Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  fseqenlem2 Structured version   Visualization version   GIF version

Theorem fseqenlem2 9440
 Description: Lemma for fseqen 9442. (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 4898 . . . . 5 (𝑦 𝑘 ∈ ω (𝐴m 𝑘) ↔ ∃𝑘 ∈ ω 𝑦 ∈ (𝐴m 𝑘))
2 elmapi 8415 . . . . . . . . . 10 (𝑦 ∈ (𝐴m 𝑘) → 𝑦:𝑘𝐴)
32ad2antll 728 . . . . . . . . 9 ((𝜑 ∧ (𝑘 ∈ ω ∧ 𝑦 ∈ (𝐴m 𝑘))) → 𝑦:𝑘𝐴)
43fdmd 6504 . . . . . . . 8 ((𝜑 ∧ (𝑘 ∈ ω ∧ 𝑦 ∈ (𝐴m 𝑘))) → dom 𝑦 = 𝑘)
5 simprl 770 . . . . . . . 8 ((𝜑 ∧ (𝑘 ∈ ω ∧ 𝑦 ∈ (𝐴m 𝑘))) → 𝑘 ∈ ω)
64, 5eqeltrd 2914 . . . . . . 7 ((𝜑 ∧ (𝑘 ∈ ω ∧ 𝑦 ∈ (𝐴m 𝑘))) → dom 𝑦 ∈ ω)
74fveq2d 6656 . . . . . . . . 9 ((𝜑 ∧ (𝑘 ∈ ω ∧ 𝑦 ∈ (𝐴m 𝑘))) → (𝐺‘dom 𝑦) = (𝐺𝑘))
87fveq1d 6654 . . . . . . . 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 9439 . . . . . . . . . . 11 ((𝜑𝑘 ∈ ω) → (𝐺𝑘):(𝐴m 𝑘)–1-1𝐴)
1413adantrr 716 . . . . . . . . . 10 ((𝜑 ∧ (𝑘 ∈ ω ∧ 𝑦 ∈ (𝐴m 𝑘))) → (𝐺𝑘):(𝐴m 𝑘)–1-1𝐴)
15 f1f 6556 . . . . . . . . . 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 772 . . . . . . . . 9 ((𝜑 ∧ (𝑘 ∈ ω ∧ 𝑦 ∈ (𝐴m 𝑘))) → 𝑦 ∈ (𝐴m 𝑘))
1816, 17ffvelrnd 6834 . . . . . . . 8 ((𝜑 ∧ (𝑘 ∈ ω ∧ 𝑦 ∈ (𝐴m 𝑘))) → ((𝐺𝑘)‘𝑦) ∈ 𝐴)
198, 18eqeltrd 2914 . . . . . . 7 ((𝜑 ∧ (𝑘 ∈ ω ∧ 𝑦 ∈ (𝐴m 𝑘))) → ((𝐺‘dom 𝑦)‘𝑦) ∈ 𝐴)
206, 19opelxpd 5570 . . . . . 6 ((𝜑 ∧ (𝑘 ∈ ω ∧ 𝑦 ∈ (𝐴m 𝑘))) → ⟨dom 𝑦, ((𝐺‘dom 𝑦)‘𝑦)⟩ ∈ (ω × 𝐴))
2120rexlimdvaa 3271 . . . . 5 (𝜑 → (∃𝑘 ∈ ω 𝑦 ∈ (𝐴m 𝑘) → ⟨dom 𝑦, ((𝐺‘dom 𝑦)‘𝑦)⟩ ∈ (ω × 𝐴)))
221, 21syl5bi 245 . . . 4 (𝜑 → (𝑦 𝑘 ∈ ω (𝐴m 𝑘) → ⟨dom 𝑦, ((𝐺‘dom 𝑦)‘𝑦)⟩ ∈ (ω × 𝐴)))
2322imp 410 . . 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 6860 . 2 (𝜑𝐾: 𝑘 ∈ ω (𝐴m 𝑘)⟶(ω × 𝐴))
26 ffun 6497 . . . . . . . . . . . . . . 15 (𝐾: 𝑘 ∈ ω (𝐴m 𝑘)⟶(ω × 𝐴) → Fun 𝐾)
27 funbrfv2b 6705 . . . . . . . . . . . . . . 15 (Fun 𝐾 → (𝑧𝐾𝑤 ↔ (𝑧 ∈ dom 𝐾 ∧ (𝐾𝑧) = 𝑤)))
2825, 26, 273syl 18 . . . . . . . . . . . . . 14 (𝜑 → (𝑧𝐾𝑤 ↔ (𝑧 ∈ dom 𝐾 ∧ (𝐾𝑧) = 𝑤)))
2928simplbda 503 . . . . . . . . . . . . 13 ((𝜑𝑧𝐾𝑤) → (𝐾𝑧) = 𝑤)
3028simprbda 502 . . . . . . . . . . . . . . 15 ((𝜑𝑧𝐾𝑤) → 𝑧 ∈ dom 𝐾)
3125fdmd 6504 . . . . . . . . . . . . . . . 16 (𝜑 → dom 𝐾 = 𝑘 ∈ ω (𝐴m 𝑘))
3231adantr 484 . . . . . . . . . . . . . . 15 ((𝜑𝑧𝐾𝑤) → dom 𝐾 = 𝑘 ∈ ω (𝐴m 𝑘))
3330, 32eleqtrd 2916 . . . . . . . . . . . . . 14 ((𝜑𝑧𝐾𝑤) → 𝑧 𝑘 ∈ ω (𝐴m 𝑘))
34 dmeq 5749 . . . . . . . . . . . . . . . 16 (𝑦 = 𝑧 → dom 𝑦 = dom 𝑧)
3534fveq2d 6656 . . . . . . . . . . . . . . . . 17 (𝑦 = 𝑧 → (𝐺‘dom 𝑦) = (𝐺‘dom 𝑧))
36 id 22 . . . . . . . . . . . . . . . . 17 (𝑦 = 𝑧𝑦 = 𝑧)
3735, 36fveq12d 6659 . . . . . . . . . . . . . . . 16 (𝑦 = 𝑧 → ((𝐺‘dom 𝑦)‘𝑦) = ((𝐺‘dom 𝑧)‘𝑧))
3834, 37opeq12d 4786 . . . . . . . . . . . . . . 15 (𝑦 = 𝑧 → ⟨dom 𝑦, ((𝐺‘dom 𝑦)‘𝑦)⟩ = ⟨dom 𝑧, ((𝐺‘dom 𝑧)‘𝑧)⟩)
39 opex 5333 . . . . . . . . . . . . . . 15 ⟨dom 𝑧, ((𝐺‘dom 𝑧)‘𝑧)⟩ ∈ V
4038, 24, 39fvmpt 6750 . . . . . . . . . . . . . 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 2859 . . . . . . . . . . . 12 ((𝜑𝑧𝐾𝑤) → 𝑤 = ⟨dom 𝑧, ((𝐺‘dom 𝑧)‘𝑧)⟩)
4342fveq2d 6656 . . . . . . . . . . 11 ((𝜑𝑧𝐾𝑤) → (1st𝑤) = (1st ‘⟨dom 𝑧, ((𝐺‘dom 𝑧)‘𝑧)⟩))
44 vex 3472 . . . . . . . . . . . . 13 𝑧 ∈ V
4544dmex 7602 . . . . . . . . . . . 12 dom 𝑧 ∈ V
46 fvex 6665 . . . . . . . . . . . 12 ((𝐺‘dom 𝑧)‘𝑧) ∈ V
4745, 46op1st 7683 . . . . . . . . . . 11 (1st ‘⟨dom 𝑧, ((𝐺‘dom 𝑧)‘𝑧)⟩) = dom 𝑧
4843, 47syl6eq 2873 . . . . . . . . . 10 ((𝜑𝑧𝐾𝑤) → (1st𝑤) = dom 𝑧)
4948fveq2d 6656 . . . . . . . . 9 ((𝜑𝑧𝐾𝑤) → (𝐺‘(1st𝑤)) = (𝐺‘dom 𝑧))
5049cnveqd 5723 . . . . . . . 8 ((𝜑𝑧𝐾𝑤) → (𝐺‘(1st𝑤)) = (𝐺‘dom 𝑧))
5142fveq2d 6656 . . . . . . . . 9 ((𝜑𝑧𝐾𝑤) → (2nd𝑤) = (2nd ‘⟨dom 𝑧, ((𝐺‘dom 𝑧)‘𝑧)⟩))
5245, 46op2nd 7684 . . . . . . . . 9 (2nd ‘⟨dom 𝑧, ((𝐺‘dom 𝑧)‘𝑧)⟩) = ((𝐺‘dom 𝑧)‘𝑧)
5351, 52syl6eq 2873 . . . . . . . 8 ((𝜑𝑧𝐾𝑤) → (2nd𝑤) = ((𝐺‘dom 𝑧)‘𝑧))
5450, 53fveq12d 6659 . . . . . . 7 ((𝜑𝑧𝐾𝑤) → ((𝐺‘(1st𝑤))‘(2nd𝑤)) = ((𝐺‘dom 𝑧)‘((𝐺‘dom 𝑧)‘𝑧)))
55 eliun 4898 . . . . . . . . . . . . 13 (𝑧 𝑘 ∈ ω (𝐴m 𝑘) ↔ ∃𝑘 ∈ ω 𝑧 ∈ (𝐴m 𝑘))
56 elmapi 8415 . . . . . . . . . . . . . . . . . 18 (𝑧 ∈ (𝐴m 𝑘) → 𝑧:𝑘𝐴)
5756adantl 485 . . . . . . . . . . . . . . . . 17 ((𝑘 ∈ ω ∧ 𝑧 ∈ (𝐴m 𝑘)) → 𝑧:𝑘𝐴)
5857fdmd 6504 . . . . . . . . . . . . . . . 16 ((𝑘 ∈ ω ∧ 𝑧 ∈ (𝐴m 𝑘)) → dom 𝑧 = 𝑘)
59 simpl 486 . . . . . . . . . . . . . . . 16 ((𝑘 ∈ ω ∧ 𝑧 ∈ (𝐴m 𝑘)) → 𝑘 ∈ ω)
6058, 59eqeltrd 2914 . . . . . . . . . . . . . . 15 ((𝑘 ∈ ω ∧ 𝑧 ∈ (𝐴m 𝑘)) → dom 𝑧 ∈ ω)
61 simpr 488 . . . . . . . . . . . . . . . 16 ((𝑘 ∈ ω ∧ 𝑧 ∈ (𝐴m 𝑘)) → 𝑧 ∈ (𝐴m 𝑘))
6258oveq2d 7156 . . . . . . . . . . . . . . . 16 ((𝑘 ∈ ω ∧ 𝑧 ∈ (𝐴m 𝑘)) → (𝐴m dom 𝑧) = (𝐴m 𝑘))
6361, 62eleqtrrd 2917 . . . . . . . . . . . . . . 15 ((𝑘 ∈ ω ∧ 𝑧 ∈ (𝐴m 𝑘)) → 𝑧 ∈ (𝐴m dom 𝑧))
6460, 63jca 515 . . . . . . . . . . . . . 14 ((𝑘 ∈ ω ∧ 𝑧 ∈ (𝐴m 𝑘)) → (dom 𝑧 ∈ ω ∧ 𝑧 ∈ (𝐴m dom 𝑧)))
6564rexlimiva 3267 . . . . . . . . . . . . 13 (∃𝑘 ∈ ω 𝑧 ∈ (𝐴m 𝑘) → (dom 𝑧 ∈ ω ∧ 𝑧 ∈ (𝐴m dom 𝑧)))
6655, 65sylbi 220 . . . . . . . . . . . 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 498 . . . . . . . . . 10 ((𝜑𝑧𝐾𝑤) → dom 𝑧 ∈ ω)
699, 10, 11, 12fseqenlem1 9439 . . . . . . . . . 10 ((𝜑 ∧ dom 𝑧 ∈ ω) → (𝐺‘dom 𝑧):(𝐴m dom 𝑧)–1-1𝐴)
7068, 69syldan 594 . . . . . . . . 9 ((𝜑𝑧𝐾𝑤) → (𝐺‘dom 𝑧):(𝐴m dom 𝑧)–1-1𝐴)
71 f1f1orn 6608 . . . . . . . . 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 499 . . . . . . . 8 ((𝜑𝑧𝐾𝑤) → 𝑧 ∈ (𝐴m dom 𝑧))
74 f1ocnvfv1 7016 . . . . . . . 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 587 . . . . . . 7 ((𝜑𝑧𝐾𝑤) → ((𝐺‘dom 𝑧)‘((𝐺‘dom 𝑧)‘𝑧)) = 𝑧)
7654, 75eqtr2d 2858 . . . . . 6 ((𝜑𝑧𝐾𝑤) → 𝑧 = ((𝐺‘(1st𝑤))‘(2nd𝑤)))
7776ex 416 . . . . 5 (𝜑 → (𝑧𝐾𝑤𝑧 = ((𝐺‘(1st𝑤))‘(2nd𝑤))))
7877alrimiv 1928 . . . 4 (𝜑 → ∀𝑧(𝑧𝐾𝑤𝑧 = ((𝐺‘(1st𝑤))‘(2nd𝑤))))
79 mo2icl 3680 . . . 4 (∀𝑧(𝑧𝐾𝑤𝑧 = ((𝐺‘(1st𝑤))‘(2nd𝑤))) → ∃*𝑧 𝑧𝐾𝑤)
8078, 79syl 17 . . 3 (𝜑 → ∃*𝑧 𝑧𝐾𝑤)
8180alrimiv 1928 . 2 (𝜑 → ∀𝑤∃*𝑧 𝑧𝐾𝑤)
82 dff12 6555 . 2 (𝐾: 𝑘 ∈ ω (𝐴m 𝑘)–1-1→(ω × 𝐴) ↔ (𝐾: 𝑘 ∈ ω (𝐴m 𝑘)⟶(ω × 𝐴) ∧ ∀𝑤∃*𝑧 𝑧𝐾𝑤))
8325, 81, 82sylanbrc 586 1 (𝜑𝐾: 𝑘 ∈ ω (𝐴m 𝑘)–1-1→(ω × 𝐴))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399  ∀wal 1536   = wceq 1538   ∈ wcel 2114  ∃*wmo 2620  ∃wrex 3131  Vcvv 3469  ∅c0 4265  {csn 4539  ⟨cop 4545  ∪ ciun 4894   class class class wbr 5042   ↦ cmpt 5122   × cxp 5530  ◡ccnv 5531  dom cdm 5532  ran crn 5533   ↾ cres 5534  suc csuc 6171  Fun wfun 6328  ⟶wf 6330  –1-1→wf1 6331  –1-1-onto→wf1o 6333  ‘cfv 6334  (class class class)co 7140   ∈ cmpo 7142  ωcom 7565  1st c1st 7673  2nd c2nd 7674  seqωcseqom 8070   ↑m cmap 8393 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2178  ax-ext 2794  ax-rep 5166  ax-sep 5179  ax-nul 5186  ax-pow 5243  ax-pr 5307  ax-un 7446 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2801  df-cleq 2815  df-clel 2894  df-nfc 2962  df-ne 3012  df-ral 3135  df-rex 3136  df-reu 3137  df-rab 3139  df-v 3471  df-sbc 3748  df-csb 3856  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-pss 3927  df-nul 4266  df-if 4440  df-pw 4513  df-sn 4540  df-pr 4542  df-tp 4544  df-op 4546  df-uni 4814  df-iun 4896  df-br 5043  df-opab 5105  df-mpt 5123  df-tr 5149  df-id 5437  df-eprel 5442  df-po 5451  df-so 5452  df-fr 5491  df-we 5493  df-xp 5538  df-rel 5539  df-cnv 5540  df-co 5541  df-dm 5542  df-rn 5543  df-res 5544  df-ima 5545  df-pred 6126  df-ord 6172  df-on 6173  df-lim 6174  df-suc 6175  df-iota 6293  df-fun 6336  df-fn 6337  df-f 6338  df-f1 6339  df-fo 6340  df-f1o 6341  df-fv 6342  df-ov 7143  df-oprab 7144  df-mpo 7145  df-om 7566  df-1st 7675  df-2nd 7676  df-wrecs 7934  df-recs 7995  df-rdg 8033  df-seqom 8071  df-1o 8089  df-map 8395 This theorem is referenced by:  fseqen  9442  pwfseqlem5  10074
 Copyright terms: Public domain W3C validator