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Theorem fseqenlem2 10062
Description: Lemma for fseqen 10064. (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 4999 . . . . 5 (𝑦 𝑘 ∈ ω (𝐴m 𝑘) ↔ ∃𝑘 ∈ ω 𝑦 ∈ (𝐴m 𝑘))
2 elmapi 8887 . . . . . . . . . 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 2838 . . . . . . 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 10061 . . . . . . . . . . 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 7104 . . . . . . . 8 ((𝜑 ∧ (𝑘 ∈ ω ∧ 𝑦 ∈ (𝐴m 𝑘))) → ((𝐺𝑘)‘𝑦) ∈ 𝐴)
198, 18eqeltrd 2838 . . . . . . 7 ((𝜑 ∧ (𝑘 ∈ ω ∧ 𝑦 ∈ (𝐴m 𝑘))) → ((𝐺‘dom 𝑦)‘𝑦) ∈ 𝐴)
206, 19opelxpd 5727 . . . . . 6 ((𝜑 ∧ (𝑘 ∈ ω ∧ 𝑦 ∈ (𝐴m 𝑘))) → ⟨dom 𝑦, ((𝐺‘dom 𝑦)‘𝑦)⟩ ∈ (ω × 𝐴))
2120rexlimdvaa 3153 . . . . 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 7133 . 2 (𝜑𝐾: 𝑘 ∈ ω (𝐴m 𝑘)⟶(ω × 𝐴))
26 ffun 6739 . . . . . . . . . . . . . . 15 (𝐾: 𝑘 ∈ ω (𝐴m 𝑘)⟶(ω × 𝐴) → Fun 𝐾)
27 funbrfv2b 6965 . . . . . . . . . . . . . . 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 2840 . . . . . . . . . . . . . 14 ((𝜑𝑧𝐾𝑤) → 𝑧 𝑘 ∈ ω (𝐴m 𝑘))
34 dmeq 5916 . . . . . . . . . . . . . . . 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 4885 . . . . . . . . . . . . . . 15 (𝑦 = 𝑧 → ⟨dom 𝑦, ((𝐺‘dom 𝑦)‘𝑦)⟩ = ⟨dom 𝑧, ((𝐺‘dom 𝑧)‘𝑧)⟩)
39 opex 5474 . . . . . . . . . . . . . . 15 ⟨dom 𝑧, ((𝐺‘dom 𝑧)‘𝑧)⟩ ∈ V
4038, 24, 39fvmpt 7015 . . . . . . . . . . . . . 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 2776 . . . . . . . . . . . 12 ((𝜑𝑧𝐾𝑤) → 𝑤 = ⟨dom 𝑧, ((𝐺‘dom 𝑧)‘𝑧)⟩)
4342fveq2d 6910 . . . . . . . . . . 11 ((𝜑𝑧𝐾𝑤) → (1st𝑤) = (1st ‘⟨dom 𝑧, ((𝐺‘dom 𝑧)‘𝑧)⟩))
44 vex 3481 . . . . . . . . . . . . 13 𝑧 ∈ V
4544dmex 7931 . . . . . . . . . . . 12 dom 𝑧 ∈ V
46 fvex 6919 . . . . . . . . . . . 12 ((𝐺‘dom 𝑧)‘𝑧) ∈ V
4745, 46op1st 8020 . . . . . . . . . . 11 (1st ‘⟨dom 𝑧, ((𝐺‘dom 𝑧)‘𝑧)⟩) = dom 𝑧
4843, 47eqtrdi 2790 . . . . . . . . . 10 ((𝜑𝑧𝐾𝑤) → (1st𝑤) = dom 𝑧)
4948fveq2d 6910 . . . . . . . . 9 ((𝜑𝑧𝐾𝑤) → (𝐺‘(1st𝑤)) = (𝐺‘dom 𝑧))
5049cnveqd 5888 . . . . . . . 8 ((𝜑𝑧𝐾𝑤) → (𝐺‘(1st𝑤)) = (𝐺‘dom 𝑧))
5142fveq2d 6910 . . . . . . . . 9 ((𝜑𝑧𝐾𝑤) → (2nd𝑤) = (2nd ‘⟨dom 𝑧, ((𝐺‘dom 𝑧)‘𝑧)⟩))
5245, 46op2nd 8021 . . . . . . . . 9 (2nd ‘⟨dom 𝑧, ((𝐺‘dom 𝑧)‘𝑧)⟩) = ((𝐺‘dom 𝑧)‘𝑧)
5351, 52eqtrdi 2790 . . . . . . . 8 ((𝜑𝑧𝐾𝑤) → (2nd𝑤) = ((𝐺‘dom 𝑧)‘𝑧))
5450, 53fveq12d 6913 . . . . . . 7 ((𝜑𝑧𝐾𝑤) → ((𝐺‘(1st𝑤))‘(2nd𝑤)) = ((𝐺‘dom 𝑧)‘((𝐺‘dom 𝑧)‘𝑧)))
55 eliun 4999 . . . . . . . . . . . . 13 (𝑧 𝑘 ∈ ω (𝐴m 𝑘) ↔ ∃𝑘 ∈ ω 𝑧 ∈ (𝐴m 𝑘))
56 elmapi 8887 . . . . . . . . . . . . . . . . . 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 2838 . . . . . . . . . . . . . . 15 ((𝑘 ∈ ω ∧ 𝑧 ∈ (𝐴m 𝑘)) → dom 𝑧 ∈ ω)
61 simpr 484 . . . . . . . . . . . . . . . 16 ((𝑘 ∈ ω ∧ 𝑧 ∈ (𝐴m 𝑘)) → 𝑧 ∈ (𝐴m 𝑘))
6258oveq2d 7446 . . . . . . . . . . . . . . . 16 ((𝑘 ∈ ω ∧ 𝑧 ∈ (𝐴m 𝑘)) → (𝐴m dom 𝑧) = (𝐴m 𝑘))
6361, 62eleqtrrd 2841 . . . . . . . . . . . . . . 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 3144 . . . . . . . . . . . . 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 10061 . . . . . . . . . 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 7295 . . . . . . . 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 2775 . . . . . 6 ((𝜑𝑧𝐾𝑤) → 𝑧 = ((𝐺‘(1st𝑤))‘(2nd𝑤)))
7776ex 412 . . . . 5 (𝜑 → (𝑧𝐾𝑤𝑧 = ((𝐺‘(1st𝑤))‘(2nd𝑤))))
7877alrimiv 1924 . . . 4 (𝜑 → ∀𝑧(𝑧𝐾𝑤𝑧 = ((𝐺‘(1st𝑤))‘(2nd𝑤))))
79 mo2icl 3722 . . . 4 (∀𝑧(𝑧𝐾𝑤𝑧 = ((𝐺‘(1st𝑤))‘(2nd𝑤))) → ∃*𝑧 𝑧𝐾𝑤)
8078, 79syl 17 . . 3 (𝜑 → ∃*𝑧 𝑧𝐾𝑤)
8180alrimiv 1924 . 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 1534   = wceq 1536  wcel 2105  ∃*wmo 2535  wrex 3067  Vcvv 3477  c0 4338  {csn 4630  cop 4636   ciun 4995   class class class wbr 5147  cmpt 5230   × cxp 5686  ccnv 5687  dom cdm 5688  ran crn 5689  cres 5690  suc csuc 6387  Fun wfun 6556  wf 6558  1-1wf1 6559  1-1-ontowf1o 6561  cfv 6562  (class class class)co 7430  cmpo 7432  ωcom 7886  1st c1st 8010  2nd c2nd 8011  seqωcseqom 8485  m cmap 8864
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-rep 5284  ax-sep 5301  ax-nul 5311  ax-pow 5370  ax-pr 5437  ax-un 7753
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-ral 3059  df-rex 3068  df-reu 3378  df-rab 3433  df-v 3479  df-sbc 3791  df-csb 3908  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-pss 3982  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-iun 4997  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5582  df-eprel 5588  df-po 5596  df-so 5597  df-fr 5640  df-we 5642  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-pred 6322  df-ord 6388  df-on 6389  df-lim 6390  df-suc 6391  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-f1 6567  df-fo 6568  df-f1o 6569  df-fv 6570  df-ov 7433  df-oprab 7434  df-mpo 7435  df-om 7887  df-1st 8012  df-2nd 8013  df-frecs 8304  df-wrecs 8335  df-recs 8409  df-rdg 8448  df-seqom 8486  df-1o 8504  df-map 8866
This theorem is referenced by:  fseqen  10064  pwfseqlem5  10700
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