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Theorem seqomlem2 8083
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 8066 . . . . . . 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 5836 . . . . . . . 8 (𝑄 ↾ ω) = (rec((𝑖 ∈ ω, 𝑣 ∈ V ↦ ⟨suc 𝑖, (𝑖𝐹𝑣)⟩), ⟨∅, ( I ‘𝐼)⟩) ↾ ω)
43fneq1i 6438 . . . . . . 7 ((𝑄 ↾ ω) Fn ω ↔ (rec((𝑖 ∈ ω, 𝑣 ∈ V ↦ ⟨suc 𝑖, (𝑖𝐹𝑣)⟩), ⟨∅, ( I ‘𝐼)⟩) ↾ ω) Fn ω)
51, 4mpbir 234 . . . . . 6 (𝑄 ↾ ω) Fn ω
6 fvres 6680 . . . . . . . . 9 (𝑏 ∈ ω → ((𝑄 ↾ ω)‘𝑏) = (𝑄𝑏))
72seqomlem1 8082 . . . . . . . . 9 (𝑏 ∈ ω → (𝑄𝑏) = ⟨𝑏, (2nd ‘(𝑄𝑏))⟩)
86, 7eqtrd 2859 . . . . . . . 8 (𝑏 ∈ ω → ((𝑄 ↾ ω)‘𝑏) = ⟨𝑏, (2nd ‘(𝑄𝑏))⟩)
9 fvex 6674 . . . . . . . . 9 (2nd ‘(𝑄𝑏)) ∈ V
10 opelxpi 5579 . . . . . . . . 9 ((𝑏 ∈ ω ∧ (2nd ‘(𝑄𝑏)) ∈ V) → ⟨𝑏, (2nd ‘(𝑄𝑏))⟩ ∈ (ω × V))
119, 10mpan2 690 . . . . . . . 8 (𝑏 ∈ ω → ⟨𝑏, (2nd ‘(𝑄𝑏))⟩ ∈ (ω × V))
128, 11eqeltrd 2916 . . . . . . 7 (𝑏 ∈ ω → ((𝑄 ↾ ω)‘𝑏) ∈ (ω × V))
1312rgen 3143 . . . . . 6 𝑏 ∈ ω ((𝑄 ↾ ω)‘𝑏) ∈ (ω × V)
14 ffnfv 6873 . . . . . 6 ((𝑄 ↾ ω):ω⟶(ω × V) ↔ ((𝑄 ↾ ω) Fn ω ∧ ∀𝑏 ∈ ω ((𝑄 ↾ ω)‘𝑏) ∈ (ω × V)))
155, 13, 14mpbir2an 710 . . . . 5 (𝑄 ↾ ω):ω⟶(ω × V)
16 frn 6509 . . . . 5 ((𝑄 ↾ ω):ω⟶(ω × V) → ran (𝑄 ↾ ω) ⊆ (ω × V))
1715, 16ax-mp 5 . . . 4 ran (𝑄 ↾ ω) ⊆ (ω × V)
18 df-br 5053 . . . . . . . . . 10 (𝑎ran (𝑄 ↾ ω)𝑏 ↔ ⟨𝑎, 𝑏⟩ ∈ ran (𝑄 ↾ ω))
19 fvelrnb 6717 . . . . . . . . . . 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 6680 . . . . . . . . . . . 12 (𝑐 ∈ ω → ((𝑄 ↾ ω)‘𝑐) = (𝑄𝑐))
2221eqeq1d 2826 . . . . . . . . . . 11 (𝑐 ∈ ω → (((𝑄 ↾ ω)‘𝑐) = ⟨𝑎, 𝑏⟩ ↔ (𝑄𝑐) = ⟨𝑎, 𝑏⟩))
2322rexbiia 3240 . . . . . . . . . 10 (∃𝑐 ∈ ω ((𝑄 ↾ ω)‘𝑐) = ⟨𝑎, 𝑏⟩ ↔ ∃𝑐 ∈ ω (𝑄𝑐) = ⟨𝑎, 𝑏⟩)
2418, 20, 233bitri 300 . . . . . . . . 9 (𝑎ran (𝑄 ↾ ω)𝑏 ↔ ∃𝑐 ∈ ω (𝑄𝑐) = ⟨𝑎, 𝑏⟩)
252seqomlem1 8082 . . . . . . . . . . . . . . . 16 (𝑐 ∈ ω → (𝑄𝑐) = ⟨𝑐, (2nd ‘(𝑄𝑐))⟩)
2625adantl 485 . . . . . . . . . . . . . . 15 ((𝑎 ∈ ω ∧ 𝑐 ∈ ω) → (𝑄𝑐) = ⟨𝑐, (2nd ‘(𝑄𝑐))⟩)
2726eqeq1d 2826 . . . . . . . . . . . . . 14 ((𝑎 ∈ ω ∧ 𝑐 ∈ ω) → ((𝑄𝑐) = ⟨𝑎, 𝑏⟩ ↔ ⟨𝑐, (2nd ‘(𝑄𝑐))⟩ = ⟨𝑎, 𝑏⟩))
28 vex 3483 . . . . . . . . . . . . . . 15 𝑐 ∈ V
29 fvex 6674 . . . . . . . . . . . . . . 15 (2nd ‘(𝑄𝑐)) ∈ V
3028, 29opth1 5354 . . . . . . . . . . . . . 14 (⟨𝑐, (2nd ‘(𝑄𝑐))⟩ = ⟨𝑎, 𝑏⟩ → 𝑐 = 𝑎)
3127, 30syl6bi 256 . . . . . . . . . . . . 13 ((𝑎 ∈ ω ∧ 𝑐 ∈ ω) → ((𝑄𝑐) = ⟨𝑎, 𝑏⟩ → 𝑐 = 𝑎))
32 fveqeq2 6670 . . . . . . . . . . . . . 14 (𝑐 = 𝑎 → ((𝑄𝑐) = ⟨𝑎, 𝑏⟩ ↔ (𝑄𝑎) = ⟨𝑎, 𝑏⟩))
3332biimpd 232 . . . . . . . . . . . . 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 6661 . . . . . . . . . . . . 13 ((𝑄𝑎) = ⟨𝑎, 𝑏⟩ → (2nd ‘(𝑄𝑎)) = (2nd ‘⟨𝑎, 𝑏⟩))
36 vex 3483 . . . . . . . . . . . . . 14 𝑎 ∈ V
37 vex 3483 . . . . . . . . . . . . . 14 𝑏 ∈ V
3836, 37op2nd 7693 . . . . . . . . . . . . 13 (2nd ‘⟨𝑎, 𝑏⟩) = 𝑏
3935, 38syl6req 2876 . . . . . . . . . . . 12 ((𝑄𝑎) = ⟨𝑎, 𝑏⟩ → 𝑏 = (2nd ‘(𝑄𝑎)))
4034, 39syl6 35 . . . . . . . . . . 11 ((𝑎 ∈ ω ∧ 𝑐 ∈ ω) → ((𝑄𝑐) = ⟨𝑎, 𝑏⟩ → 𝑏 = (2nd ‘(𝑄𝑎))))
4140rexlimdva 3276 . . . . . . . . . 10 (𝑎 ∈ ω → (∃𝑐 ∈ ω (𝑄𝑐) = ⟨𝑎, 𝑏⟩ → 𝑏 = (2nd ‘(𝑄𝑎))))
422seqomlem1 8082 . . . . . . . . . . . 12 (𝑎 ∈ ω → (𝑄𝑎) = ⟨𝑎, (2nd ‘(𝑄𝑎))⟩)
43 fveqeq2 6670 . . . . . . . . . . . . 13 (𝑐 = 𝑎 → ((𝑄𝑐) = ⟨𝑎, (2nd ‘(𝑄𝑎))⟩ ↔ (𝑄𝑎) = ⟨𝑎, (2nd ‘(𝑄𝑎))⟩))
4443rspcev 3609 . . . . . . . . . . . 12 ((𝑎 ∈ ω ∧ (𝑄𝑎) = ⟨𝑎, (2nd ‘(𝑄𝑎))⟩) → ∃𝑐 ∈ ω (𝑄𝑐) = ⟨𝑎, (2nd ‘(𝑄𝑎))⟩)
4542, 44mpdan 686 . . . . . . . . . . 11 (𝑎 ∈ ω → ∃𝑐 ∈ ω (𝑄𝑐) = ⟨𝑎, (2nd ‘(𝑄𝑎))⟩)
46 opeq2 4789 . . . . . . . . . . . . 13 (𝑏 = (2nd ‘(𝑄𝑎)) → ⟨𝑎, 𝑏⟩ = ⟨𝑎, (2nd ‘(𝑄𝑎))⟩)
4746eqeq2d 2835 . . . . . . . . . . . 12 (𝑏 = (2nd ‘(𝑄𝑎)) → ((𝑄𝑐) = ⟨𝑎, 𝑏⟩ ↔ (𝑄𝑐) = ⟨𝑎, (2nd ‘(𝑄𝑎))⟩))
4847rexbidv 3289 . . . . . . . . . . 11 (𝑏 = (2nd ‘(𝑄𝑎)) → (∃𝑐 ∈ ω (𝑄𝑐) = ⟨𝑎, 𝑏⟩ ↔ ∃𝑐 ∈ ω (𝑄𝑐) = ⟨𝑎, (2nd ‘(𝑄𝑎))⟩))
4945, 48syl5ibrcom 250 . . . . . . . . . 10 (𝑎 ∈ ω → (𝑏 = (2nd ‘(𝑄𝑎)) → ∃𝑐 ∈ ω (𝑄𝑐) = ⟨𝑎, 𝑏⟩))
5041, 49impbid 215 . . . . . . . . 9 (𝑎 ∈ ω → (∃𝑐 ∈ ω (𝑄𝑐) = ⟨𝑎, 𝑏⟩ ↔ 𝑏 = (2nd ‘(𝑄𝑎))))
5124, 50syl5bb 286 . . . . . . . 8 (𝑎 ∈ ω → (𝑎ran (𝑄 ↾ ω)𝑏𝑏 = (2nd ‘(𝑄𝑎))))
5251alrimiv 1929 . . . . . . 7 (𝑎 ∈ ω → ∀𝑏(𝑎ran (𝑄 ↾ ω)𝑏𝑏 = (2nd ‘(𝑄𝑎))))
53 fvex 6674 . . . . . . . 8 (2nd ‘(𝑄𝑎)) ∈ V
54 eqeq2 2836 . . . . . . . . . 10 (𝑐 = (2nd ‘(𝑄𝑎)) → (𝑏 = 𝑐𝑏 = (2nd ‘(𝑄𝑎))))
5554bibi2d 346 . . . . . . . . 9 (𝑐 = (2nd ‘(𝑄𝑎)) → ((𝑎ran (𝑄 ↾ ω)𝑏𝑏 = 𝑐) ↔ (𝑎ran (𝑄 ↾ ω)𝑏𝑏 = (2nd ‘(𝑄𝑎)))))
5655albidv 1922 . . . . . . . 8 (𝑐 = (2nd ‘(𝑄𝑎)) → (∀𝑏(𝑎ran (𝑄 ↾ ω)𝑏𝑏 = 𝑐) ↔ ∀𝑏(𝑎ran (𝑄 ↾ ω)𝑏𝑏 = (2nd ‘(𝑄𝑎)))))
5753, 56spcev 3593 . . . . . . 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 2660 . . . . . 6 (∃!𝑏 𝑎ran (𝑄 ↾ ω)𝑏 ↔ ∃𝑐𝑏(𝑎ran (𝑄 ↾ ω)𝑏𝑏 = 𝑐))
6058, 59sylibr 237 . . . . 5 (𝑎 ∈ ω → ∃!𝑏 𝑎ran (𝑄 ↾ ω)𝑏)
6160rgen 3143 . . . 4 𝑎 ∈ ω ∃!𝑏 𝑎ran (𝑄 ↾ ω)𝑏
62 dff3 6857 . . . 4 (ran (𝑄 ↾ ω):ω⟶V ↔ (ran (𝑄 ↾ ω) ⊆ (ω × V) ∧ ∀𝑎 ∈ ω ∃!𝑏 𝑎ran (𝑄 ↾ ω)𝑏))
6317, 61, 62mpbir2an 710 . . 3 ran (𝑄 ↾ ω):ω⟶V
64 df-ima 5555 . . . 4 (𝑄 “ ω) = ran (𝑄 ↾ ω)
6564feq1i 6494 . . 3 ((𝑄 “ ω):ω⟶V ↔ ran (𝑄 ↾ ω):ω⟶V)
6663, 65mpbir 234 . 2 (𝑄 “ ω):ω⟶V
67 dffn2 6505 . 2 ((𝑄 “ ω) Fn ω ↔ (𝑄 “ ω):ω⟶V)
6866, 67mpbir 234 1 (𝑄 “ ω) Fn ω
Colors of variables: wff setvar class
Syntax hints:  wb 209  wa 399  wal 1536   = wceq 1538  wex 1781  wcel 2115  ∃!weu 2654  wral 3133  wrex 3134  Vcvv 3480  wss 3919  c0 4276  cop 4556   class class class wbr 5052   I cid 5446   × cxp 5540  ran crn 5543  cres 5544  cima 5545  suc csuc 6180   Fn wfn 6338  wf 6339  cfv 6343  (class class class)co 7149  cmpo 7151  ωcom 7574  2nd c2nd 7683  reccrdg 8041
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 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5189  ax-nul 5196  ax-pow 5253  ax-pr 5317  ax-un 7455
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 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-ral 3138  df-rex 3139  df-reu 3140  df-rab 3142  df-v 3482  df-sbc 3759  df-csb 3867  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-pss 3938  df-nul 4277  df-if 4451  df-pw 4524  df-sn 4551  df-pr 4553  df-tp 4555  df-op 4557  df-uni 4825  df-iun 4907  df-br 5053  df-opab 5115  df-mpt 5133  df-tr 5159  df-id 5447  df-eprel 5452  df-po 5461  df-so 5462  df-fr 5501  df-we 5503  df-xp 5548  df-rel 5549  df-cnv 5550  df-co 5551  df-dm 5552  df-rn 5553  df-res 5554  df-ima 5555  df-pred 6135  df-ord 6181  df-on 6182  df-lim 6183  df-suc 6184  df-iota 6302  df-fun 6345  df-fn 6346  df-f 6347  df-f1 6348  df-fo 6349  df-f1o 6350  df-fv 6351  df-ov 7152  df-oprab 7153  df-mpo 7154  df-om 7575  df-2nd 7685  df-wrecs 7943  df-recs 8004  df-rdg 8042
This theorem is referenced by:  seqomlem3  8084  seqomlem4  8085  fnseqom  8087
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