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Theorem fseqenlem2 10050
Description: Lemma for fseqen 10052. (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 5001 . . . . 5 (𝑦 𝑘 ∈ ω (𝐴m 𝑘) ↔ ∃𝑘 ∈ ω 𝑦 ∈ (𝐴m 𝑘))
2 elmapi 8868 . . . . . . . . . 10 (𝑦 ∈ (𝐴m 𝑘) → 𝑦:𝑘𝐴)
32ad2antll 727 . . . . . . . . 9 ((𝜑 ∧ (𝑘 ∈ ω ∧ 𝑦 ∈ (𝐴m 𝑘))) → 𝑦:𝑘𝐴)
43fdmd 6733 . . . . . . . 8 ((𝜑 ∧ (𝑘 ∈ ω ∧ 𝑦 ∈ (𝐴m 𝑘))) → dom 𝑦 = 𝑘)
5 simprl 769 . . . . . . . 8 ((𝜑 ∧ (𝑘 ∈ ω ∧ 𝑦 ∈ (𝐴m 𝑘))) → 𝑘 ∈ ω)
64, 5eqeltrd 2825 . . . . . . 7 ((𝜑 ∧ (𝑘 ∈ ω ∧ 𝑦 ∈ (𝐴m 𝑘))) → dom 𝑦 ∈ ω)
74fveq2d 6900 . . . . . . . . 9 ((𝜑 ∧ (𝑘 ∈ ω ∧ 𝑦 ∈ (𝐴m 𝑘))) → (𝐺‘dom 𝑦) = (𝐺𝑘))
87fveq1d 6898 . . . . . . . 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 10049 . . . . . . . . . . 11 ((𝜑𝑘 ∈ ω) → (𝐺𝑘):(𝐴m 𝑘)–1-1𝐴)
1413adantrr 715 . . . . . . . . . 10 ((𝜑 ∧ (𝑘 ∈ ω ∧ 𝑦 ∈ (𝐴m 𝑘))) → (𝐺𝑘):(𝐴m 𝑘)–1-1𝐴)
15 f1f 6793 . . . . . . . . . 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 771 . . . . . . . . 9 ((𝜑 ∧ (𝑘 ∈ ω ∧ 𝑦 ∈ (𝐴m 𝑘))) → 𝑦 ∈ (𝐴m 𝑘))
1816, 17ffvelcdmd 7094 . . . . . . . 8 ((𝜑 ∧ (𝑘 ∈ ω ∧ 𝑦 ∈ (𝐴m 𝑘))) → ((𝐺𝑘)‘𝑦) ∈ 𝐴)
198, 18eqeltrd 2825 . . . . . . 7 ((𝜑 ∧ (𝑘 ∈ ω ∧ 𝑦 ∈ (𝐴m 𝑘))) → ((𝐺‘dom 𝑦)‘𝑦) ∈ 𝐴)
206, 19opelxpd 5717 . . . . . 6 ((𝜑 ∧ (𝑘 ∈ ω ∧ 𝑦 ∈ (𝐴m 𝑘))) → ⟨dom 𝑦, ((𝐺‘dom 𝑦)‘𝑦)⟩ ∈ (ω × 𝐴))
2120rexlimdvaa 3145 . . . . 5 (𝜑 → (∃𝑘 ∈ ω 𝑦 ∈ (𝐴m 𝑘) → ⟨dom 𝑦, ((𝐺‘dom 𝑦)‘𝑦)⟩ ∈ (ω × 𝐴)))
221, 21biimtrid 241 . . . 4 (𝜑 → (𝑦 𝑘 ∈ ω (𝐴m 𝑘) → ⟨dom 𝑦, ((𝐺‘dom 𝑦)‘𝑦)⟩ ∈ (ω × 𝐴)))
2322imp 405 . . 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 7123 . 2 (𝜑𝐾: 𝑘 ∈ ω (𝐴m 𝑘)⟶(ω × 𝐴))
26 ffun 6726 . . . . . . . . . . . . . . 15 (𝐾: 𝑘 ∈ ω (𝐴m 𝑘)⟶(ω × 𝐴) → Fun 𝐾)
27 funbrfv2b 6955 . . . . . . . . . . . . . . 15 (Fun 𝐾 → (𝑧𝐾𝑤 ↔ (𝑧 ∈ dom 𝐾 ∧ (𝐾𝑧) = 𝑤)))
2825, 26, 273syl 18 . . . . . . . . . . . . . 14 (𝜑 → (𝑧𝐾𝑤 ↔ (𝑧 ∈ dom 𝐾 ∧ (𝐾𝑧) = 𝑤)))
2928simplbda 498 . . . . . . . . . . . . 13 ((𝜑𝑧𝐾𝑤) → (𝐾𝑧) = 𝑤)
3028simprbda 497 . . . . . . . . . . . . . . 15 ((𝜑𝑧𝐾𝑤) → 𝑧 ∈ dom 𝐾)
3125fdmd 6733 . . . . . . . . . . . . . . . 16 (𝜑 → dom 𝐾 = 𝑘 ∈ ω (𝐴m 𝑘))
3231adantr 479 . . . . . . . . . . . . . . 15 ((𝜑𝑧𝐾𝑤) → dom 𝐾 = 𝑘 ∈ ω (𝐴m 𝑘))
3330, 32eleqtrd 2827 . . . . . . . . . . . . . 14 ((𝜑𝑧𝐾𝑤) → 𝑧 𝑘 ∈ ω (𝐴m 𝑘))
34 dmeq 5906 . . . . . . . . . . . . . . . 16 (𝑦 = 𝑧 → dom 𝑦 = dom 𝑧)
3534fveq2d 6900 . . . . . . . . . . . . . . . . 17 (𝑦 = 𝑧 → (𝐺‘dom 𝑦) = (𝐺‘dom 𝑧))
36 id 22 . . . . . . . . . . . . . . . . 17 (𝑦 = 𝑧𝑦 = 𝑧)
3735, 36fveq12d 6903 . . . . . . . . . . . . . . . 16 (𝑦 = 𝑧 → ((𝐺‘dom 𝑦)‘𝑦) = ((𝐺‘dom 𝑧)‘𝑧))
3834, 37opeq12d 4883 . . . . . . . . . . . . . . 15 (𝑦 = 𝑧 → ⟨dom 𝑦, ((𝐺‘dom 𝑦)‘𝑦)⟩ = ⟨dom 𝑧, ((𝐺‘dom 𝑧)‘𝑧)⟩)
39 opex 5466 . . . . . . . . . . . . . . 15 ⟨dom 𝑧, ((𝐺‘dom 𝑧)‘𝑧)⟩ ∈ V
4038, 24, 39fvmpt 7004 . . . . . . . . . . . . . 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 2767 . . . . . . . . . . . 12 ((𝜑𝑧𝐾𝑤) → 𝑤 = ⟨dom 𝑧, ((𝐺‘dom 𝑧)‘𝑧)⟩)
4342fveq2d 6900 . . . . . . . . . . 11 ((𝜑𝑧𝐾𝑤) → (1st𝑤) = (1st ‘⟨dom 𝑧, ((𝐺‘dom 𝑧)‘𝑧)⟩))
44 vex 3465 . . . . . . . . . . . . 13 𝑧 ∈ V
4544dmex 7917 . . . . . . . . . . . 12 dom 𝑧 ∈ V
46 fvex 6909 . . . . . . . . . . . 12 ((𝐺‘dom 𝑧)‘𝑧) ∈ V
4745, 46op1st 8002 . . . . . . . . . . 11 (1st ‘⟨dom 𝑧, ((𝐺‘dom 𝑧)‘𝑧)⟩) = dom 𝑧
4843, 47eqtrdi 2781 . . . . . . . . . 10 ((𝜑𝑧𝐾𝑤) → (1st𝑤) = dom 𝑧)
4948fveq2d 6900 . . . . . . . . 9 ((𝜑𝑧𝐾𝑤) → (𝐺‘(1st𝑤)) = (𝐺‘dom 𝑧))
5049cnveqd 5878 . . . . . . . 8 ((𝜑𝑧𝐾𝑤) → (𝐺‘(1st𝑤)) = (𝐺‘dom 𝑧))
5142fveq2d 6900 . . . . . . . . 9 ((𝜑𝑧𝐾𝑤) → (2nd𝑤) = (2nd ‘⟨dom 𝑧, ((𝐺‘dom 𝑧)‘𝑧)⟩))
5245, 46op2nd 8003 . . . . . . . . 9 (2nd ‘⟨dom 𝑧, ((𝐺‘dom 𝑧)‘𝑧)⟩) = ((𝐺‘dom 𝑧)‘𝑧)
5351, 52eqtrdi 2781 . . . . . . . 8 ((𝜑𝑧𝐾𝑤) → (2nd𝑤) = ((𝐺‘dom 𝑧)‘𝑧))
5450, 53fveq12d 6903 . . . . . . 7 ((𝜑𝑧𝐾𝑤) → ((𝐺‘(1st𝑤))‘(2nd𝑤)) = ((𝐺‘dom 𝑧)‘((𝐺‘dom 𝑧)‘𝑧)))
55 eliun 5001 . . . . . . . . . . . . 13 (𝑧 𝑘 ∈ ω (𝐴m 𝑘) ↔ ∃𝑘 ∈ ω 𝑧 ∈ (𝐴m 𝑘))
56 elmapi 8868 . . . . . . . . . . . . . . . . . 18 (𝑧 ∈ (𝐴m 𝑘) → 𝑧:𝑘𝐴)
5756adantl 480 . . . . . . . . . . . . . . . . 17 ((𝑘 ∈ ω ∧ 𝑧 ∈ (𝐴m 𝑘)) → 𝑧:𝑘𝐴)
5857fdmd 6733 . . . . . . . . . . . . . . . 16 ((𝑘 ∈ ω ∧ 𝑧 ∈ (𝐴m 𝑘)) → dom 𝑧 = 𝑘)
59 simpl 481 . . . . . . . . . . . . . . . 16 ((𝑘 ∈ ω ∧ 𝑧 ∈ (𝐴m 𝑘)) → 𝑘 ∈ ω)
6058, 59eqeltrd 2825 . . . . . . . . . . . . . . 15 ((𝑘 ∈ ω ∧ 𝑧 ∈ (𝐴m 𝑘)) → dom 𝑧 ∈ ω)
61 simpr 483 . . . . . . . . . . . . . . . 16 ((𝑘 ∈ ω ∧ 𝑧 ∈ (𝐴m 𝑘)) → 𝑧 ∈ (𝐴m 𝑘))
6258oveq2d 7435 . . . . . . . . . . . . . . . 16 ((𝑘 ∈ ω ∧ 𝑧 ∈ (𝐴m 𝑘)) → (𝐴m dom 𝑧) = (𝐴m 𝑘))
6361, 62eleqtrrd 2828 . . . . . . . . . . . . . . 15 ((𝑘 ∈ ω ∧ 𝑧 ∈ (𝐴m 𝑘)) → 𝑧 ∈ (𝐴m dom 𝑧))
6460, 63jca 510 . . . . . . . . . . . . . 14 ((𝑘 ∈ ω ∧ 𝑧 ∈ (𝐴m 𝑘)) → (dom 𝑧 ∈ ω ∧ 𝑧 ∈ (𝐴m dom 𝑧)))
6564rexlimiva 3136 . . . . . . . . . . . . 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 493 . . . . . . . . . 10 ((𝜑𝑧𝐾𝑤) → dom 𝑧 ∈ ω)
699, 10, 11, 12fseqenlem1 10049 . . . . . . . . . 10 ((𝜑 ∧ dom 𝑧 ∈ ω) → (𝐺‘dom 𝑧):(𝐴m dom 𝑧)–1-1𝐴)
7068, 69syldan 589 . . . . . . . . 9 ((𝜑𝑧𝐾𝑤) → (𝐺‘dom 𝑧):(𝐴m dom 𝑧)–1-1𝐴)
71 f1f1orn 6849 . . . . . . . . 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 494 . . . . . . . 8 ((𝜑𝑧𝐾𝑤) → 𝑧 ∈ (𝐴m dom 𝑧))
74 f1ocnvfv1 7285 . . . . . . . 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 582 . . . . . . 7 ((𝜑𝑧𝐾𝑤) → ((𝐺‘dom 𝑧)‘((𝐺‘dom 𝑧)‘𝑧)) = 𝑧)
7654, 75eqtr2d 2766 . . . . . 6 ((𝜑𝑧𝐾𝑤) → 𝑧 = ((𝐺‘(1st𝑤))‘(2nd𝑤)))
7776ex 411 . . . . 5 (𝜑 → (𝑧𝐾𝑤𝑧 = ((𝐺‘(1st𝑤))‘(2nd𝑤))))
7877alrimiv 1922 . . . 4 (𝜑 → ∀𝑧(𝑧𝐾𝑤𝑧 = ((𝐺‘(1st𝑤))‘(2nd𝑤))))
79 mo2icl 3706 . . . 4 (∀𝑧(𝑧𝐾𝑤𝑧 = ((𝐺‘(1st𝑤))‘(2nd𝑤))) → ∃*𝑧 𝑧𝐾𝑤)
8078, 79syl 17 . . 3 (𝜑 → ∃*𝑧 𝑧𝐾𝑤)
8180alrimiv 1922 . 2 (𝜑 → ∀𝑤∃*𝑧 𝑧𝐾𝑤)
82 dff12 6792 . 2 (𝐾: 𝑘 ∈ ω (𝐴m 𝑘)–1-1→(ω × 𝐴) ↔ (𝐾: 𝑘 ∈ ω (𝐴m 𝑘)⟶(ω × 𝐴) ∧ ∀𝑤∃*𝑧 𝑧𝐾𝑤))
8325, 81, 82sylanbrc 581 1 (𝜑𝐾: 𝑘 ∈ ω (𝐴m 𝑘)–1-1→(ω × 𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 394  wal 1531   = wceq 1533  wcel 2098  ∃*wmo 2526  wrex 3059  Vcvv 3461  c0 4322  {csn 4630  cop 4636   ciun 4997   class class class wbr 5149  cmpt 5232   × cxp 5676  ccnv 5677  dom cdm 5678  ran crn 5679  cres 5680  suc csuc 6373  Fun wfun 6543  wf 6545  1-1wf1 6546  1-1-ontowf1o 6548  cfv 6549  (class class class)co 7419  cmpo 7421  ωcom 7871  1st c1st 7992  2nd c2nd 7993  seqωcseqom 8468  m cmap 8845
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5365  ax-pr 5429  ax-un 7741
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2930  df-ral 3051  df-rex 3060  df-reu 3364  df-rab 3419  df-v 3463  df-sbc 3774  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-pss 3964  df-nul 4323  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4910  df-iun 4999  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5576  df-eprel 5582  df-po 5590  df-so 5591  df-fr 5633  df-we 5635  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-pred 6307  df-ord 6374  df-on 6375  df-lim 6376  df-suc 6377  df-iota 6501  df-fun 6551  df-fn 6552  df-f 6553  df-f1 6554  df-fo 6555  df-f1o 6556  df-fv 6557  df-ov 7422  df-oprab 7423  df-mpo 7424  df-om 7872  df-1st 7994  df-2nd 7995  df-frecs 8287  df-wrecs 8318  df-recs 8392  df-rdg 8431  df-seqom 8469  df-1o 8487  df-map 8847
This theorem is referenced by:  fseqen  10052  pwfseqlem5  10688
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