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Theorem fseqenlem2 8833
Description: Lemma for fseqen 8835. (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 4515 . . . . 5 (𝑦 𝑘 ∈ ω (𝐴𝑚 𝑘) ↔ ∃𝑘 ∈ ω 𝑦 ∈ (𝐴𝑚 𝑘))
2 elmapi 7864 . . . . . . . . . 10 (𝑦 ∈ (𝐴𝑚 𝑘) → 𝑦:𝑘𝐴)
32ad2antll 764 . . . . . . . . 9 ((𝜑 ∧ (𝑘 ∈ ω ∧ 𝑦 ∈ (𝐴𝑚 𝑘))) → 𝑦:𝑘𝐴)
4 fdm 6038 . . . . . . . . 9 (𝑦:𝑘𝐴 → dom 𝑦 = 𝑘)
53, 4syl 17 . . . . . . . 8 ((𝜑 ∧ (𝑘 ∈ ω ∧ 𝑦 ∈ (𝐴𝑚 𝑘))) → dom 𝑦 = 𝑘)
6 simprl 793 . . . . . . . 8 ((𝜑 ∧ (𝑘 ∈ ω ∧ 𝑦 ∈ (𝐴𝑚 𝑘))) → 𝑘 ∈ ω)
75, 6eqeltrd 2699 . . . . . . 7 ((𝜑 ∧ (𝑘 ∈ ω ∧ 𝑦 ∈ (𝐴𝑚 𝑘))) → dom 𝑦 ∈ ω)
85fveq2d 6182 . . . . . . . . 9 ((𝜑 ∧ (𝑘 ∈ ω ∧ 𝑦 ∈ (𝐴𝑚 𝑘))) → (𝐺‘dom 𝑦) = (𝐺𝑘))
98fveq1d 6180 . . . . . . . 8 ((𝜑 ∧ (𝑘 ∈ ω ∧ 𝑦 ∈ (𝐴𝑚 𝑘))) → ((𝐺‘dom 𝑦)‘𝑦) = ((𝐺𝑘)‘𝑦))
10 fseqenlem.a . . . . . . . . . . . 12 (𝜑𝐴𝑉)
11 fseqenlem.b . . . . . . . . . . . 12 (𝜑𝐵𝐴)
12 fseqenlem.f . . . . . . . . . . . 12 (𝜑𝐹:(𝐴 × 𝐴)–1-1-onto𝐴)
13 fseqenlem.g . . . . . . . . . . . 12 𝐺 = seq𝜔((𝑛 ∈ V, 𝑓 ∈ V ↦ (𝑥 ∈ (𝐴𝑚 suc 𝑛) ↦ ((𝑓‘(𝑥𝑛))𝐹(𝑥𝑛)))), {⟨∅, 𝐵⟩})
1410, 11, 12, 13fseqenlem1 8832 . . . . . . . . . . 11 ((𝜑𝑘 ∈ ω) → (𝐺𝑘):(𝐴𝑚 𝑘)–1-1𝐴)
1514adantrr 752 . . . . . . . . . 10 ((𝜑 ∧ (𝑘 ∈ ω ∧ 𝑦 ∈ (𝐴𝑚 𝑘))) → (𝐺𝑘):(𝐴𝑚 𝑘)–1-1𝐴)
16 f1f 6088 . . . . . . . . . 10 ((𝐺𝑘):(𝐴𝑚 𝑘)–1-1𝐴 → (𝐺𝑘):(𝐴𝑚 𝑘)⟶𝐴)
1715, 16syl 17 . . . . . . . . 9 ((𝜑 ∧ (𝑘 ∈ ω ∧ 𝑦 ∈ (𝐴𝑚 𝑘))) → (𝐺𝑘):(𝐴𝑚 𝑘)⟶𝐴)
18 simprr 795 . . . . . . . . 9 ((𝜑 ∧ (𝑘 ∈ ω ∧ 𝑦 ∈ (𝐴𝑚 𝑘))) → 𝑦 ∈ (𝐴𝑚 𝑘))
1917, 18ffvelrnd 6346 . . . . . . . 8 ((𝜑 ∧ (𝑘 ∈ ω ∧ 𝑦 ∈ (𝐴𝑚 𝑘))) → ((𝐺𝑘)‘𝑦) ∈ 𝐴)
209, 19eqeltrd 2699 . . . . . . 7 ((𝜑 ∧ (𝑘 ∈ ω ∧ 𝑦 ∈ (𝐴𝑚 𝑘))) → ((𝐺‘dom 𝑦)‘𝑦) ∈ 𝐴)
21 opelxpi 5138 . . . . . . 7 ((dom 𝑦 ∈ ω ∧ ((𝐺‘dom 𝑦)‘𝑦) ∈ 𝐴) → ⟨dom 𝑦, ((𝐺‘dom 𝑦)‘𝑦)⟩ ∈ (ω × 𝐴))
227, 20, 21syl2anc 692 . . . . . 6 ((𝜑 ∧ (𝑘 ∈ ω ∧ 𝑦 ∈ (𝐴𝑚 𝑘))) → ⟨dom 𝑦, ((𝐺‘dom 𝑦)‘𝑦)⟩ ∈ (ω × 𝐴))
2322rexlimdvaa 3028 . . . . 5 (𝜑 → (∃𝑘 ∈ ω 𝑦 ∈ (𝐴𝑚 𝑘) → ⟨dom 𝑦, ((𝐺‘dom 𝑦)‘𝑦)⟩ ∈ (ω × 𝐴)))
241, 23syl5bi 232 . . . 4 (𝜑 → (𝑦 𝑘 ∈ ω (𝐴𝑚 𝑘) → ⟨dom 𝑦, ((𝐺‘dom 𝑦)‘𝑦)⟩ ∈ (ω × 𝐴)))
2524imp 445 . . 3 ((𝜑𝑦 𝑘 ∈ ω (𝐴𝑚 𝑘)) → ⟨dom 𝑦, ((𝐺‘dom 𝑦)‘𝑦)⟩ ∈ (ω × 𝐴))
26 fseqenlem.k . . 3 𝐾 = (𝑦 𝑘 ∈ ω (𝐴𝑚 𝑘) ↦ ⟨dom 𝑦, ((𝐺‘dom 𝑦)‘𝑦)⟩)
2725, 26fmptd 6371 . 2 (𝜑𝐾: 𝑘 ∈ ω (𝐴𝑚 𝑘)⟶(ω × 𝐴))
28 ffun 6035 . . . . . . . . . . . . . . 15 (𝐾: 𝑘 ∈ ω (𝐴𝑚 𝑘)⟶(ω × 𝐴) → Fun 𝐾)
29 funbrfv2b 6227 . . . . . . . . . . . . . . 15 (Fun 𝐾 → (𝑧𝐾𝑤 ↔ (𝑧 ∈ dom 𝐾 ∧ (𝐾𝑧) = 𝑤)))
3027, 28, 293syl 18 . . . . . . . . . . . . . 14 (𝜑 → (𝑧𝐾𝑤 ↔ (𝑧 ∈ dom 𝐾 ∧ (𝐾𝑧) = 𝑤)))
3130simplbda 653 . . . . . . . . . . . . 13 ((𝜑𝑧𝐾𝑤) → (𝐾𝑧) = 𝑤)
3230simprbda 652 . . . . . . . . . . . . . . 15 ((𝜑𝑧𝐾𝑤) → 𝑧 ∈ dom 𝐾)
33 fdm 6038 . . . . . . . . . . . . . . . . 17 (𝐾: 𝑘 ∈ ω (𝐴𝑚 𝑘)⟶(ω × 𝐴) → dom 𝐾 = 𝑘 ∈ ω (𝐴𝑚 𝑘))
3427, 33syl 17 . . . . . . . . . . . . . . . 16 (𝜑 → dom 𝐾 = 𝑘 ∈ ω (𝐴𝑚 𝑘))
3534adantr 481 . . . . . . . . . . . . . . 15 ((𝜑𝑧𝐾𝑤) → dom 𝐾 = 𝑘 ∈ ω (𝐴𝑚 𝑘))
3632, 35eleqtrd 2701 . . . . . . . . . . . . . 14 ((𝜑𝑧𝐾𝑤) → 𝑧 𝑘 ∈ ω (𝐴𝑚 𝑘))
37 dmeq 5313 . . . . . . . . . . . . . . . 16 (𝑦 = 𝑧 → dom 𝑦 = dom 𝑧)
3837fveq2d 6182 . . . . . . . . . . . . . . . . 17 (𝑦 = 𝑧 → (𝐺‘dom 𝑦) = (𝐺‘dom 𝑧))
39 id 22 . . . . . . . . . . . . . . . . 17 (𝑦 = 𝑧𝑦 = 𝑧)
4038, 39fveq12d 6184 . . . . . . . . . . . . . . . 16 (𝑦 = 𝑧 → ((𝐺‘dom 𝑦)‘𝑦) = ((𝐺‘dom 𝑧)‘𝑧))
4137, 40opeq12d 4401 . . . . . . . . . . . . . . 15 (𝑦 = 𝑧 → ⟨dom 𝑦, ((𝐺‘dom 𝑦)‘𝑦)⟩ = ⟨dom 𝑧, ((𝐺‘dom 𝑧)‘𝑧)⟩)
42 opex 4923 . . . . . . . . . . . . . . 15 ⟨dom 𝑧, ((𝐺‘dom 𝑧)‘𝑧)⟩ ∈ V
4341, 26, 42fvmpt 6269 . . . . . . . . . . . . . 14 (𝑧 𝑘 ∈ ω (𝐴𝑚 𝑘) → (𝐾𝑧) = ⟨dom 𝑧, ((𝐺‘dom 𝑧)‘𝑧)⟩)
4436, 43syl 17 . . . . . . . . . . . . 13 ((𝜑𝑧𝐾𝑤) → (𝐾𝑧) = ⟨dom 𝑧, ((𝐺‘dom 𝑧)‘𝑧)⟩)
4531, 44eqtr3d 2656 . . . . . . . . . . . 12 ((𝜑𝑧𝐾𝑤) → 𝑤 = ⟨dom 𝑧, ((𝐺‘dom 𝑧)‘𝑧)⟩)
4645fveq2d 6182 . . . . . . . . . . 11 ((𝜑𝑧𝐾𝑤) → (1st𝑤) = (1st ‘⟨dom 𝑧, ((𝐺‘dom 𝑧)‘𝑧)⟩))
47 vex 3198 . . . . . . . . . . . . 13 𝑧 ∈ V
4847dmex 7084 . . . . . . . . . . . 12 dom 𝑧 ∈ V
49 fvex 6188 . . . . . . . . . . . 12 ((𝐺‘dom 𝑧)‘𝑧) ∈ V
5048, 49op1st 7161 . . . . . . . . . . 11 (1st ‘⟨dom 𝑧, ((𝐺‘dom 𝑧)‘𝑧)⟩) = dom 𝑧
5146, 50syl6eq 2670 . . . . . . . . . 10 ((𝜑𝑧𝐾𝑤) → (1st𝑤) = dom 𝑧)
5251fveq2d 6182 . . . . . . . . 9 ((𝜑𝑧𝐾𝑤) → (𝐺‘(1st𝑤)) = (𝐺‘dom 𝑧))
5352cnveqd 5287 . . . . . . . 8 ((𝜑𝑧𝐾𝑤) → (𝐺‘(1st𝑤)) = (𝐺‘dom 𝑧))
5445fveq2d 6182 . . . . . . . . 9 ((𝜑𝑧𝐾𝑤) → (2nd𝑤) = (2nd ‘⟨dom 𝑧, ((𝐺‘dom 𝑧)‘𝑧)⟩))
5548, 49op2nd 7162 . . . . . . . . 9 (2nd ‘⟨dom 𝑧, ((𝐺‘dom 𝑧)‘𝑧)⟩) = ((𝐺‘dom 𝑧)‘𝑧)
5654, 55syl6eq 2670 . . . . . . . 8 ((𝜑𝑧𝐾𝑤) → (2nd𝑤) = ((𝐺‘dom 𝑧)‘𝑧))
5753, 56fveq12d 6184 . . . . . . 7 ((𝜑𝑧𝐾𝑤) → ((𝐺‘(1st𝑤))‘(2nd𝑤)) = ((𝐺‘dom 𝑧)‘((𝐺‘dom 𝑧)‘𝑧)))
58 eliun 4515 . . . . . . . . . . . . 13 (𝑧 𝑘 ∈ ω (𝐴𝑚 𝑘) ↔ ∃𝑘 ∈ ω 𝑧 ∈ (𝐴𝑚 𝑘))
59 elmapi 7864 . . . . . . . . . . . . . . . . . 18 (𝑧 ∈ (𝐴𝑚 𝑘) → 𝑧:𝑘𝐴)
6059adantl 482 . . . . . . . . . . . . . . . . 17 ((𝑘 ∈ ω ∧ 𝑧 ∈ (𝐴𝑚 𝑘)) → 𝑧:𝑘𝐴)
61 fdm 6038 . . . . . . . . . . . . . . . . 17 (𝑧:𝑘𝐴 → dom 𝑧 = 𝑘)
6260, 61syl 17 . . . . . . . . . . . . . . . 16 ((𝑘 ∈ ω ∧ 𝑧 ∈ (𝐴𝑚 𝑘)) → dom 𝑧 = 𝑘)
63 simpl 473 . . . . . . . . . . . . . . . 16 ((𝑘 ∈ ω ∧ 𝑧 ∈ (𝐴𝑚 𝑘)) → 𝑘 ∈ ω)
6462, 63eqeltrd 2699 . . . . . . . . . . . . . . 15 ((𝑘 ∈ ω ∧ 𝑧 ∈ (𝐴𝑚 𝑘)) → dom 𝑧 ∈ ω)
65 simpr 477 . . . . . . . . . . . . . . . 16 ((𝑘 ∈ ω ∧ 𝑧 ∈ (𝐴𝑚 𝑘)) → 𝑧 ∈ (𝐴𝑚 𝑘))
6662oveq2d 6651 . . . . . . . . . . . . . . . 16 ((𝑘 ∈ ω ∧ 𝑧 ∈ (𝐴𝑚 𝑘)) → (𝐴𝑚 dom 𝑧) = (𝐴𝑚 𝑘))
6765, 66eleqtrrd 2702 . . . . . . . . . . . . . . 15 ((𝑘 ∈ ω ∧ 𝑧 ∈ (𝐴𝑚 𝑘)) → 𝑧 ∈ (𝐴𝑚 dom 𝑧))
6864, 67jca 554 . . . . . . . . . . . . . 14 ((𝑘 ∈ ω ∧ 𝑧 ∈ (𝐴𝑚 𝑘)) → (dom 𝑧 ∈ ω ∧ 𝑧 ∈ (𝐴𝑚 dom 𝑧)))
6968rexlimiva 3024 . . . . . . . . . . . . 13 (∃𝑘 ∈ ω 𝑧 ∈ (𝐴𝑚 𝑘) → (dom 𝑧 ∈ ω ∧ 𝑧 ∈ (𝐴𝑚 dom 𝑧)))
7058, 69sylbi 207 . . . . . . . . . . . 12 (𝑧 𝑘 ∈ ω (𝐴𝑚 𝑘) → (dom 𝑧 ∈ ω ∧ 𝑧 ∈ (𝐴𝑚 dom 𝑧)))
7136, 70syl 17 . . . . . . . . . . 11 ((𝜑𝑧𝐾𝑤) → (dom 𝑧 ∈ ω ∧ 𝑧 ∈ (𝐴𝑚 dom 𝑧)))
7271simpld 475 . . . . . . . . . 10 ((𝜑𝑧𝐾𝑤) → dom 𝑧 ∈ ω)
7310, 11, 12, 13fseqenlem1 8832 . . . . . . . . . 10 ((𝜑 ∧ dom 𝑧 ∈ ω) → (𝐺‘dom 𝑧):(𝐴𝑚 dom 𝑧)–1-1𝐴)
7472, 73syldan 487 . . . . . . . . 9 ((𝜑𝑧𝐾𝑤) → (𝐺‘dom 𝑧):(𝐴𝑚 dom 𝑧)–1-1𝐴)
75 f1f1orn 6135 . . . . . . . . 9 ((𝐺‘dom 𝑧):(𝐴𝑚 dom 𝑧)–1-1𝐴 → (𝐺‘dom 𝑧):(𝐴𝑚 dom 𝑧)–1-1-onto→ran (𝐺‘dom 𝑧))
7674, 75syl 17 . . . . . . . 8 ((𝜑𝑧𝐾𝑤) → (𝐺‘dom 𝑧):(𝐴𝑚 dom 𝑧)–1-1-onto→ran (𝐺‘dom 𝑧))
7771simprd 479 . . . . . . . 8 ((𝜑𝑧𝐾𝑤) → 𝑧 ∈ (𝐴𝑚 dom 𝑧))
78 f1ocnvfv1 6517 . . . . . . . 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)
7976, 77, 78syl2anc 692 . . . . . . 7 ((𝜑𝑧𝐾𝑤) → ((𝐺‘dom 𝑧)‘((𝐺‘dom 𝑧)‘𝑧)) = 𝑧)
8057, 79eqtr2d 2655 . . . . . 6 ((𝜑𝑧𝐾𝑤) → 𝑧 = ((𝐺‘(1st𝑤))‘(2nd𝑤)))
8180ex 450 . . . . 5 (𝜑 → (𝑧𝐾𝑤𝑧 = ((𝐺‘(1st𝑤))‘(2nd𝑤))))
8281alrimiv 1853 . . . 4 (𝜑 → ∀𝑧(𝑧𝐾𝑤𝑧 = ((𝐺‘(1st𝑤))‘(2nd𝑤))))
83 mo2icl 3379 . . . 4 (∀𝑧(𝑧𝐾𝑤𝑧 = ((𝐺‘(1st𝑤))‘(2nd𝑤))) → ∃*𝑧 𝑧𝐾𝑤)
8482, 83syl 17 . . 3 (𝜑 → ∃*𝑧 𝑧𝐾𝑤)
8584alrimiv 1853 . 2 (𝜑 → ∀𝑤∃*𝑧 𝑧𝐾𝑤)
86 dff12 6087 . 2 (𝐾: 𝑘 ∈ ω (𝐴𝑚 𝑘)–1-1→(ω × 𝐴) ↔ (𝐾: 𝑘 ∈ ω (𝐴𝑚 𝑘)⟶(ω × 𝐴) ∧ ∀𝑤∃*𝑧 𝑧𝐾𝑤))
8727, 85, 86sylanbrc 697 1 (𝜑𝐾: 𝑘 ∈ ω (𝐴𝑚 𝑘)–1-1→(ω × 𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384  wal 1479   = wceq 1481  wcel 1988  ∃*wmo 2469  wrex 2910  Vcvv 3195  c0 3907  {csn 4168  cop 4174   ciun 4511   class class class wbr 4644  cmpt 4720   × cxp 5102  ccnv 5103  dom cdm 5104  ran crn 5105  cres 5106  suc csuc 5713  Fun wfun 5870  wf 5872  1-1wf1 5873  1-1-ontowf1o 5875  cfv 5876  (class class class)co 6635  cmpt2 6637  ωcom 7050  1st c1st 7151  2nd c2nd 7152  seq𝜔cseqom 7527  𝑚 cmap 7842
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-8 1990  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-rep 4762  ax-sep 4772  ax-nul 4780  ax-pow 4834  ax-pr 4897  ax-un 6934
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ne 2792  df-ral 2914  df-rex 2915  df-reu 2916  df-rab 2918  df-v 3197  df-sbc 3430  df-csb 3527  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-pss 3583  df-nul 3908  df-if 4078  df-pw 4151  df-sn 4169  df-pr 4171  df-tp 4173  df-op 4175  df-uni 4428  df-iun 4513  df-br 4645  df-opab 4704  df-mpt 4721  df-tr 4744  df-id 5014  df-eprel 5019  df-po 5025  df-so 5026  df-fr 5063  df-we 5065  df-xp 5110  df-rel 5111  df-cnv 5112  df-co 5113  df-dm 5114  df-rn 5115  df-res 5116  df-ima 5117  df-pred 5668  df-ord 5714  df-on 5715  df-lim 5716  df-suc 5717  df-iota 5839  df-fun 5878  df-fn 5879  df-f 5880  df-f1 5881  df-fo 5882  df-f1o 5883  df-fv 5884  df-ov 6638  df-oprab 6639  df-mpt2 6640  df-om 7051  df-1st 7153  df-2nd 7154  df-wrecs 7392  df-recs 7453  df-rdg 7491  df-seqom 7528  df-1o 7545  df-map 7844
This theorem is referenced by:  fseqen  8835  pwfseqlem5  9470
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