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Theorem seqomlem2 7591
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 7575 . . . . . . 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 5424 . . . . . . . 8 (𝑄 ↾ ω) = (rec((𝑖 ∈ ω, 𝑣 ∈ V ↦ ⟨suc 𝑖, (𝑖𝐹𝑣)⟩), ⟨∅, ( I ‘𝐼)⟩) ↾ ω)
43fneq1i 6023 . . . . . . 7 ((𝑄 ↾ ω) Fn ω ↔ (rec((𝑖 ∈ ω, 𝑣 ∈ V ↦ ⟨suc 𝑖, (𝑖𝐹𝑣)⟩), ⟨∅, ( I ‘𝐼)⟩) ↾ ω) Fn ω)
51, 4mpbir 221 . . . . . 6 (𝑄 ↾ ω) Fn ω
6 fvres 6245 . . . . . . . . 9 (𝑏 ∈ ω → ((𝑄 ↾ ω)‘𝑏) = (𝑄𝑏))
72seqomlem1 7590 . . . . . . . . 9 (𝑏 ∈ ω → (𝑄𝑏) = ⟨𝑏, (2nd ‘(𝑄𝑏))⟩)
86, 7eqtrd 2685 . . . . . . . 8 (𝑏 ∈ ω → ((𝑄 ↾ ω)‘𝑏) = ⟨𝑏, (2nd ‘(𝑄𝑏))⟩)
9 fvex 6239 . . . . . . . . 9 (2nd ‘(𝑄𝑏)) ∈ V
10 opelxpi 5182 . . . . . . . . 9 ((𝑏 ∈ ω ∧ (2nd ‘(𝑄𝑏)) ∈ V) → ⟨𝑏, (2nd ‘(𝑄𝑏))⟩ ∈ (ω × V))
119, 10mpan2 707 . . . . . . . 8 (𝑏 ∈ ω → ⟨𝑏, (2nd ‘(𝑄𝑏))⟩ ∈ (ω × V))
128, 11eqeltrd 2730 . . . . . . 7 (𝑏 ∈ ω → ((𝑄 ↾ ω)‘𝑏) ∈ (ω × V))
1312rgen 2951 . . . . . 6 𝑏 ∈ ω ((𝑄 ↾ ω)‘𝑏) ∈ (ω × V)
14 ffnfv 6428 . . . . . 6 ((𝑄 ↾ ω):ω⟶(ω × V) ↔ ((𝑄 ↾ ω) Fn ω ∧ ∀𝑏 ∈ ω ((𝑄 ↾ ω)‘𝑏) ∈ (ω × V)))
155, 13, 14mpbir2an 975 . . . . 5 (𝑄 ↾ ω):ω⟶(ω × V)
16 frn 6091 . . . . 5 ((𝑄 ↾ ω):ω⟶(ω × V) → ran (𝑄 ↾ ω) ⊆ (ω × V))
1715, 16ax-mp 5 . . . 4 ran (𝑄 ↾ ω) ⊆ (ω × V)
18 df-br 4686 . . . . . . . . . 10 (𝑎ran (𝑄 ↾ ω)𝑏 ↔ ⟨𝑎, 𝑏⟩ ∈ ran (𝑄 ↾ ω))
19 fvelrnb 6282 . . . . . . . . . . 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 6245 . . . . . . . . . . . 12 (𝑐 ∈ ω → ((𝑄 ↾ ω)‘𝑐) = (𝑄𝑐))
2221eqeq1d 2653 . . . . . . . . . . 11 (𝑐 ∈ ω → (((𝑄 ↾ ω)‘𝑐) = ⟨𝑎, 𝑏⟩ ↔ (𝑄𝑐) = ⟨𝑎, 𝑏⟩))
2322rexbiia 3069 . . . . . . . . . 10 (∃𝑐 ∈ ω ((𝑄 ↾ ω)‘𝑐) = ⟨𝑎, 𝑏⟩ ↔ ∃𝑐 ∈ ω (𝑄𝑐) = ⟨𝑎, 𝑏⟩)
2418, 20, 233bitri 286 . . . . . . . . 9 (𝑎ran (𝑄 ↾ ω)𝑏 ↔ ∃𝑐 ∈ ω (𝑄𝑐) = ⟨𝑎, 𝑏⟩)
252seqomlem1 7590 . . . . . . . . . . . . . . . 16 (𝑐 ∈ ω → (𝑄𝑐) = ⟨𝑐, (2nd ‘(𝑄𝑐))⟩)
2625adantl 481 . . . . . . . . . . . . . . 15 ((𝑎 ∈ ω ∧ 𝑐 ∈ ω) → (𝑄𝑐) = ⟨𝑐, (2nd ‘(𝑄𝑐))⟩)
2726eqeq1d 2653 . . . . . . . . . . . . . 14 ((𝑎 ∈ ω ∧ 𝑐 ∈ ω) → ((𝑄𝑐) = ⟨𝑎, 𝑏⟩ ↔ ⟨𝑐, (2nd ‘(𝑄𝑐))⟩ = ⟨𝑎, 𝑏⟩))
28 vex 3234 . . . . . . . . . . . . . . 15 𝑐 ∈ V
29 fvex 6239 . . . . . . . . . . . . . . 15 (2nd ‘(𝑄𝑐)) ∈ V
3028, 29opth1 4973 . . . . . . . . . . . . . 14 (⟨𝑐, (2nd ‘(𝑄𝑐))⟩ = ⟨𝑎, 𝑏⟩ → 𝑐 = 𝑎)
3127, 30syl6bi 243 . . . . . . . . . . . . 13 ((𝑎 ∈ ω ∧ 𝑐 ∈ ω) → ((𝑄𝑐) = ⟨𝑎, 𝑏⟩ → 𝑐 = 𝑎))
32 fveq2 6229 . . . . . . . . . . . . . . 15 (𝑐 = 𝑎 → (𝑄𝑐) = (𝑄𝑎))
3332eqeq1d 2653 . . . . . . . . . . . . . 14 (𝑐 = 𝑎 → ((𝑄𝑐) = ⟨𝑎, 𝑏⟩ ↔ (𝑄𝑎) = ⟨𝑎, 𝑏⟩))
3433biimpd 219 . . . . . . . . . . . . 13 (𝑐 = 𝑎 → ((𝑄𝑐) = ⟨𝑎, 𝑏⟩ → (𝑄𝑎) = ⟨𝑎, 𝑏⟩))
3531, 34syli 39 . . . . . . . . . . . 12 ((𝑎 ∈ ω ∧ 𝑐 ∈ ω) → ((𝑄𝑐) = ⟨𝑎, 𝑏⟩ → (𝑄𝑎) = ⟨𝑎, 𝑏⟩))
36 fveq2 6229 . . . . . . . . . . . . 13 ((𝑄𝑎) = ⟨𝑎, 𝑏⟩ → (2nd ‘(𝑄𝑎)) = (2nd ‘⟨𝑎, 𝑏⟩))
37 vex 3234 . . . . . . . . . . . . . 14 𝑎 ∈ V
38 vex 3234 . . . . . . . . . . . . . 14 𝑏 ∈ V
3937, 38op2nd 7219 . . . . . . . . . . . . 13 (2nd ‘⟨𝑎, 𝑏⟩) = 𝑏
4036, 39syl6req 2702 . . . . . . . . . . . 12 ((𝑄𝑎) = ⟨𝑎, 𝑏⟩ → 𝑏 = (2nd ‘(𝑄𝑎)))
4135, 40syl6 35 . . . . . . . . . . 11 ((𝑎 ∈ ω ∧ 𝑐 ∈ ω) → ((𝑄𝑐) = ⟨𝑎, 𝑏⟩ → 𝑏 = (2nd ‘(𝑄𝑎))))
4241rexlimdva 3060 . . . . . . . . . 10 (𝑎 ∈ ω → (∃𝑐 ∈ ω (𝑄𝑐) = ⟨𝑎, 𝑏⟩ → 𝑏 = (2nd ‘(𝑄𝑎))))
432seqomlem1 7590 . . . . . . . . . . . 12 (𝑎 ∈ ω → (𝑄𝑎) = ⟨𝑎, (2nd ‘(𝑄𝑎))⟩)
4432eqeq1d 2653 . . . . . . . . . . . . 13 (𝑐 = 𝑎 → ((𝑄𝑐) = ⟨𝑎, (2nd ‘(𝑄𝑎))⟩ ↔ (𝑄𝑎) = ⟨𝑎, (2nd ‘(𝑄𝑎))⟩))
4544rspcev 3340 . . . . . . . . . . . 12 ((𝑎 ∈ ω ∧ (𝑄𝑎) = ⟨𝑎, (2nd ‘(𝑄𝑎))⟩) → ∃𝑐 ∈ ω (𝑄𝑐) = ⟨𝑎, (2nd ‘(𝑄𝑎))⟩)
4643, 45mpdan 703 . . . . . . . . . . 11 (𝑎 ∈ ω → ∃𝑐 ∈ ω (𝑄𝑐) = ⟨𝑎, (2nd ‘(𝑄𝑎))⟩)
47 opeq2 4434 . . . . . . . . . . . . 13 (𝑏 = (2nd ‘(𝑄𝑎)) → ⟨𝑎, 𝑏⟩ = ⟨𝑎, (2nd ‘(𝑄𝑎))⟩)
4847eqeq2d 2661 . . . . . . . . . . . 12 (𝑏 = (2nd ‘(𝑄𝑎)) → ((𝑄𝑐) = ⟨𝑎, 𝑏⟩ ↔ (𝑄𝑐) = ⟨𝑎, (2nd ‘(𝑄𝑎))⟩))
4948rexbidv 3081 . . . . . . . . . . 11 (𝑏 = (2nd ‘(𝑄𝑎)) → (∃𝑐 ∈ ω (𝑄𝑐) = ⟨𝑎, 𝑏⟩ ↔ ∃𝑐 ∈ ω (𝑄𝑐) = ⟨𝑎, (2nd ‘(𝑄𝑎))⟩))
5046, 49syl5ibrcom 237 . . . . . . . . . 10 (𝑎 ∈ ω → (𝑏 = (2nd ‘(𝑄𝑎)) → ∃𝑐 ∈ ω (𝑄𝑐) = ⟨𝑎, 𝑏⟩))
5142, 50impbid 202 . . . . . . . . 9 (𝑎 ∈ ω → (∃𝑐 ∈ ω (𝑄𝑐) = ⟨𝑎, 𝑏⟩ ↔ 𝑏 = (2nd ‘(𝑄𝑎))))
5224, 51syl5bb 272 . . . . . . . 8 (𝑎 ∈ ω → (𝑎ran (𝑄 ↾ ω)𝑏𝑏 = (2nd ‘(𝑄𝑎))))
5352alrimiv 1895 . . . . . . 7 (𝑎 ∈ ω → ∀𝑏(𝑎ran (𝑄 ↾ ω)𝑏𝑏 = (2nd ‘(𝑄𝑎))))
54 fvex 6239 . . . . . . . 8 (2nd ‘(𝑄𝑎)) ∈ V
55 eqeq2 2662 . . . . . . . . . 10 (𝑐 = (2nd ‘(𝑄𝑎)) → (𝑏 = 𝑐𝑏 = (2nd ‘(𝑄𝑎))))
5655bibi2d 331 . . . . . . . . 9 (𝑐 = (2nd ‘(𝑄𝑎)) → ((𝑎ran (𝑄 ↾ ω)𝑏𝑏 = 𝑐) ↔ (𝑎ran (𝑄 ↾ ω)𝑏𝑏 = (2nd ‘(𝑄𝑎)))))
5756albidv 1889 . . . . . . . 8 (𝑐 = (2nd ‘(𝑄𝑎)) → (∀𝑏(𝑎ran (𝑄 ↾ ω)𝑏𝑏 = 𝑐) ↔ ∀𝑏(𝑎ran (𝑄 ↾ ω)𝑏𝑏 = (2nd ‘(𝑄𝑎)))))
5854, 57spcev 3331 . . . . . . 7 (∀𝑏(𝑎ran (𝑄 ↾ ω)𝑏𝑏 = (2nd ‘(𝑄𝑎))) → ∃𝑐𝑏(𝑎ran (𝑄 ↾ ω)𝑏𝑏 = 𝑐))
5953, 58syl 17 . . . . . 6 (𝑎 ∈ ω → ∃𝑐𝑏(𝑎ran (𝑄 ↾ ω)𝑏𝑏 = 𝑐))
60 df-eu 2502 . . . . . 6 (∃!𝑏 𝑎ran (𝑄 ↾ ω)𝑏 ↔ ∃𝑐𝑏(𝑎ran (𝑄 ↾ ω)𝑏𝑏 = 𝑐))
6159, 60sylibr 224 . . . . 5 (𝑎 ∈ ω → ∃!𝑏 𝑎ran (𝑄 ↾ ω)𝑏)
6261rgen 2951 . . . 4 𝑎 ∈ ω ∃!𝑏 𝑎ran (𝑄 ↾ ω)𝑏
63 dff3 6412 . . . 4 (ran (𝑄 ↾ ω):ω⟶V ↔ (ran (𝑄 ↾ ω) ⊆ (ω × V) ∧ ∀𝑎 ∈ ω ∃!𝑏 𝑎ran (𝑄 ↾ ω)𝑏))
6417, 62, 63mpbir2an 975 . . 3 ran (𝑄 ↾ ω):ω⟶V
65 df-ima 5156 . . . 4 (𝑄 “ ω) = ran (𝑄 ↾ ω)
6665feq1i 6074 . . 3 ((𝑄 “ ω):ω⟶V ↔ ran (𝑄 ↾ ω):ω⟶V)
6764, 66mpbir 221 . 2 (𝑄 “ ω):ω⟶V
68 dffn2 6085 . 2 ((𝑄 “ ω) Fn ω ↔ (𝑄 “ ω):ω⟶V)
6967, 68mpbir 221 1 (𝑄 “ ω) Fn ω
Colors of variables: wff setvar class
Syntax hints:  wb 196  wa 383  wal 1521   = wceq 1523  wex 1744  wcel 2030  ∃!weu 2498  wral 2941  wrex 2942  Vcvv 3231  wss 3607  c0 3948  cop 4216   class class class wbr 4685   I cid 5052   × cxp 5141  ran crn 5144  cres 5145  cima 5146  suc csuc 5763   Fn wfn 5921  wf 5922  cfv 5926  (class class class)co 6690  cmpt2 6692  ωcom 7107  2nd c2nd 7209  reccrdg 7550
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-om 7108  df-2nd 7211  df-wrecs 7452  df-recs 7513  df-rdg 7551
This theorem is referenced by:  seqomlem3  7592  seqomlem4  7593  fnseqom  7595
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