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Theorem seqomlem2 7433
 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 7417 . . . . . . 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 5313 . . . . . . . 8 (𝑄 ↾ ω) = (rec((𝑖 ∈ ω, 𝑣 ∈ V ↦ ⟨suc 𝑖, (𝑖𝐹𝑣)⟩), ⟨∅, ( I ‘𝐼)⟩) ↾ ω)
43fneq1i 5899 . . . . . . 7 ((𝑄 ↾ ω) Fn ω ↔ (rec((𝑖 ∈ ω, 𝑣 ∈ V ↦ ⟨suc 𝑖, (𝑖𝐹𝑣)⟩), ⟨∅, ( I ‘𝐼)⟩) ↾ ω) Fn ω)
51, 4mpbir 220 . . . . . 6 (𝑄 ↾ ω) Fn ω
6 fvres 6117 . . . . . . . . 9 (𝑏 ∈ ω → ((𝑄 ↾ ω)‘𝑏) = (𝑄𝑏))
72seqomlem1 7432 . . . . . . . . 9 (𝑏 ∈ ω → (𝑄𝑏) = ⟨𝑏, (2nd ‘(𝑄𝑏))⟩)
86, 7eqtrd 2644 . . . . . . . 8 (𝑏 ∈ ω → ((𝑄 ↾ ω)‘𝑏) = ⟨𝑏, (2nd ‘(𝑄𝑏))⟩)
9 fvex 6113 . . . . . . . . 9 (2nd ‘(𝑄𝑏)) ∈ V
10 opelxpi 5072 . . . . . . . . 9 ((𝑏 ∈ ω ∧ (2nd ‘(𝑄𝑏)) ∈ V) → ⟨𝑏, (2nd ‘(𝑄𝑏))⟩ ∈ (ω × V))
119, 10mpan2 703 . . . . . . . 8 (𝑏 ∈ ω → ⟨𝑏, (2nd ‘(𝑄𝑏))⟩ ∈ (ω × V))
128, 11eqeltrd 2688 . . . . . . 7 (𝑏 ∈ ω → ((𝑄 ↾ ω)‘𝑏) ∈ (ω × V))
1312rgen 2906 . . . . . 6 𝑏 ∈ ω ((𝑄 ↾ ω)‘𝑏) ∈ (ω × V)
14 ffnfv 6295 . . . . . 6 ((𝑄 ↾ ω):ω⟶(ω × V) ↔ ((𝑄 ↾ ω) Fn ω ∧ ∀𝑏 ∈ ω ((𝑄 ↾ ω)‘𝑏) ∈ (ω × V)))
155, 13, 14mpbir2an 957 . . . . 5 (𝑄 ↾ ω):ω⟶(ω × V)
16 frn 5966 . . . . 5 ((𝑄 ↾ ω):ω⟶(ω × V) → ran (𝑄 ↾ ω) ⊆ (ω × V))
1715, 16ax-mp 5 . . . 4 ran (𝑄 ↾ ω) ⊆ (ω × V)
18 df-br 4584 . . . . . . . . . 10 (𝑎ran (𝑄 ↾ ω)𝑏 ↔ ⟨𝑎, 𝑏⟩ ∈ ran (𝑄 ↾ ω))
19 fvelrnb 6153 . . . . . . . . . . 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 6117 . . . . . . . . . . . 12 (𝑐 ∈ ω → ((𝑄 ↾ ω)‘𝑐) = (𝑄𝑐))
2221eqeq1d 2612 . . . . . . . . . . 11 (𝑐 ∈ ω → (((𝑄 ↾ ω)‘𝑐) = ⟨𝑎, 𝑏⟩ ↔ (𝑄𝑐) = ⟨𝑎, 𝑏⟩))
2322rexbiia 3022 . . . . . . . . . 10 (∃𝑐 ∈ ω ((𝑄 ↾ ω)‘𝑐) = ⟨𝑎, 𝑏⟩ ↔ ∃𝑐 ∈ ω (𝑄𝑐) = ⟨𝑎, 𝑏⟩)
2418, 20, 233bitri 285 . . . . . . . . 9 (𝑎ran (𝑄 ↾ ω)𝑏 ↔ ∃𝑐 ∈ ω (𝑄𝑐) = ⟨𝑎, 𝑏⟩)
252seqomlem1 7432 . . . . . . . . . . . . . . . 16 (𝑐 ∈ ω → (𝑄𝑐) = ⟨𝑐, (2nd ‘(𝑄𝑐))⟩)
2625adantl 481 . . . . . . . . . . . . . . 15 ((𝑎 ∈ ω ∧ 𝑐 ∈ ω) → (𝑄𝑐) = ⟨𝑐, (2nd ‘(𝑄𝑐))⟩)
2726eqeq1d 2612 . . . . . . . . . . . . . 14 ((𝑎 ∈ ω ∧ 𝑐 ∈ ω) → ((𝑄𝑐) = ⟨𝑎, 𝑏⟩ ↔ ⟨𝑐, (2nd ‘(𝑄𝑐))⟩ = ⟨𝑎, 𝑏⟩))
28 vex 3176 . . . . . . . . . . . . . . 15 𝑐 ∈ V
29 fvex 6113 . . . . . . . . . . . . . . 15 (2nd ‘(𝑄𝑐)) ∈ V
3028, 29opth1 4870 . . . . . . . . . . . . . 14 (⟨𝑐, (2nd ‘(𝑄𝑐))⟩ = ⟨𝑎, 𝑏⟩ → 𝑐 = 𝑎)
3127, 30syl6bi 242 . . . . . . . . . . . . 13 ((𝑎 ∈ ω ∧ 𝑐 ∈ ω) → ((𝑄𝑐) = ⟨𝑎, 𝑏⟩ → 𝑐 = 𝑎))
32 fveq2 6103 . . . . . . . . . . . . . . 15 (𝑐 = 𝑎 → (𝑄𝑐) = (𝑄𝑎))
3332eqeq1d 2612 . . . . . . . . . . . . . 14 (𝑐 = 𝑎 → ((𝑄𝑐) = ⟨𝑎, 𝑏⟩ ↔ (𝑄𝑎) = ⟨𝑎, 𝑏⟩))
3433biimpd 218 . . . . . . . . . . . . 13 (𝑐 = 𝑎 → ((𝑄𝑐) = ⟨𝑎, 𝑏⟩ → (𝑄𝑎) = ⟨𝑎, 𝑏⟩))
3531, 34syli 38 . . . . . . . . . . . 12 ((𝑎 ∈ ω ∧ 𝑐 ∈ ω) → ((𝑄𝑐) = ⟨𝑎, 𝑏⟩ → (𝑄𝑎) = ⟨𝑎, 𝑏⟩))
36 fveq2 6103 . . . . . . . . . . . . 13 ((𝑄𝑎) = ⟨𝑎, 𝑏⟩ → (2nd ‘(𝑄𝑎)) = (2nd ‘⟨𝑎, 𝑏⟩))
37 vex 3176 . . . . . . . . . . . . . 14 𝑎 ∈ V
38 vex 3176 . . . . . . . . . . . . . 14 𝑏 ∈ V
3937, 38op2nd 7068 . . . . . . . . . . . . 13 (2nd ‘⟨𝑎, 𝑏⟩) = 𝑏
4036, 39syl6req 2661 . . . . . . . . . . . 12 ((𝑄𝑎) = ⟨𝑎, 𝑏⟩ → 𝑏 = (2nd ‘(𝑄𝑎)))
4135, 40syl6 34 . . . . . . . . . . 11 ((𝑎 ∈ ω ∧ 𝑐 ∈ ω) → ((𝑄𝑐) = ⟨𝑎, 𝑏⟩ → 𝑏 = (2nd ‘(𝑄𝑎))))
4241rexlimdva 3013 . . . . . . . . . 10 (𝑎 ∈ ω → (∃𝑐 ∈ ω (𝑄𝑐) = ⟨𝑎, 𝑏⟩ → 𝑏 = (2nd ‘(𝑄𝑎))))
432seqomlem1 7432 . . . . . . . . . . . 12 (𝑎 ∈ ω → (𝑄𝑎) = ⟨𝑎, (2nd ‘(𝑄𝑎))⟩)
4432eqeq1d 2612 . . . . . . . . . . . . 13 (𝑐 = 𝑎 → ((𝑄𝑐) = ⟨𝑎, (2nd ‘(𝑄𝑎))⟩ ↔ (𝑄𝑎) = ⟨𝑎, (2nd ‘(𝑄𝑎))⟩))
4544rspcev 3282 . . . . . . . . . . . 12 ((𝑎 ∈ ω ∧ (𝑄𝑎) = ⟨𝑎, (2nd ‘(𝑄𝑎))⟩) → ∃𝑐 ∈ ω (𝑄𝑐) = ⟨𝑎, (2nd ‘(𝑄𝑎))⟩)
4643, 45mpdan 699 . . . . . . . . . . 11 (𝑎 ∈ ω → ∃𝑐 ∈ ω (𝑄𝑐) = ⟨𝑎, (2nd ‘(𝑄𝑎))⟩)
47 opeq2 4341 . . . . . . . . . . . . 13 (𝑏 = (2nd ‘(𝑄𝑎)) → ⟨𝑎, 𝑏⟩ = ⟨𝑎, (2nd ‘(𝑄𝑎))⟩)
4847eqeq2d 2620 . . . . . . . . . . . 12 (𝑏 = (2nd ‘(𝑄𝑎)) → ((𝑄𝑐) = ⟨𝑎, 𝑏⟩ ↔ (𝑄𝑐) = ⟨𝑎, (2nd ‘(𝑄𝑎))⟩))
4948rexbidv 3034 . . . . . . . . . . 11 (𝑏 = (2nd ‘(𝑄𝑎)) → (∃𝑐 ∈ ω (𝑄𝑐) = ⟨𝑎, 𝑏⟩ ↔ ∃𝑐 ∈ ω (𝑄𝑐) = ⟨𝑎, (2nd ‘(𝑄𝑎))⟩))
5046, 49syl5ibrcom 236 . . . . . . . . . 10 (𝑎 ∈ ω → (𝑏 = (2nd ‘(𝑄𝑎)) → ∃𝑐 ∈ ω (𝑄𝑐) = ⟨𝑎, 𝑏⟩))
5142, 50impbid 201 . . . . . . . . 9 (𝑎 ∈ ω → (∃𝑐 ∈ ω (𝑄𝑐) = ⟨𝑎, 𝑏⟩ ↔ 𝑏 = (2nd ‘(𝑄𝑎))))
5224, 51syl5bb 271 . . . . . . . 8 (𝑎 ∈ ω → (𝑎ran (𝑄 ↾ ω)𝑏𝑏 = (2nd ‘(𝑄𝑎))))
5352alrimiv 1842 . . . . . . 7 (𝑎 ∈ ω → ∀𝑏(𝑎ran (𝑄 ↾ ω)𝑏𝑏 = (2nd ‘(𝑄𝑎))))
54 fvex 6113 . . . . . . . 8 (2nd ‘(𝑄𝑎)) ∈ V
55 eqeq2 2621 . . . . . . . . . 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 1836 . . . . . . . 8 (𝑐 = (2nd ‘(𝑄𝑎)) → (∀𝑏(𝑎ran (𝑄 ↾ ω)𝑏𝑏 = 𝑐) ↔ ∀𝑏(𝑎ran (𝑄 ↾ ω)𝑏𝑏 = (2nd ‘(𝑄𝑎)))))
5854, 57spcev 3273 . . . . . . 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 2462 . . . . . 6 (∃!𝑏 𝑎ran (𝑄 ↾ ω)𝑏 ↔ ∃𝑐𝑏(𝑎ran (𝑄 ↾ ω)𝑏𝑏 = 𝑐))
6159, 60sylibr 223 . . . . 5 (𝑎 ∈ ω → ∃!𝑏 𝑎ran (𝑄 ↾ ω)𝑏)
6261rgen 2906 . . . 4 𝑎 ∈ ω ∃!𝑏 𝑎ran (𝑄 ↾ ω)𝑏
63 dff3 6280 . . . 4 (ran (𝑄 ↾ ω):ω⟶V ↔ (ran (𝑄 ↾ ω) ⊆ (ω × V) ∧ ∀𝑎 ∈ ω ∃!𝑏 𝑎ran (𝑄 ↾ ω)𝑏))
6417, 62, 63mpbir2an 957 . . 3 ran (𝑄 ↾ ω):ω⟶V
65 df-ima 5051 . . . 4 (𝑄 “ ω) = ran (𝑄 ↾ ω)
6665feq1i 5949 . . 3 ((𝑄 “ ω):ω⟶V ↔ ran (𝑄 ↾ ω):ω⟶V)
6764, 66mpbir 220 . 2 (𝑄 “ ω):ω⟶V
68 dffn2 5960 . 2 ((𝑄 “ ω) Fn ω ↔ (𝑄 “ ω):ω⟶V)
6967, 68mpbir 220 1 (𝑄 “ ω) Fn ω
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 195   ∧ wa 383  ∀wal 1473   = wceq 1475  ∃wex 1695   ∈ wcel 1977  ∃!weu 2458  ∀wral 2896  ∃wrex 2897  Vcvv 3173   ⊆ wss 3540  ∅c0 3874  ⟨cop 4131   class class class wbr 4583   I cid 4948   × cxp 5036  ran crn 5039   ↾ cres 5040   “ cima 5041  suc csuc 5642   Fn wfn 5799  ⟶wf 5800  ‘cfv 5804  (class class class)co 6549   ↦ cmpt2 6551  ωcom 6957  2nd c2nd 7058  reccrdg 7392 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393 This theorem is referenced by:  seqomlem3  7434  seqomlem4  7435  fnseqom  7437
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