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Theorem frecuzrdgfunlem 10143
Description: The recursive definition generator on upper integers produces a a function. (Contributed by Jim Kingdon, 24-Apr-2022.)
Hypotheses
Ref Expression
frecuzrdgrclt.c (𝜑𝐶 ∈ ℤ)
frecuzrdgrclt.a (𝜑𝐴𝑆)
frecuzrdgrclt.t (𝜑𝑆𝑇)
frecuzrdgrclt.f ((𝜑 ∧ (𝑥 ∈ (ℤ𝐶) ∧ 𝑦𝑆)) → (𝑥𝐹𝑦) ∈ 𝑆)
frecuzrdgrclt.r 𝑅 = frec((𝑥 ∈ (ℤ𝐶), 𝑦𝑇 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩), ⟨𝐶, 𝐴⟩)
frecuzrdgfunlem.g 𝐺 = frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 𝐶)
Assertion
Ref Expression
frecuzrdgfunlem (𝜑 → Fun ran 𝑅)
Distinct variable groups:   𝑥,𝐶,𝑦   𝑥,𝐹,𝑦   𝑥,𝑆,𝑦   𝑥,𝑇,𝑦   𝜑,𝑥,𝑦   𝑥,𝐺,𝑦   𝑥,𝑅,𝑦
Allowed substitution hints:   𝐴(𝑥,𝑦)

Proof of Theorem frecuzrdgfunlem
Dummy variables 𝑧 𝑤 𝑣 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 frecuzrdgrclt.c . . . . . 6 (𝜑𝐶 ∈ ℤ)
2 frecuzrdgrclt.a . . . . . 6 (𝜑𝐴𝑆)
3 frecuzrdgrclt.t . . . . . 6 (𝜑𝑆𝑇)
4 frecuzrdgrclt.f . . . . . 6 ((𝜑 ∧ (𝑥 ∈ (ℤ𝐶) ∧ 𝑦𝑆)) → (𝑥𝐹𝑦) ∈ 𝑆)
5 frecuzrdgrclt.r . . . . . 6 𝑅 = frec((𝑥 ∈ (ℤ𝐶), 𝑦𝑇 ↦ ⟨(𝑥 + 1), (𝑥𝐹𝑦)⟩), ⟨𝐶, 𝐴⟩)
61, 2, 3, 4, 5frecuzrdgrclt 10139 . . . . 5 (𝜑𝑅:ω⟶((ℤ𝐶) × 𝑆))
7 frn 5249 . . . . 5 (𝑅:ω⟶((ℤ𝐶) × 𝑆) → ran 𝑅 ⊆ ((ℤ𝐶) × 𝑆))
86, 7syl 14 . . . 4 (𝜑 → ran 𝑅 ⊆ ((ℤ𝐶) × 𝑆))
9 xpss 4615 . . . 4 ((ℤ𝐶) × 𝑆) ⊆ (V × V)
108, 9sstrdi 3077 . . 3 (𝜑 → ran 𝑅 ⊆ (V × V))
11 df-rel 4514 . . 3 (Rel ran 𝑅 ↔ ran 𝑅 ⊆ (V × V))
1210, 11sylibr 133 . 2 (𝜑 → Rel ran 𝑅)
13 frecuzrdgfunlem.g . . . . . . . . . 10 𝐺 = frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 𝐶)
141, 13frec2uzf1od 10130 . . . . . . . . 9 (𝜑𝐺:ω–1-1-onto→(ℤ𝐶))
15 f1ocnvdm 5648 . . . . . . . . 9 ((𝐺:ω–1-1-onto→(ℤ𝐶) ∧ 𝑣 ∈ (ℤ𝐶)) → (𝐺𝑣) ∈ ω)
1614, 15sylan 279 . . . . . . . 8 ((𝜑𝑣 ∈ (ℤ𝐶)) → (𝐺𝑣) ∈ ω)
176ffvelrnda 5521 . . . . . . . 8 ((𝜑 ∧ (𝐺𝑣) ∈ ω) → (𝑅‘(𝐺𝑣)) ∈ ((ℤ𝐶) × 𝑆))
1816, 17syldan 278 . . . . . . 7 ((𝜑𝑣 ∈ (ℤ𝐶)) → (𝑅‘(𝐺𝑣)) ∈ ((ℤ𝐶) × 𝑆))
19 xp2nd 6030 . . . . . . 7 ((𝑅‘(𝐺𝑣)) ∈ ((ℤ𝐶) × 𝑆) → (2nd ‘(𝑅‘(𝐺𝑣))) ∈ 𝑆)
2018, 19syl 14 . . . . . 6 ((𝜑𝑣 ∈ (ℤ𝐶)) → (2nd ‘(𝑅‘(𝐺𝑣))) ∈ 𝑆)
21 ffn 5240 . . . . . . . . . 10 (𝑅:ω⟶((ℤ𝐶) × 𝑆) → 𝑅 Fn ω)
22 fvelrnb 5435 . . . . . . . . . 10 (𝑅 Fn ω → (⟨𝑣, 𝑧⟩ ∈ ran 𝑅 ↔ ∃𝑤 ∈ ω (𝑅𝑤) = ⟨𝑣, 𝑧⟩))
236, 21, 223syl 17 . . . . . . . . 9 (𝜑 → (⟨𝑣, 𝑧⟩ ∈ ran 𝑅 ↔ ∃𝑤 ∈ ω (𝑅𝑤) = ⟨𝑣, 𝑧⟩))
246ffvelrnda 5521 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑤 ∈ ω) → (𝑅𝑤) ∈ ((ℤ𝐶) × 𝑆))
25 1st2nd2 6039 . . . . . . . . . . . . . . . . . . 19 ((𝑅𝑤) ∈ ((ℤ𝐶) × 𝑆) → (𝑅𝑤) = ⟨(1st ‘(𝑅𝑤)), (2nd ‘(𝑅𝑤))⟩)
2624, 25syl 14 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑤 ∈ ω) → (𝑅𝑤) = ⟨(1st ‘(𝑅𝑤)), (2nd ‘(𝑅𝑤))⟩)
271adantr 272 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑤 ∈ ω) → 𝐶 ∈ ℤ)
282adantr 272 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑤 ∈ ω) → 𝐴𝑆)
293adantr 272 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑤 ∈ ω) → 𝑆𝑇)
304adantlr 466 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑤 ∈ ω) ∧ (𝑥 ∈ (ℤ𝐶) ∧ 𝑦𝑆)) → (𝑥𝐹𝑦) ∈ 𝑆)
31 simpr 109 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑤 ∈ ω) → 𝑤 ∈ ω)
3227, 28, 29, 30, 5, 31, 13frecuzrdgg 10140 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑤 ∈ ω) → (1st ‘(𝑅𝑤)) = (𝐺𝑤))
3332opeq1d 3679 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑤 ∈ ω) → ⟨(1st ‘(𝑅𝑤)), (2nd ‘(𝑅𝑤))⟩ = ⟨(𝐺𝑤), (2nd ‘(𝑅𝑤))⟩)
3426, 33eqtrd 2148 . . . . . . . . . . . . . . . . 17 ((𝜑𝑤 ∈ ω) → (𝑅𝑤) = ⟨(𝐺𝑤), (2nd ‘(𝑅𝑤))⟩)
3534eqeq1d 2124 . . . . . . . . . . . . . . . 16 ((𝜑𝑤 ∈ ω) → ((𝑅𝑤) = ⟨𝑣, 𝑧⟩ ↔ ⟨(𝐺𝑤), (2nd ‘(𝑅𝑤))⟩ = ⟨𝑣, 𝑧⟩))
36 vex 2661 . . . . . . . . . . . . . . . . . 18 𝑣 ∈ V
37 vex 2661 . . . . . . . . . . . . . . . . . 18 𝑧 ∈ V
3836, 37opth2 4130 . . . . . . . . . . . . . . . . 17 (⟨(𝐺𝑤), (2nd ‘(𝑅𝑤))⟩ = ⟨𝑣, 𝑧⟩ ↔ ((𝐺𝑤) = 𝑣 ∧ (2nd ‘(𝑅𝑤)) = 𝑧))
3938simplbi 270 . . . . . . . . . . . . . . . 16 (⟨(𝐺𝑤), (2nd ‘(𝑅𝑤))⟩ = ⟨𝑣, 𝑧⟩ → (𝐺𝑤) = 𝑣)
4035, 39syl6bi 162 . . . . . . . . . . . . . . 15 ((𝜑𝑤 ∈ ω) → ((𝑅𝑤) = ⟨𝑣, 𝑧⟩ → (𝐺𝑤) = 𝑣))
41 f1ocnvfv 5646 . . . . . . . . . . . . . . . 16 ((𝐺:ω–1-1-onto→(ℤ𝐶) ∧ 𝑤 ∈ ω) → ((𝐺𝑤) = 𝑣 → (𝐺𝑣) = 𝑤))
4214, 41sylan 279 . . . . . . . . . . . . . . 15 ((𝜑𝑤 ∈ ω) → ((𝐺𝑤) = 𝑣 → (𝐺𝑣) = 𝑤))
4340, 42syld 45 . . . . . . . . . . . . . 14 ((𝜑𝑤 ∈ ω) → ((𝑅𝑤) = ⟨𝑣, 𝑧⟩ → (𝐺𝑣) = 𝑤))
44 fveq2 5387 . . . . . . . . . . . . . . 15 ((𝐺𝑣) = 𝑤 → (𝑅‘(𝐺𝑣)) = (𝑅𝑤))
4544fveq2d 5391 . . . . . . . . . . . . . 14 ((𝐺𝑣) = 𝑤 → (2nd ‘(𝑅‘(𝐺𝑣))) = (2nd ‘(𝑅𝑤)))
4643, 45syl6 33 . . . . . . . . . . . . 13 ((𝜑𝑤 ∈ ω) → ((𝑅𝑤) = ⟨𝑣, 𝑧⟩ → (2nd ‘(𝑅‘(𝐺𝑣))) = (2nd ‘(𝑅𝑤))))
4746imp 123 . . . . . . . . . . . 12 (((𝜑𝑤 ∈ ω) ∧ (𝑅𝑤) = ⟨𝑣, 𝑧⟩) → (2nd ‘(𝑅‘(𝐺𝑣))) = (2nd ‘(𝑅𝑤)))
4836, 37op2ndd 6013 . . . . . . . . . . . . 13 ((𝑅𝑤) = ⟨𝑣, 𝑧⟩ → (2nd ‘(𝑅𝑤)) = 𝑧)
4948adantl 273 . . . . . . . . . . . 12 (((𝜑𝑤 ∈ ω) ∧ (𝑅𝑤) = ⟨𝑣, 𝑧⟩) → (2nd ‘(𝑅𝑤)) = 𝑧)
5047, 49eqtr2d 2149 . . . . . . . . . . 11 (((𝜑𝑤 ∈ ω) ∧ (𝑅𝑤) = ⟨𝑣, 𝑧⟩) → 𝑧 = (2nd ‘(𝑅‘(𝐺𝑣))))
5150ex 114 . . . . . . . . . 10 ((𝜑𝑤 ∈ ω) → ((𝑅𝑤) = ⟨𝑣, 𝑧⟩ → 𝑧 = (2nd ‘(𝑅‘(𝐺𝑣)))))
5251rexlimdva 2524 . . . . . . . . 9 (𝜑 → (∃𝑤 ∈ ω (𝑅𝑤) = ⟨𝑣, 𝑧⟩ → 𝑧 = (2nd ‘(𝑅‘(𝐺𝑣)))))
5323, 52sylbid 149 . . . . . . . 8 (𝜑 → (⟨𝑣, 𝑧⟩ ∈ ran 𝑅𝑧 = (2nd ‘(𝑅‘(𝐺𝑣)))))
5453alrimiv 1828 . . . . . . 7 (𝜑 → ∀𝑧(⟨𝑣, 𝑧⟩ ∈ ran 𝑅𝑧 = (2nd ‘(𝑅‘(𝐺𝑣)))))
5554adantr 272 . . . . . 6 ((𝜑𝑣 ∈ (ℤ𝐶)) → ∀𝑧(⟨𝑣, 𝑧⟩ ∈ ran 𝑅𝑧 = (2nd ‘(𝑅‘(𝐺𝑣)))))
56 eqeq2 2125 . . . . . . . . 9 (𝑤 = (2nd ‘(𝑅‘(𝐺𝑣))) → (𝑧 = 𝑤𝑧 = (2nd ‘(𝑅‘(𝐺𝑣)))))
5756imbi2d 229 . . . . . . . 8 (𝑤 = (2nd ‘(𝑅‘(𝐺𝑣))) → ((⟨𝑣, 𝑧⟩ ∈ ran 𝑅𝑧 = 𝑤) ↔ (⟨𝑣, 𝑧⟩ ∈ ran 𝑅𝑧 = (2nd ‘(𝑅‘(𝐺𝑣))))))
5857albidv 1778 . . . . . . 7 (𝑤 = (2nd ‘(𝑅‘(𝐺𝑣))) → (∀𝑧(⟨𝑣, 𝑧⟩ ∈ ran 𝑅𝑧 = 𝑤) ↔ ∀𝑧(⟨𝑣, 𝑧⟩ ∈ ran 𝑅𝑧 = (2nd ‘(𝑅‘(𝐺𝑣))))))
5958spcegv 2746 . . . . . 6 ((2nd ‘(𝑅‘(𝐺𝑣))) ∈ 𝑆 → (∀𝑧(⟨𝑣, 𝑧⟩ ∈ ran 𝑅𝑧 = (2nd ‘(𝑅‘(𝐺𝑣)))) → ∃𝑤𝑧(⟨𝑣, 𝑧⟩ ∈ ran 𝑅𝑧 = 𝑤)))
6020, 55, 59sylc 62 . . . . 5 ((𝜑𝑣 ∈ (ℤ𝐶)) → ∃𝑤𝑧(⟨𝑣, 𝑧⟩ ∈ ran 𝑅𝑧 = 𝑤))
61 nfv 1491 . . . . . 6 𝑤𝑣, 𝑧⟩ ∈ ran 𝑅
6261mo2r 2027 . . . . 5 (∃𝑤𝑧(⟨𝑣, 𝑧⟩ ∈ ran 𝑅𝑧 = 𝑤) → ∃*𝑧𝑣, 𝑧⟩ ∈ ran 𝑅)
6360, 62syl 14 . . . 4 ((𝜑𝑣 ∈ (ℤ𝐶)) → ∃*𝑧𝑣, 𝑧⟩ ∈ ran 𝑅)
641, 2, 3, 4, 5frecuzrdgdom 10142 . . . . . 6 (𝜑 → dom ran 𝑅 = (ℤ𝐶))
6564eleq2d 2185 . . . . 5 (𝜑 → (𝑣 ∈ dom ran 𝑅𝑣 ∈ (ℤ𝐶)))
6665pm5.32i 447 . . . 4 ((𝜑𝑣 ∈ dom ran 𝑅) ↔ (𝜑𝑣 ∈ (ℤ𝐶)))
67 df-br 3898 . . . . 5 (𝑣ran 𝑅 𝑧 ↔ ⟨𝑣, 𝑧⟩ ∈ ran 𝑅)
6867mobii 2012 . . . 4 (∃*𝑧 𝑣ran 𝑅 𝑧 ↔ ∃*𝑧𝑣, 𝑧⟩ ∈ ran 𝑅)
6963, 66, 683imtr4i 200 . . 3 ((𝜑𝑣 ∈ dom ran 𝑅) → ∃*𝑧 𝑣ran 𝑅 𝑧)
7069ralrimiva 2480 . 2 (𝜑 → ∀𝑣 ∈ dom ran 𝑅∃*𝑧 𝑣ran 𝑅 𝑧)
71 dffun7 5118 . 2 (Fun ran 𝑅 ↔ (Rel ran 𝑅 ∧ ∀𝑣 ∈ dom ran 𝑅∃*𝑧 𝑣ran 𝑅 𝑧))
7212, 70, 71sylanbrc 411 1 (𝜑 → Fun ran 𝑅)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wb 104  wal 1312   = wceq 1314  wex 1451  wcel 1463  ∃*wmo 1976  wral 2391  wrex 2392  Vcvv 2658  wss 3039  cop 3498   class class class wbr 3897  cmpt 3957  ωcom 4472   × cxp 4505  ccnv 4506  dom cdm 4507  ran crn 4508  Rel wrel 4512  Fun wfun 5085   Fn wfn 5086  wf 5087  1-1-ontowf1o 5090  cfv 5091  (class class class)co 5740  cmpo 5742  1st c1st 6002  2nd c2nd 6003  freccfrec 6253  1c1 7585   + caddc 7587  cz 9008  cuz 9278
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 586  ax-in2 587  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-13 1474  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-coll 4011  ax-sep 4014  ax-nul 4022  ax-pow 4066  ax-pr 4099  ax-un 4323  ax-setind 4420  ax-iinf 4470  ax-cnex 7675  ax-resscn 7676  ax-1cn 7677  ax-1re 7678  ax-icn 7679  ax-addcl 7680  ax-addrcl 7681  ax-mulcl 7682  ax-addcom 7684  ax-addass 7686  ax-distr 7688  ax-i2m1 7689  ax-0lt1 7690  ax-0id 7692  ax-rnegex 7693  ax-cnre 7695  ax-pre-ltirr 7696  ax-pre-ltwlin 7697  ax-pre-lttrn 7698  ax-pre-ltadd 7700
This theorem depends on definitions:  df-bi 116  df-3or 946  df-3an 947  df-tru 1317  df-fal 1320  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-ne 2284  df-nel 2379  df-ral 2396  df-rex 2397  df-reu 2398  df-rab 2400  df-v 2660  df-sbc 2881  df-csb 2974  df-dif 3041  df-un 3043  df-in 3045  df-ss 3052  df-nul 3332  df-pw 3480  df-sn 3501  df-pr 3502  df-op 3504  df-uni 3705  df-int 3740  df-iun 3783  df-br 3898  df-opab 3958  df-mpt 3959  df-tr 3995  df-id 4183  df-iord 4256  df-on 4258  df-ilim 4259  df-suc 4261  df-iom 4473  df-xp 4513  df-rel 4514  df-cnv 4515  df-co 4516  df-dm 4517  df-rn 4518  df-res 4519  df-ima 4520  df-iota 5056  df-fun 5093  df-fn 5094  df-f 5095  df-f1 5096  df-fo 5097  df-f1o 5098  df-fv 5099  df-riota 5696  df-ov 5743  df-oprab 5744  df-mpo 5745  df-1st 6004  df-2nd 6005  df-recs 6168  df-frec 6254  df-pnf 7766  df-mnf 7767  df-xr 7768  df-ltxr 7769  df-le 7770  df-sub 7899  df-neg 7900  df-inn 8681  df-n0 8932  df-z 9009  df-uz 9279
This theorem is referenced by:  frecuzrdgfun  10144
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