘 ∈ ω ∧ 𝑦 ∈ (𝐴m 𝑘))) → 𝑦:𝑘𝐴)
43fdmd 6607 . . . . . . . 8 ((𝜑 ∧ (𝑘 ∈ ω ∧ 𝑦 ∈ (𝐴m 𝑘))) → dom 𝑦 = 𝑘)
5 simprl 767 . . . . . . . 8 ((𝜑 ∧ (𝑘 ∈ ω ∧ 𝑦 ∈ (𝐴m 𝑘))) → 𝑘 ∈ ω)
64, 5eqeltrd 2840 . . . . . . 7 ((𝜑 ∧ (𝑘 ∈ ω ∧ 𝑦 ∈ (𝐴m 𝑘))) → dom 𝑦 ∈ ω)
74fveq2d 6772 . . . . . . . . 9 ((𝜑 ∧ (𝑘 ∈ ω ∧ 𝑦 ∈ (𝐴m 𝑘))) → (𝐺‘dom 𝑦) = (𝐺𝑘))
87fveq1d 6770 . . . . . . . 8 ((𝜑 ∧ (𝑘 ∈ ω ∧ 𝑦 ∈ (𝐴m 𝑘))) → ((𝐺‘dom 𝑦)‘𝑦) = ((𝐺𝑘)‘𝑦))
9 fseqenlem.a . . . . . . . . . . . 12 (𝜑𝐴𝑉)
10 fseqenlem.b . . . . . . . . . . . 12 (𝜑𝐵𝐴)
11 fseqenlem.f . . . . . . . . . . . 12 (𝜑𝐹:(𝐴 × 𝐴)–1-1-onto𝐴)
12 fseqenlem.g . . . . . . . . . . . 12 𝐺 = seqω((𝑛 ∈ V, 𝑓 ∈ V ↦ (𝑥 ∈ (𝐴m suc 𝑛) ↦ ((𝑓‘(𝑥𝑛))𝐹(𝑥𝑛)))), {⟨∅, 𝐵⟩})
139, 10, 11, 12fseqenlem1 9764 . . . . . . . . . . 11 ((𝜑𝑘 ∈ ω) → (𝐺𝑘):(𝐴m 𝑘)–1-1𝐴)
1413adantrr 713 . . . . . . . . . 10 ((𝜑 ∧ (𝑘 ∈ ω ∧ 𝑦 ∈ (𝐴m 𝑘))) → (𝐺𝑘):(𝐴m 𝑘)–1-1𝐴)
15 f1f 6666 . . . . . . . . . 10 ((𝐺𝑘):(𝐴m 𝑘)–1-1𝐴 → (𝐺𝑘):(𝐴m 𝑘)⟶𝐴)
1614, 15syl 17 . . . . . . . . 9 ((𝜑 ∧ (𝑘 ∈ ω ∧ 𝑦 ∈ (𝐴m 𝑘))) → (𝐺𝑘):(𝐴m 𝑘)⟶𝐴)
17 simprr 769 . . . . . . . . 9 ((𝜑 ∧ (𝑘 ∈ ω ∧ 𝑦 ∈ (𝐴m 𝑘))) → 𝑦 ∈ (𝐴m 𝑘))
1816, 17ffvelrnd 6956 . . . . . . . 8 ((𝜑 ∧ (𝑘 ∈ ω ∧ 𝑦 ∈ (𝐴m 𝑘))) → ((𝐺𝑘)‘𝑦) ∈ 𝐴)
198, 18eqeltrd 2840 . . . . . . 7 ((𝜑 ∧ (𝑘 ∈ ω ∧ 𝑦 ∈ (𝐴m 𝑘))) → ((𝐺‘dom 𝑦)‘𝑦) ∈ 𝐴)
206, 19opelxpd 5626 . . . . . 6 ((𝜑 ∧ (𝑘 ∈ ω ∧ 𝑦 ∈ (𝐴m 𝑘))) → ⟨dom 𝑦, ((𝐺‘dom 𝑦)‘𝑦)⟩ ∈ (ω × 𝐴))
2120rexlimdvaa 3215 . . . . 5 (𝜑 → (∃𝑘 ∈ ω 𝑦 ∈ (𝐴m 𝑘) → ⟨dom 𝑦, ((𝐺‘dom 𝑦)‘𝑦)⟩ ∈ (ω × 𝐴)))
221, 21syl5bi 241 . . . 4 (𝜑 → (𝑦 𝑘 ∈ ω (𝐴m 𝑘) → ⟨dom 𝑦, ((𝐺‘dom 𝑦)‘𝑦)⟩ ∈ (ω × 𝐴)))
2322imp 406 . . 3 ((𝜑𝑦 𝑘 ∈ ω (𝐴m 𝑘)) → ⟨dom 𝑦, ((𝐺‘dom 𝑦)‘𝑦)⟩ ∈ (ω × 𝐴))
24 fseqenlem.k . . 3 𝐾 = (𝑦 𝑘 ∈ ω (𝐴m 𝑘) ↦ ⟨dom 𝑦, ((𝐺‘dom 𝑦)‘𝑦)⟩)
2523, 24fmptd 6982 . 2 (𝜑𝐾: 𝑘 ∈ ω (𝐴m 𝑘)⟶(ω × 𝐴))
26 ffun 6599 . . . . . . . . . . . . . . 15 (𝐾: 𝑘 ∈ ω (𝐴m 𝑘)⟶(ω × 𝐴) → Fun 𝐾)
27 funbrfv2b 6821 . . . . . . . . . . . . . . 15 (Fun 𝐾 → (𝑧𝐾𝑤 ↔ (𝑧 ∈ dom 𝐾 ∧ (𝐾𝑧) = 𝑤)))
2825, 26, 273syl 18 . . . . . . . . . . . . . 14 (𝜑 → (𝑧𝐾𝑤 ↔ (𝑧 ∈ dom 𝐾 ∧ (𝐾𝑧) = 𝑤)))
2928simplbda 499 . . . . . . . . . . . . 13 ((𝜑𝑧𝐾𝑤) → (𝐾𝑧) = 𝑤)
3028simprbda 498 . . . . . . . . . . . . . . 15 ((𝜑𝑧𝐾𝑤) → 𝑧 ∈ dom 𝐾)
3125fdmd 6607 . . . . . . . . . . . . . . . 16 (𝜑 → dom 𝐾 = 𝑘 ∈ ω (𝐴m 𝑘))
3231adantr 480 . . . . . . . . . . . . . . 15 ((𝜑𝑧𝐾𝑤) → dom 𝐾 = 𝑘 ∈ ω (𝐴m 𝑘))
3330, 32eleqtrd 2842 . . . . . . . . . . . . . 14 ((𝜑𝑧𝐾𝑤) → 𝑧 𝑘 ∈ ω (𝐴m 𝑘))
34 dmeq 5809 . . . . . . . . . . . . . . . 16 (𝑦 = 𝑧 → dom 𝑦 = dom 𝑧)
3534fveq2d 6772 . . . . . . . . . . . . . . . . 17 (𝑦 = 𝑧 → (𝐺‘dom 𝑦) = (𝐺‘dom 𝑧))
36 id 22 . . . . . . . . . . . . . . . . 17 (𝑦 = 𝑧𝑦 = 𝑧)
3735, 36fveq12d 6775 . . . . . . . . . . . . . . . 16 (𝑦 = 𝑧 → ((𝐺‘dom 𝑦)‘𝑦) = ((𝐺‘dom 𝑧)‘𝑧))
3834, 37opeq12d 4817 . . . . . . . . . . . . . . 15 (𝑦 = 𝑧 → ⟨dom 𝑦, ((𝐺‘dom 𝑦)‘𝑦)⟩ = ⟨dom 𝑧, ((𝐺‘dom 𝑧)‘𝑧)⟩)
39 opex 5381 . . . . . . . . . . . . . . 15 ⟨dom 𝑧, ((𝐺‘dom 𝑧)‘𝑧)⟩ ∈ V
4038, 24, 39fvmpt 6869 . . . . . . . . . . . . . 14 (𝑧 𝑘 ∈ ω (𝐴m 𝑘) → (𝐾𝑧) = ⟨dom 𝑧, ((𝐺‘dom 𝑧)‘𝑧)⟩)
4133, 40syl 17 . . . . . . . . . . . . 13 ((𝜑𝑧𝐾𝑤) → (𝐾𝑧) = ⟨dom 𝑧, ((𝐺‘dom 𝑧)‘𝑧)⟩)
4229, 41eqtr3d 2781 . . . . . . . . . . . 12 ((𝜑𝑧𝐾𝑤) → 𝑤 = ⟨dom 𝑧, ((𝐺‘dom 𝑧)‘𝑧)⟩)
4342fveq2d 6772 . . . . . . . . . . 11 ((𝜑𝑧𝐾𝑤) → (1st𝑤) = (1st ‘⟨dom 𝑧, ((𝐺‘dom 𝑧)‘𝑧)⟩))
44 vex 3434 . . . . . . . . . . . . 13 𝑧 ∈ V
4544dmex 7745 . . . . . . . . . . . 12 dom 𝑧 ∈ V
46 fvex 6781 . . . . . . . . . . . 12 ((𝐺‘dom 𝑧)‘𝑧) ∈ V
4745, 46op1st 7825 . . . . . . . . . . 11 (1st ‘⟨dom 𝑧, ((𝐺‘dom 𝑧)‘𝑧)⟩) = dom 𝑧
4843, 47eqtrdi 2795 . . . . . . . . . 10 ((𝜑𝑧𝐾𝑤) → (1st𝑤) = dom 𝑧)
4948fveq2d 6772 . . . . . . . . 9 ((𝜑𝑧𝐾𝑤) → (𝐺‘(1st𝑤)) = (𝐺‘dom 𝑧))
5049cnveqd 5781 . . . . . . . 8 ((𝜑𝑧𝐾𝑤) → (𝐺‘(1st𝑤)) = (𝐺‘dom 𝑧))
5142fveq2d 6772 . . . . . . . . 9 ((𝜑𝑧𝐾𝑤) → (2nd𝑤) = (2nd ‘⟨dom 𝑧, ((𝐺‘dom 𝑧)‘𝑧)⟩))
5245, 46op2nd 7826 . . . . . . . . 9 (2nd ‘⟨dom 𝑧, ((𝐺‘dom 𝑧)‘𝑧)⟩) = ((𝐺‘dom 𝑧)‘𝑧)
5351, 52eqtrdi 2795 . . . . . . . 8 ((𝜑𝑧𝐾𝑤) → (2nd𝑤) = ((𝐺‘dom 𝑧)‘𝑧))
5450, 53fveq12d 6775 . . . . . . 7 ((𝜑𝑧𝐾𝑤) → ((𝐺‘(1st𝑤))‘(2nd𝑤)) = ((𝐺‘dom 𝑧)‘((𝐺‘dom 𝑧)‘𝑧)))
55 eliun 4933 . . . . . . . . . . . . 13 (𝑧 𝑘 ∈ ω (𝐴m 𝑘) ↔ ∃𝑘 ∈ ω 𝑧 ∈ (𝐴m 𝑘))
56 elmapi 8611 . . . . . . . . . . . . . . . . . 18 (𝑧 ∈ (𝐴m 𝑘) → 𝑧:𝑘𝐴)
5756adantl 481 . . . . . . . . . . . . . . . . 17 ((𝑘 ∈ ω ∧ 𝑧 ∈ (𝐴m 𝑘)) → 𝑧:𝑘𝐴)
5857fdmd 6607 . . . . . . . . . . . . . . . 16 ((𝑘 ∈ ω ∧ 𝑧 ∈ (𝐴m 𝑘)) → dom 𝑧 = 𝑘)
59 simpl 482 . . . . . . . . . . . . . . . 16 ((𝑘 ∈ ω ∧ 𝑧 ∈ (𝐴m 𝑘)) → 𝑘 ∈ ω)
6058, 59eqeltrd 2840 . . . . . . . . . . . . . . 15 ((𝑘 ∈ ω ∧ 𝑧 ∈ (𝐴m 𝑘)) → dom 𝑧 ∈ ω)
61 simpr 484 . . . . . . . . . . . . . . . 16 ((𝑘 ∈ ω ∧ 𝑧 ∈ (𝐴m 𝑘)) → 𝑧 ∈ (𝐴m 𝑘))
6258oveq2d 7284 . . . . . . . . . . . . . . . 16 ((𝑘 ∈ ω ∧ 𝑧 ∈ (𝐴m 𝑘)) → (𝐴m dom 𝑧) = (𝐴m 𝑘))
6361, 62eleqtrrd 2843 . . . . . . . . . . . . . . 15 ((𝑘 ∈ ω ∧ 𝑧 ∈ (𝐴m 𝑘)) → 𝑧 ∈ (𝐴m dom 𝑧))
6460, 63jca 511 . . . . . . . . . . . . . 14 ((𝑘 ∈ ω ∧ 𝑧 ∈ (𝐴m 𝑘)) → (dom 𝑧 ∈ ω ∧ 𝑧 ∈ (𝐴m dom 𝑧)))
6564rexlimiva 3211 . . . . . . . . . . . . 13 (∃𝑘 ∈ ω 𝑧 ∈ (𝐴m 𝑘) → (dom 𝑧 ∈ ω ∧ 𝑧 ∈ (𝐴m dom 𝑧)))
6655, 65sylbi 216 . . . . . . . . . . . 12 (𝑧 𝑘 ∈ ω (𝐴m 𝑘) → (dom 𝑧 ∈ ω ∧ 𝑧 ∈ (𝐴m dom 𝑧)))
6733, 66syl 17 . . . . . . . . . . 11 ((𝜑𝑧𝐾𝑤) → (dom 𝑧 ∈ ω ∧ 𝑧 ∈ (𝐴m dom 𝑧)))
6867simpld 494 . . . . . . . . . 10 ((𝜑𝑧𝐾𝑤) → dom 𝑧 ∈ ω)
699, 10, 11, 12fseqenlem1 9764 . . . . . . . . . 10 ((𝜑 ∧ dom 𝑧 ∈ ω) → (𝐺‘dom 𝑧):(𝐴m dom 𝑧)–1-1𝐴)
7068, 69syldan 590 . . . . . . . . 9 ((𝜑𝑧𝐾𝑤) → (𝐺‘dom 𝑧):(𝐴m dom 𝑧)–1-1𝐴)
71 f1f1orn 6723 . . . . . . . . 9 ((𝐺‘dom 𝑧):(𝐴m dom 𝑧)–1-1𝐴 → (𝐺‘dom 𝑧):(𝐴m dom 𝑧)–1-1-onto→ran (𝐺‘dom 𝑧))
7270, 71syl 17 . . . . . . . 8 ((𝜑𝑧𝐾𝑤) → (𝐺‘dom 𝑧):(𝐴m dom 𝑧)–1-1-onto→ran (𝐺‘dom 𝑧))
7367simprd 495 . . . . . . . 8 ((𝜑𝑧𝐾𝑤) → 𝑧 ∈ (𝐴m dom 𝑧))
74 f1ocnvfv1 7142 . . . . . . . 8 (((𝐺‘dom 𝑧):(𝐴m dom 𝑧)–1-1-onto→ran (𝐺‘dom 𝑧) ∧ 𝑧 ∈ (𝐴m dom 𝑧)) → ((𝐺‘dom 𝑧)‘((𝐺‘dom 𝑧)‘𝑧)) = 𝑧)
7572, 73, 74syl2anc 583 . . . . . . 7 ((𝜑𝑧𝐾𝑤) → ((𝐺‘dom 𝑧)‘((𝐺‘dom 𝑧)‘𝑧)) = 𝑧)
7654, 75eqtr2d 2780 . . . . . 6 ((𝜑𝑧𝐾𝑤) → 𝑧 = ((𝐺‘(1st𝑤))‘(2nd𝑤)))
7776ex 412 . . . . 5 (𝜑 → (𝑧𝐾𝑤𝑧 = ((𝐺‘(1st𝑤))‘(2nd𝑤))))
7877alrimiv 1933 . . . 4 (𝜑 → ∀𝑧(𝑧𝐾𝑤𝑧 = ((𝐺‘(1st𝑤))‘(2nd𝑤))))
79 mo2icl 3652 . . . 4 (∀𝑧(𝑧𝐾𝑤𝑧 = ((𝐺‘(1st𝑤))‘(2nd𝑤))) → ∃*𝑧 𝑧𝐾𝑤)
8078, 79syl 17 . . 3 (𝜑 → ∃*𝑧 𝑧𝐾𝑤)
8180alrimiv 1933 . 2 (𝜑 → ∀𝑤∃*𝑧 𝑧𝐾𝑤)
82 dff12 6665 . 2 (𝐾: 𝑘 ∈ ω (𝐴m 𝑘)–1-1→(ω × 𝐴) ↔ (𝐾: 𝑘 ∈ ω (𝐴m 𝑘)⟶(ω × 𝐴) ∧ ∀𝑤∃*𝑧 𝑧𝐾𝑤))
8325, 81, 82sylanbrc 582 1 (𝜑𝐾: 𝑘 ∈ ω (𝐴m 𝑘)–1-1→(ω × 𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 395  wal 1539   = wceq 1541  wcel 2109  ∃*wmo 2539  wrex 3066  Vcvv 3430  c0 4261  {csn 4566  cop 4572   ciun 4929   class class class wbr 5078  cmpt 5161   × cxp 5586  ccnv 5587  dom cdm 5588  ran crn 5589  cres 5590  suc csuc 6265  Fun wfun 6424  wf 6426  1-1wf1 6427  1-1-ontowf1o 6429  cfv 6430  (class class class)co 7268  cmpo 7270  ωcom 7700  1st c1st 7815  2nd c2nd 7816  seqωcseqom 8262  m cmap 8589
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1801  ax-4 1815  ax-5 1916  ax-6 1974  ax-7 2014  ax-8 2111  ax-9 2119  ax-10 2140  ax-11 2157  ax-12 2174  ax-ext 2710  ax-rep 5213  ax-sep 5226  ax-nul 5233  ax-pow 5291  ax-pr 5355  ax-un 7579
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3or 1086  df-3an 1087  df-tru 1544  df-fal 1554  df-ex 1786  df-nf 1790  df-sb 2071  df-mo 2541  df-eu 2570  df-clab 2717  df-cleq 2731  df-clel 2817  df-nfc 2890  df-ne 2945  df-ral 3070  df-rex 3071  df-reu 3072  df-rab 3074  df-v 3432  df-sbc 3720  df-csb 3837  df-dif 3894  df-un 3896  df-in 3898  df-ss 3908  df-pss 3910  df-nul 4262  df-if 4465  df-pw 4540  df-sn 4567  df-pr 4569  df-tp 4571  df-op 4573  df-uni 4845  df-iun 4931  df-br 5079  df-opab 5141  df-mpt 5162  df-tr 5196  df-id 5488  df-eprel 5494  df-po 5502  df-so 5503  df-fr 5543  df-we 5545  df-xp 5594  df-rel 5595  df-cnv 5596  df-co 5597  df-dm 5598  df-rn 5599  df-res 5600  df-ima 5601  df-pred 6199  df-ord 6266  df-on 6267  df-lim 6268  df-suc 6269  df-iota 6388  df-fun 6432  df-fn 6433  df-f 6434  df-f1 6435  df-fo 6436  df-f1o 6437  df-fv 6438  df-ov 7271  df-oprab 7272  df-mpo 7273  df-om 7701  df-1st 7817  df-2nd 7818  df-frecs 8081  df-wrecs 8112  df-recs 8186  df-rdg 8225  df-seqom 8263  df-1o 8281  df-map 8591
This theorem is referenced by:  fseqen  9767  pwfseqlem5  10403
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