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Theorem seqomlem2 8491
Description: Lemma for seqω. (Contributed by Stefan O'Rear, 1-Nov-2014.) (Revised by Mario Carneiro, 23-Jun-2015.)
Hypothesis
Ref Expression
seqomlem.a 𝑄 = rec((𝑖 ∈ ω, 𝑣 ∈ V ↦ ⟨suc 𝑖, (𝑖𝐹𝑣)⟩), ⟨∅, ( I ‘𝐼)⟩)
Assertion
Ref Expression
seqomlem2 (𝑄 “ ω) Fn ω
Distinct variable groups:   𝑄,𝑖,𝑣   𝑖,𝐹,𝑣
Allowed substitution hints:   𝐼(𝑣,𝑖)

Proof of Theorem seqomlem2
Dummy variables 𝑎 𝑏 𝑐 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 frfnom 8475 . . . . . . 7 (rec((𝑖 ∈ ω, 𝑣 ∈ V ↦ ⟨suc 𝑖, (𝑖𝐹𝑣)⟩), ⟨∅, ( I ‘𝐼)⟩) ↾ ω) Fn ω
2 seqomlem.a . . . . . . . . 9 𝑄 = rec((𝑖 ∈ ω, 𝑣 ∈ V ↦ ⟨suc 𝑖, (𝑖𝐹𝑣)⟩), ⟨∅, ( I ‘𝐼)⟩)
32reseq1i 5993 . . . . . . . 8 (𝑄 ↾ ω) = (rec((𝑖 ∈ ω, 𝑣 ∈ V ↦ ⟨suc 𝑖, (𝑖𝐹𝑣)⟩), ⟨∅, ( I ‘𝐼)⟩) ↾ ω)
43fneq1i 6665 . . . . . . 7 ((𝑄 ↾ ω) Fn ω ↔ (rec((𝑖 ∈ ω, 𝑣 ∈ V ↦ ⟨suc 𝑖, (𝑖𝐹𝑣)⟩), ⟨∅, ( I ‘𝐼)⟩) ↾ ω) Fn ω)
51, 4mpbir 231 . . . . . 6 (𝑄 ↾ ω) Fn ω
6 fvres 6925 . . . . . . . . 9 (𝑏 ∈ ω → ((𝑄 ↾ ω)‘𝑏) = (𝑄𝑏))
72seqomlem1 8490 . . . . . . . . 9 (𝑏 ∈ ω → (𝑄𝑏) = ⟨𝑏, (2nd ‘(𝑄𝑏))⟩)
86, 7eqtrd 2777 . . . . . . . 8 (𝑏 ∈ ω → ((𝑄 ↾ ω)‘𝑏) = ⟨𝑏, (2nd ‘(𝑄𝑏))⟩)
9 fvex 6919 . . . . . . . . 9 (2nd ‘(𝑄𝑏)) ∈ V
10 opelxpi 5722 . . . . . . . . 9 ((𝑏 ∈ ω ∧ (2nd ‘(𝑄𝑏)) ∈ V) → ⟨𝑏, (2nd ‘(𝑄𝑏))⟩ ∈ (ω × V))
119, 10mpan2 691 . . . . . . . 8 (𝑏 ∈ ω → ⟨𝑏, (2nd ‘(𝑄𝑏))⟩ ∈ (ω × V))
128, 11eqeltrd 2841 . . . . . . 7 (𝑏 ∈ ω → ((𝑄 ↾ ω)‘𝑏) ∈ (ω × V))
1312rgen 3063 . . . . . 6 𝑏 ∈ ω ((𝑄 ↾ ω)‘𝑏) ∈ (ω × V)
14 ffnfv 7139 . . . . . 6 ((𝑄 ↾ ω):ω⟶(ω × V) ↔ ((𝑄 ↾ ω) Fn ω ∧ ∀𝑏 ∈ ω ((𝑄 ↾ ω)‘𝑏) ∈ (ω × V)))
155, 13, 14mpbir2an 711 . . . . 5 (𝑄 ↾ ω):ω⟶(ω × V)
16 frn 6743 . . . . 5 ((𝑄 ↾ ω):ω⟶(ω × V) → ran (𝑄 ↾ ω) ⊆ (ω × V))
1715, 16ax-mp 5 . . . 4 ran (𝑄 ↾ ω) ⊆ (ω × V)
18 df-br 5144 . . . . . . . . . 10 (𝑎ran (𝑄 ↾ ω)𝑏 ↔ ⟨𝑎, 𝑏⟩ ∈ ran (𝑄 ↾ ω))
19 fvelrnb 6969 . . . . . . . . . . 11 ((𝑄 ↾ ω) Fn ω → (⟨𝑎, 𝑏⟩ ∈ ran (𝑄 ↾ ω) ↔ ∃𝑐 ∈ ω ((𝑄 ↾ ω)‘𝑐) = ⟨𝑎, 𝑏⟩))
205, 19ax-mp 5 . . . . . . . . . 10 (⟨𝑎, 𝑏⟩ ∈ ran (𝑄 ↾ ω) ↔ ∃𝑐 ∈ ω ((𝑄 ↾ ω)‘𝑐) = ⟨𝑎, 𝑏⟩)
21 fvres 6925 . . . . . . . . . . . 12 (𝑐 ∈ ω → ((𝑄 ↾ ω)‘𝑐) = (𝑄𝑐))
2221eqeq1d 2739 . . . . . . . . . . 11 (𝑐 ∈ ω → (((𝑄 ↾ ω)‘𝑐) = ⟨𝑎, 𝑏⟩ ↔ (𝑄𝑐) = ⟨𝑎, 𝑏⟩))
2322rexbiia 3092 . . . . . . . . . 10 (∃𝑐 ∈ ω ((𝑄 ↾ ω)‘𝑐) = ⟨𝑎, 𝑏⟩ ↔ ∃𝑐 ∈ ω (𝑄𝑐) = ⟨𝑎, 𝑏⟩)
2418, 20, 233bitri 297 . . . . . . . . 9 (𝑎ran (𝑄 ↾ ω)𝑏 ↔ ∃𝑐 ∈ ω (𝑄𝑐) = ⟨𝑎, 𝑏⟩)
252seqomlem1 8490 . . . . . . . . . . . . . . . 16 (𝑐 ∈ ω → (𝑄𝑐) = ⟨𝑐, (2nd ‘(𝑄𝑐))⟩)
2625adantl 481 . . . . . . . . . . . . . . 15 ((𝑎 ∈ ω ∧ 𝑐 ∈ ω) → (𝑄𝑐) = ⟨𝑐, (2nd ‘(𝑄𝑐))⟩)
2726eqeq1d 2739 . . . . . . . . . . . . . 14 ((𝑎 ∈ ω ∧ 𝑐 ∈ ω) → ((𝑄𝑐) = ⟨𝑎, 𝑏⟩ ↔ ⟨𝑐, (2nd ‘(𝑄𝑐))⟩ = ⟨𝑎, 𝑏⟩))
28 vex 3484 . . . . . . . . . . . . . . 15 𝑐 ∈ V
29 fvex 6919 . . . . . . . . . . . . . . 15 (2nd ‘(𝑄𝑐)) ∈ V
3028, 29opth1 5480 . . . . . . . . . . . . . 14 (⟨𝑐, (2nd ‘(𝑄𝑐))⟩ = ⟨𝑎, 𝑏⟩ → 𝑐 = 𝑎)
3127, 30biimtrdi 253 . . . . . . . . . . . . 13 ((𝑎 ∈ ω ∧ 𝑐 ∈ ω) → ((𝑄𝑐) = ⟨𝑎, 𝑏⟩ → 𝑐 = 𝑎))
32 fveqeq2 6915 . . . . . . . . . . . . . 14 (𝑐 = 𝑎 → ((𝑄𝑐) = ⟨𝑎, 𝑏⟩ ↔ (𝑄𝑎) = ⟨𝑎, 𝑏⟩))
3332biimpd 229 . . . . . . . . . . . . 13 (𝑐 = 𝑎 → ((𝑄𝑐) = ⟨𝑎, 𝑏⟩ → (𝑄𝑎) = ⟨𝑎, 𝑏⟩))
3431, 33syli 39 . . . . . . . . . . . 12 ((𝑎 ∈ ω ∧ 𝑐 ∈ ω) → ((𝑄𝑐) = ⟨𝑎, 𝑏⟩ → (𝑄𝑎) = ⟨𝑎, 𝑏⟩))
35 fveq2 6906 . . . . . . . . . . . . 13 ((𝑄𝑎) = ⟨𝑎, 𝑏⟩ → (2nd ‘(𝑄𝑎)) = (2nd ‘⟨𝑎, 𝑏⟩))
36 vex 3484 . . . . . . . . . . . . . 14 𝑎 ∈ V
37 vex 3484 . . . . . . . . . . . . . 14 𝑏 ∈ V
3836, 37op2nd 8023 . . . . . . . . . . . . 13 (2nd ‘⟨𝑎, 𝑏⟩) = 𝑏
3935, 38eqtr2di 2794 . . . . . . . . . . . 12 ((𝑄𝑎) = ⟨𝑎, 𝑏⟩ → 𝑏 = (2nd ‘(𝑄𝑎)))
4034, 39syl6 35 . . . . . . . . . . 11 ((𝑎 ∈ ω ∧ 𝑐 ∈ ω) → ((𝑄𝑐) = ⟨𝑎, 𝑏⟩ → 𝑏 = (2nd ‘(𝑄𝑎))))
4140rexlimdva 3155 . . . . . . . . . 10 (𝑎 ∈ ω → (∃𝑐 ∈ ω (𝑄𝑐) = ⟨𝑎, 𝑏⟩ → 𝑏 = (2nd ‘(𝑄𝑎))))
422seqomlem1 8490 . . . . . . . . . . . 12 (𝑎 ∈ ω → (𝑄𝑎) = ⟨𝑎, (2nd ‘(𝑄𝑎))⟩)
43 fveqeq2 6915 . . . . . . . . . . . . 13 (𝑐 = 𝑎 → ((𝑄𝑐) = ⟨𝑎, (2nd ‘(𝑄𝑎))⟩ ↔ (𝑄𝑎) = ⟨𝑎, (2nd ‘(𝑄𝑎))⟩))
4443rspcev 3622 . . . . . . . . . . . 12 ((𝑎 ∈ ω ∧ (𝑄𝑎) = ⟨𝑎, (2nd ‘(𝑄𝑎))⟩) → ∃𝑐 ∈ ω (𝑄𝑐) = ⟨𝑎, (2nd ‘(𝑄𝑎))⟩)
4542, 44mpdan 687 . . . . . . . . . . 11 (𝑎 ∈ ω → ∃𝑐 ∈ ω (𝑄𝑐) = ⟨𝑎, (2nd ‘(𝑄𝑎))⟩)
46 opeq2 4874 . . . . . . . . . . . . 13 (𝑏 = (2nd ‘(𝑄𝑎)) → ⟨𝑎, 𝑏⟩ = ⟨𝑎, (2nd ‘(𝑄𝑎))⟩)
4746eqeq2d 2748 . . . . . . . . . . . 12 (𝑏 = (2nd ‘(𝑄𝑎)) → ((𝑄𝑐) = ⟨𝑎, 𝑏⟩ ↔ (𝑄𝑐) = ⟨𝑎, (2nd ‘(𝑄𝑎))⟩))
4847rexbidv 3179 . . . . . . . . . . 11 (𝑏 = (2nd ‘(𝑄𝑎)) → (∃𝑐 ∈ ω (𝑄𝑐) = ⟨𝑎, 𝑏⟩ ↔ ∃𝑐 ∈ ω (𝑄𝑐) = ⟨𝑎, (2nd ‘(𝑄𝑎))⟩))
4945, 48syl5ibrcom 247 . . . . . . . . . 10 (𝑎 ∈ ω → (𝑏 = (2nd ‘(𝑄𝑎)) → ∃𝑐 ∈ ω (𝑄𝑐) = ⟨𝑎, 𝑏⟩))
5041, 49impbid 212 . . . . . . . . 9 (𝑎 ∈ ω → (∃𝑐 ∈ ω (𝑄𝑐) = ⟨𝑎, 𝑏⟩ ↔ 𝑏 = (2nd ‘(𝑄𝑎))))
5124, 50bitrid 283 . . . . . . . 8 (𝑎 ∈ ω → (𝑎ran (𝑄 ↾ ω)𝑏𝑏 = (2nd ‘(𝑄𝑎))))
5251alrimiv 1927 . . . . . . 7 (𝑎 ∈ ω → ∀𝑏(𝑎ran (𝑄 ↾ ω)𝑏𝑏 = (2nd ‘(𝑄𝑎))))
53 fvex 6919 . . . . . . . 8 (2nd ‘(𝑄𝑎)) ∈ V
54 eqeq2 2749 . . . . . . . . . 10 (𝑐 = (2nd ‘(𝑄𝑎)) → (𝑏 = 𝑐𝑏 = (2nd ‘(𝑄𝑎))))
5554bibi2d 342 . . . . . . . . 9 (𝑐 = (2nd ‘(𝑄𝑎)) → ((𝑎ran (𝑄 ↾ ω)𝑏𝑏 = 𝑐) ↔ (𝑎ran (𝑄 ↾ ω)𝑏𝑏 = (2nd ‘(𝑄𝑎)))))
5655albidv 1920 . . . . . . . 8 (𝑐 = (2nd ‘(𝑄𝑎)) → (∀𝑏(𝑎ran (𝑄 ↾ ω)𝑏𝑏 = 𝑐) ↔ ∀𝑏(𝑎ran (𝑄 ↾ ω)𝑏𝑏 = (2nd ‘(𝑄𝑎)))))
5753, 56spcev 3606 . . . . . . 7 (∀𝑏(𝑎ran (𝑄 ↾ ω)𝑏𝑏 = (2nd ‘(𝑄𝑎))) → ∃𝑐𝑏(𝑎ran (𝑄 ↾ ω)𝑏𝑏 = 𝑐))
5852, 57syl 17 . . . . . 6 (𝑎 ∈ ω → ∃𝑐𝑏(𝑎ran (𝑄 ↾ ω)𝑏𝑏 = 𝑐))
59 eu6 2574 . . . . . 6 (∃!𝑏 𝑎ran (𝑄 ↾ ω)𝑏 ↔ ∃𝑐𝑏(𝑎ran (𝑄 ↾ ω)𝑏𝑏 = 𝑐))
6058, 59sylibr 234 . . . . 5 (𝑎 ∈ ω → ∃!𝑏 𝑎ran (𝑄 ↾ ω)𝑏)
6160rgen 3063 . . . 4 𝑎 ∈ ω ∃!𝑏 𝑎ran (𝑄 ↾ ω)𝑏
62 dff3 7120 . . . 4 (ran (𝑄 ↾ ω):ω⟶V ↔ (ran (𝑄 ↾ ω) ⊆ (ω × V) ∧ ∀𝑎 ∈ ω ∃!𝑏 𝑎ran (𝑄 ↾ ω)𝑏))
6317, 61, 62mpbir2an 711 . . 3 ran (𝑄 ↾ ω):ω⟶V
64 df-ima 5698 . . . 4 (𝑄 “ ω) = ran (𝑄 ↾ ω)
6564feq1i 6727 . . 3 ((𝑄 “ ω):ω⟶V ↔ ran (𝑄 ↾ ω):ω⟶V)
6663, 65mpbir 231 . 2 (𝑄 “ ω):ω⟶V
67 dffn2 6738 . 2 ((𝑄 “ ω) Fn ω ↔ (𝑄 “ ω):ω⟶V)
6866, 67mpbir 231 1 (𝑄 “ ω) Fn ω
Colors of variables: wff setvar class
Syntax hints:  wb 206  wa 395  wal 1538   = wceq 1540  wex 1779  wcel 2108  ∃!weu 2568  wral 3061  wrex 3070  Vcvv 3480  wss 3951  c0 4333  cop 4632   class class class wbr 5143   I cid 5577   × cxp 5683  ran crn 5686  cres 5687  cima 5688  suc csuc 6386   Fn wfn 6556  wf 6557  cfv 6561  (class class class)co 7431  cmpo 7433  ωcom 7887  2nd c2nd 8013  reccrdg 8449
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-sep 5296  ax-nul 5306  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-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450
This theorem is referenced by:  seqomlem3  8492  seqomlem4  8493  fnseqom  8495
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