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Theorem fseqenlem2 10065
Description: Lemma for fseqen 10067. (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 4995 . . . . 5 (𝑦 𝑘 ∈ ω (𝐴m 𝑘) ↔ ∃𝑘 ∈ ω 𝑦 ∈ (𝐴m 𝑘))
2 elmapi 8889 . . . . . . . . . 10 (𝑦 ∈ (𝐴m 𝑘) → 𝑦:𝑘𝐴)
32ad2antll 729 . . . . . . . . 9 ((𝜑 ∧ (𝑘 ∈ ω ∧ 𝑦 ∈ (𝐴m 𝑘))) → 𝑦:𝑘𝐴)
43fdmd 6746 . . . . . . . 8 ((𝜑 ∧ (𝑘 ∈ ω ∧ 𝑦 ∈ (𝐴m 𝑘))) → dom 𝑦 = 𝑘)
5 simprl 771 . . . . . . . 8 ((𝜑 ∧ (𝑘 ∈ ω ∧ 𝑦 ∈ (𝐴m 𝑘))) → 𝑘 ∈ ω)
64, 5eqeltrd 2841 . . . . . . 7 ((𝜑 ∧ (𝑘 ∈ ω ∧ 𝑦 ∈ (𝐴m 𝑘))) → dom 𝑦 ∈ ω)
74fveq2d 6910 . . . . . . . . 9 ((𝜑 ∧ (𝑘 ∈ ω ∧ 𝑦 ∈ (𝐴m 𝑘))) → (𝐺‘dom 𝑦) = (𝐺𝑘))
87fveq1d 6908 . . . . . . . 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 10064 . . . . . . . . . . 11 ((𝜑𝑘 ∈ ω) → (𝐺𝑘):(𝐴m 𝑘)–1-1𝐴)
1413adantrr 717 . . . . . . . . . 10 ((𝜑 ∧ (𝑘 ∈ ω ∧ 𝑦 ∈ (𝐴m 𝑘))) → (𝐺𝑘):(𝐴m 𝑘)–1-1𝐴)
15 f1f 6804 . . . . . . . . . 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 773 . . . . . . . . 9 ((𝜑 ∧ (𝑘 ∈ ω ∧ 𝑦 ∈ (𝐴m 𝑘))) → 𝑦 ∈ (𝐴m 𝑘))
1816, 17ffvelcdmd 7105 . . . . . . . 8 ((𝜑 ∧ (𝑘 ∈ ω ∧ 𝑦 ∈ (𝐴m 𝑘))) → ((𝐺𝑘)‘𝑦) ∈ 𝐴)
198, 18eqeltrd 2841 . . . . . . 7 ((𝜑 ∧ (𝑘 ∈ ω ∧ 𝑦 ∈ (𝐴m 𝑘))) → ((𝐺‘dom 𝑦)‘𝑦) ∈ 𝐴)
206, 19opelxpd 5724 . . . . . 6 ((𝜑 ∧ (𝑘 ∈ ω ∧ 𝑦 ∈ (𝐴m 𝑘))) → ⟨dom 𝑦, ((𝐺‘dom 𝑦)‘𝑦)⟩ ∈ (ω × 𝐴))
2120rexlimdvaa 3156 . . . . 5 (𝜑 → (∃𝑘 ∈ ω 𝑦 ∈ (𝐴m 𝑘) → ⟨dom 𝑦, ((𝐺‘dom 𝑦)‘𝑦)⟩ ∈ (ω × 𝐴)))
221, 21biimtrid 242 . . . 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 7134 . 2 (𝜑𝐾: 𝑘 ∈ ω (𝐴m 𝑘)⟶(ω × 𝐴))
26 ffun 6739 . . . . . . . . . . . . . . 15 (𝐾: 𝑘 ∈ ω (𝐴m 𝑘)⟶(ω × 𝐴) → Fun 𝐾)
27 funbrfv2b 6966 . . . . . . . . . . . . . . 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 6746 . . . . . . . . . . . . . . . 16 (𝜑 → dom 𝐾 = 𝑘 ∈ ω (𝐴m 𝑘))
3231adantr 480 . . . . . . . . . . . . . . 15 ((𝜑𝑧𝐾𝑤) → dom 𝐾 = 𝑘 ∈ ω (𝐴m 𝑘))
3330, 32eleqtrd 2843 . . . . . . . . . . . . . 14 ((𝜑𝑧𝐾𝑤) → 𝑧 𝑘 ∈ ω (𝐴m 𝑘))
34 dmeq 5914 . . . . . . . . . . . . . . . 16 (𝑦 = 𝑧 → dom 𝑦 = dom 𝑧)
3534fveq2d 6910 . . . . . . . . . . . . . . . . 17 (𝑦 = 𝑧 → (𝐺‘dom 𝑦) = (𝐺‘dom 𝑧))
36 id 22 . . . . . . . . . . . . . . . . 17 (𝑦 = 𝑧𝑦 = 𝑧)
3735, 36fveq12d 6913 . . . . . . . . . . . . . . . 16 (𝑦 = 𝑧 → ((𝐺‘dom 𝑦)‘𝑦) = ((𝐺‘dom 𝑧)‘𝑧))
3834, 37opeq12d 4881 . . . . . . . . . . . . . . 15 (𝑦 = 𝑧 → ⟨dom 𝑦, ((𝐺‘dom 𝑦)‘𝑦)⟩ = ⟨dom 𝑧, ((𝐺‘dom 𝑧)‘𝑧)⟩)
39 opex 5469 . . . . . . . . . . . . . . 15 ⟨dom 𝑧, ((𝐺‘dom 𝑧)‘𝑧)⟩ ∈ V
4038, 24, 39fvmpt 7016 . . . . . . . . . . . . . 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 2779 . . . . . . . . . . . 12 ((𝜑𝑧𝐾𝑤) → 𝑤 = ⟨dom 𝑧, ((𝐺‘dom 𝑧)‘𝑧)⟩)
4342fveq2d 6910 . . . . . . . . . . 11 ((𝜑𝑧𝐾𝑤) → (1st𝑤) = (1st ‘⟨dom 𝑧, ((𝐺‘dom 𝑧)‘𝑧)⟩))
44 vex 3484 . . . . . . . . . . . . 13 𝑧 ∈ V
4544dmex 7931 . . . . . . . . . . . 12 dom 𝑧 ∈ V
46 fvex 6919 . . . . . . . . . . . 12 ((𝐺‘dom 𝑧)‘𝑧) ∈ V
4745, 46op1st 8022 . . . . . . . . . . 11 (1st ‘⟨dom 𝑧, ((𝐺‘dom 𝑧)‘𝑧)⟩) = dom 𝑧
4843, 47eqtrdi 2793 . . . . . . . . . 10 ((𝜑𝑧𝐾𝑤) → (1st𝑤) = dom 𝑧)
4948fveq2d 6910 . . . . . . . . 9 ((𝜑𝑧𝐾𝑤) → (𝐺‘(1st𝑤)) = (𝐺‘dom 𝑧))
5049cnveqd 5886 . . . . . . . 8 ((𝜑𝑧𝐾𝑤) → (𝐺‘(1st𝑤)) = (𝐺‘dom 𝑧))
5142fveq2d 6910 . . . . . . . . 9 ((𝜑𝑧𝐾𝑤) → (2nd𝑤) = (2nd ‘⟨dom 𝑧, ((𝐺‘dom 𝑧)‘𝑧)⟩))
5245, 46op2nd 8023 . . . . . . . . 9 (2nd ‘⟨dom 𝑧, ((𝐺‘dom 𝑧)‘𝑧)⟩) = ((𝐺‘dom 𝑧)‘𝑧)
5351, 52eqtrdi 2793 . . . . . . . 8 ((𝜑𝑧𝐾𝑤) → (2nd𝑤) = ((𝐺‘dom 𝑧)‘𝑧))
5450, 53fveq12d 6913 . . . . . . 7 ((𝜑𝑧𝐾𝑤) → ((𝐺‘(1st𝑤))‘(2nd𝑤)) = ((𝐺‘dom 𝑧)‘((𝐺‘dom 𝑧)‘𝑧)))
55 eliun 4995 . . . . . . . . . . . . 13 (𝑧 𝑘 ∈ ω (𝐴m 𝑘) ↔ ∃𝑘 ∈ ω 𝑧 ∈ (𝐴m 𝑘))
56 elmapi 8889 . . . . . . . . . . . . . . . . . 18 (𝑧 ∈ (𝐴m 𝑘) → 𝑧:𝑘𝐴)
5756adantl 481 . . . . . . . . . . . . . . . . 17 ((𝑘 ∈ ω ∧ 𝑧 ∈ (𝐴m 𝑘)) → 𝑧:𝑘𝐴)
5857fdmd 6746 . . . . . . . . . . . . . . . 16 ((𝑘 ∈ ω ∧ 𝑧 ∈ (𝐴m 𝑘)) → dom 𝑧 = 𝑘)
59 simpl 482 . . . . . . . . . . . . . . . 16 ((𝑘 ∈ ω ∧ 𝑧 ∈ (𝐴m 𝑘)) → 𝑘 ∈ ω)
6058, 59eqeltrd 2841 . . . . . . . . . . . . . . 15 ((𝑘 ∈ ω ∧ 𝑧 ∈ (𝐴m 𝑘)) → dom 𝑧 ∈ ω)
61 simpr 484 . . . . . . . . . . . . . . . 16 ((𝑘 ∈ ω ∧ 𝑧 ∈ (𝐴m 𝑘)) → 𝑧 ∈ (𝐴m 𝑘))
6258oveq2d 7447 . . . . . . . . . . . . . . . 16 ((𝑘 ∈ ω ∧ 𝑧 ∈ (𝐴m 𝑘)) → (𝐴m dom 𝑧) = (𝐴m 𝑘))
6361, 62eleqtrrd 2844 . . . . . . . . . . . . . . 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 3147 . . . . . . . . . . . . 13 (∃𝑘 ∈ ω 𝑧 ∈ (𝐴m 𝑘) → (dom 𝑧 ∈ ω ∧ 𝑧 ∈ (𝐴m dom 𝑧)))
6655, 65sylbi 217 . . . . . . . . . . . 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 10064 . . . . . . . . . 10 ((𝜑 ∧ dom 𝑧 ∈ ω) → (𝐺‘dom 𝑧):(𝐴m dom 𝑧)–1-1𝐴)
7068, 69syldan 591 . . . . . . . . 9 ((𝜑𝑧𝐾𝑤) → (𝐺‘dom 𝑧):(𝐴m dom 𝑧)–1-1𝐴)
71 f1f1orn 6859 . . . . . . . . 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 7296 . . . . . . . 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 584 . . . . . . 7 ((𝜑𝑧𝐾𝑤) → ((𝐺‘dom 𝑧)‘((𝐺‘dom 𝑧)‘𝑧)) = 𝑧)
7654, 75eqtr2d 2778 . . . . . 6 ((𝜑𝑧𝐾𝑤) → 𝑧 = ((𝐺‘(1st𝑤))‘(2nd𝑤)))
7776ex 412 . . . . 5 (𝜑 → (𝑧𝐾𝑤𝑧 = ((𝐺‘(1st𝑤))‘(2nd𝑤))))
7877alrimiv 1927 . . . 4 (𝜑 → ∀𝑧(𝑧𝐾𝑤𝑧 = ((𝐺‘(1st𝑤))‘(2nd𝑤))))
79 mo2icl 3720 . . . 4 (∀𝑧(𝑧𝐾𝑤𝑧 = ((𝐺‘(1st𝑤))‘(2nd𝑤))) → ∃*𝑧 𝑧𝐾𝑤)
8078, 79syl 17 . . 3 (𝜑 → ∃*𝑧 𝑧𝐾𝑤)
8180alrimiv 1927 . 2 (𝜑 → ∀𝑤∃*𝑧 𝑧𝐾𝑤)
82 dff12 6803 . 2 (𝐾: 𝑘 ∈ ω (𝐴m 𝑘)–1-1→(ω × 𝐴) ↔ (𝐾: 𝑘 ∈ ω (𝐴m 𝑘)⟶(ω × 𝐴) ∧ ∀𝑤∃*𝑧 𝑧𝐾𝑤))
8325, 81, 82sylanbrc 583 1 (𝜑𝐾: 𝑘 ∈ ω (𝐴m 𝑘)–1-1→(ω × 𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  wal 1538   = wceq 1540  wcel 2108  ∃*wmo 2538  wrex 3070  Vcvv 3480  c0 4333  {csn 4626  cop 4632   ciun 4991   class class class wbr 5143  cmpt 5225   × cxp 5683  ccnv 5684  dom cdm 5685  ran crn 5686  cres 5687  suc csuc 6386  Fun wfun 6555  wf 6557  1-1wf1 6558  1-1-ontowf1o 6560  cfv 6561  (class class class)co 7431  cmpo 7433  ωcom 7887  1st c1st 8012  2nd c2nd 8013  seqωcseqom 8487  m cmap 8866
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8014  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-seqom 8488  df-1o 8506  df-map 8868
This theorem is referenced by:  fseqen  10067  pwfseqlem5  10703
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