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Theorem frecuzrdgfunlem 9791
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 9787 . . . . 5 (𝜑𝑅:ω⟶((ℤ𝐶) × 𝑆))
7 frn 5155 . . . . 5 (𝑅:ω⟶((ℤ𝐶) × 𝑆) → ran 𝑅 ⊆ ((ℤ𝐶) × 𝑆))
86, 7syl 14 . . . 4 (𝜑 → ran 𝑅 ⊆ ((ℤ𝐶) × 𝑆))
9 xpss 4534 . . . 4 ((ℤ𝐶) × 𝑆) ⊆ (V × V)
108, 9syl6ss 3035 . . 3 (𝜑 → ran 𝑅 ⊆ (V × V))
11 df-rel 4435 . . 3 (Rel ran 𝑅 ↔ ran 𝑅 ⊆ (V × V))
1210, 11sylibr 132 . 2 (𝜑 → Rel ran 𝑅)
13 frecuzrdgfunlem.g . . . . . . . . . 10 𝐺 = frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 𝐶)
141, 13frec2uzf1od 9778 . . . . . . . . 9 (𝜑𝐺:ω–1-1-onto→(ℤ𝐶))
15 f1ocnvdm 5542 . . . . . . . . 9 ((𝐺:ω–1-1-onto→(ℤ𝐶) ∧ 𝑣 ∈ (ℤ𝐶)) → (𝐺𝑣) ∈ ω)
1614, 15sylan 277 . . . . . . . 8 ((𝜑𝑣 ∈ (ℤ𝐶)) → (𝐺𝑣) ∈ ω)
176ffvelrnda 5418 . . . . . . . 8 ((𝜑 ∧ (𝐺𝑣) ∈ ω) → (𝑅‘(𝐺𝑣)) ∈ ((ℤ𝐶) × 𝑆))
1816, 17syldan 276 . . . . . . 7 ((𝜑𝑣 ∈ (ℤ𝐶)) → (𝑅‘(𝐺𝑣)) ∈ ((ℤ𝐶) × 𝑆))
19 xp2nd 5919 . . . . . . 7 ((𝑅‘(𝐺𝑣)) ∈ ((ℤ𝐶) × 𝑆) → (2nd ‘(𝑅‘(𝐺𝑣))) ∈ 𝑆)
2018, 19syl 14 . . . . . 6 ((𝜑𝑣 ∈ (ℤ𝐶)) → (2nd ‘(𝑅‘(𝐺𝑣))) ∈ 𝑆)
21 ffn 5147 . . . . . . . . . 10 (𝑅:ω⟶((ℤ𝐶) × 𝑆) → 𝑅 Fn ω)
22 fvelrnb 5336 . . . . . . . . . 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 5418 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑤 ∈ ω) → (𝑅𝑤) ∈ ((ℤ𝐶) × 𝑆))
25 1st2nd2 5927 . . . . . . . . . . . . . . . . . . 19 ((𝑅𝑤) ∈ ((ℤ𝐶) × 𝑆) → (𝑅𝑤) = ⟨(1st ‘(𝑅𝑤)), (2nd ‘(𝑅𝑤))⟩)
2624, 25syl 14 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑤 ∈ ω) → (𝑅𝑤) = ⟨(1st ‘(𝑅𝑤)), (2nd ‘(𝑅𝑤))⟩)
271adantr 270 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑤 ∈ ω) → 𝐶 ∈ ℤ)
282adantr 270 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑤 ∈ ω) → 𝐴𝑆)
293adantr 270 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑤 ∈ ω) → 𝑆𝑇)
304adantlr 461 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑤 ∈ ω) ∧ (𝑥 ∈ (ℤ𝐶) ∧ 𝑦𝑆)) → (𝑥𝐹𝑦) ∈ 𝑆)
31 simpr 108 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑤 ∈ ω) → 𝑤 ∈ ω)
3227, 28, 29, 30, 5, 31, 13frecuzrdgg 9788 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑤 ∈ ω) → (1st ‘(𝑅𝑤)) = (𝐺𝑤))
3332opeq1d 3623 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑤 ∈ ω) → ⟨(1st ‘(𝑅𝑤)), (2nd ‘(𝑅𝑤))⟩ = ⟨(𝐺𝑤), (2nd ‘(𝑅𝑤))⟩)
3426, 33eqtrd 2120 . . . . . . . . . . . . . . . . 17 ((𝜑𝑤 ∈ ω) → (𝑅𝑤) = ⟨(𝐺𝑤), (2nd ‘(𝑅𝑤))⟩)
3534eqeq1d 2096 . . . . . . . . . . . . . . . 16 ((𝜑𝑤 ∈ ω) → ((𝑅𝑤) = ⟨𝑣, 𝑧⟩ ↔ ⟨(𝐺𝑤), (2nd ‘(𝑅𝑤))⟩ = ⟨𝑣, 𝑧⟩))
36 vex 2622 . . . . . . . . . . . . . . . . . 18 𝑣 ∈ V
37 vex 2622 . . . . . . . . . . . . . . . . . 18 𝑧 ∈ V
3836, 37opth2 4058 . . . . . . . . . . . . . . . . 17 (⟨(𝐺𝑤), (2nd ‘(𝑅𝑤))⟩ = ⟨𝑣, 𝑧⟩ ↔ ((𝐺𝑤) = 𝑣 ∧ (2nd ‘(𝑅𝑤)) = 𝑧))
3938simplbi 268 . . . . . . . . . . . . . . . 16 (⟨(𝐺𝑤), (2nd ‘(𝑅𝑤))⟩ = ⟨𝑣, 𝑧⟩ → (𝐺𝑤) = 𝑣)
4035, 39syl6bi 161 . . . . . . . . . . . . . . 15 ((𝜑𝑤 ∈ ω) → ((𝑅𝑤) = ⟨𝑣, 𝑧⟩ → (𝐺𝑤) = 𝑣))
41 f1ocnvfv 5540 . . . . . . . . . . . . . . . 16 ((𝐺:ω–1-1-onto→(ℤ𝐶) ∧ 𝑤 ∈ ω) → ((𝐺𝑤) = 𝑣 → (𝐺𝑣) = 𝑤))
4214, 41sylan 277 . . . . . . . . . . . . . . 15 ((𝜑𝑤 ∈ ω) → ((𝐺𝑤) = 𝑣 → (𝐺𝑣) = 𝑤))
4340, 42syld 44 . . . . . . . . . . . . . 14 ((𝜑𝑤 ∈ ω) → ((𝑅𝑤) = ⟨𝑣, 𝑧⟩ → (𝐺𝑣) = 𝑤))
44 fveq2 5289 . . . . . . . . . . . . . . 15 ((𝐺𝑣) = 𝑤 → (𝑅‘(𝐺𝑣)) = (𝑅𝑤))
4544fveq2d 5293 . . . . . . . . . . . . . 14 ((𝐺𝑣) = 𝑤 → (2nd ‘(𝑅‘(𝐺𝑣))) = (2nd ‘(𝑅𝑤)))
4643, 45syl6 33 . . . . . . . . . . . . 13 ((𝜑𝑤 ∈ ω) → ((𝑅𝑤) = ⟨𝑣, 𝑧⟩ → (2nd ‘(𝑅‘(𝐺𝑣))) = (2nd ‘(𝑅𝑤))))
4746imp 122 . . . . . . . . . . . 12 (((𝜑𝑤 ∈ ω) ∧ (𝑅𝑤) = ⟨𝑣, 𝑧⟩) → (2nd ‘(𝑅‘(𝐺𝑣))) = (2nd ‘(𝑅𝑤)))
4836, 37op2ndd 5902 . . . . . . . . . . . . 13 ((𝑅𝑤) = ⟨𝑣, 𝑧⟩ → (2nd ‘(𝑅𝑤)) = 𝑧)
4948adantl 271 . . . . . . . . . . . 12 (((𝜑𝑤 ∈ ω) ∧ (𝑅𝑤) = ⟨𝑣, 𝑧⟩) → (2nd ‘(𝑅𝑤)) = 𝑧)
5047, 49eqtr2d 2121 . . . . . . . . . . 11 (((𝜑𝑤 ∈ ω) ∧ (𝑅𝑤) = ⟨𝑣, 𝑧⟩) → 𝑧 = (2nd ‘(𝑅‘(𝐺𝑣))))
5150ex 113 . . . . . . . . . 10 ((𝜑𝑤 ∈ ω) → ((𝑅𝑤) = ⟨𝑣, 𝑧⟩ → 𝑧 = (2nd ‘(𝑅‘(𝐺𝑣)))))
5251rexlimdva 2489 . . . . . . . . 9 (𝜑 → (∃𝑤 ∈ ω (𝑅𝑤) = ⟨𝑣, 𝑧⟩ → 𝑧 = (2nd ‘(𝑅‘(𝐺𝑣)))))
5323, 52sylbid 148 . . . . . . . 8 (𝜑 → (⟨𝑣, 𝑧⟩ ∈ ran 𝑅𝑧 = (2nd ‘(𝑅‘(𝐺𝑣)))))
5453alrimiv 1802 . . . . . . 7 (𝜑 → ∀𝑧(⟨𝑣, 𝑧⟩ ∈ ran 𝑅𝑧 = (2nd ‘(𝑅‘(𝐺𝑣)))))
5554adantr 270 . . . . . 6 ((𝜑𝑣 ∈ (ℤ𝐶)) → ∀𝑧(⟨𝑣, 𝑧⟩ ∈ ran 𝑅𝑧 = (2nd ‘(𝑅‘(𝐺𝑣)))))
56 eqeq2 2097 . . . . . . . . 9 (𝑤 = (2nd ‘(𝑅‘(𝐺𝑣))) → (𝑧 = 𝑤𝑧 = (2nd ‘(𝑅‘(𝐺𝑣)))))
5756imbi2d 228 . . . . . . . 8 (𝑤 = (2nd ‘(𝑅‘(𝐺𝑣))) → ((⟨𝑣, 𝑧⟩ ∈ ran 𝑅𝑧 = 𝑤) ↔ (⟨𝑣, 𝑧⟩ ∈ ran 𝑅𝑧 = (2nd ‘(𝑅‘(𝐺𝑣))))))
5857albidv 1752 . . . . . . 7 (𝑤 = (2nd ‘(𝑅‘(𝐺𝑣))) → (∀𝑧(⟨𝑣, 𝑧⟩ ∈ ran 𝑅𝑧 = 𝑤) ↔ ∀𝑧(⟨𝑣, 𝑧⟩ ∈ ran 𝑅𝑧 = (2nd ‘(𝑅‘(𝐺𝑣))))))
5958spcegv 2707 . . . . . 6 ((2nd ‘(𝑅‘(𝐺𝑣))) ∈ 𝑆 → (∀𝑧(⟨𝑣, 𝑧⟩ ∈ ran 𝑅𝑧 = (2nd ‘(𝑅‘(𝐺𝑣)))) → ∃𝑤𝑧(⟨𝑣, 𝑧⟩ ∈ ran 𝑅𝑧 = 𝑤)))
6020, 55, 59sylc 61 . . . . 5 ((𝜑𝑣 ∈ (ℤ𝐶)) → ∃𝑤𝑧(⟨𝑣, 𝑧⟩ ∈ ran 𝑅𝑧 = 𝑤))
61 nfv 1466 . . . . . 6 𝑤𝑣, 𝑧⟩ ∈ ran 𝑅
6261mo2r 2000 . . . . 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 9790 . . . . . 6 (𝜑 → dom ran 𝑅 = (ℤ𝐶))
6564eleq2d 2157 . . . . 5 (𝜑 → (𝑣 ∈ dom ran 𝑅𝑣 ∈ (ℤ𝐶)))
6665pm5.32i 442 . . . 4 ((𝜑𝑣 ∈ dom ran 𝑅) ↔ (𝜑𝑣 ∈ (ℤ𝐶)))
67 df-br 3838 . . . . 5 (𝑣ran 𝑅 𝑧 ↔ ⟨𝑣, 𝑧⟩ ∈ ran 𝑅)
6867mobii 1985 . . . 4 (∃*𝑧 𝑣ran 𝑅 𝑧 ↔ ∃*𝑧𝑣, 𝑧⟩ ∈ ran 𝑅)
6963, 66, 683imtr4i 199 . . 3 ((𝜑𝑣 ∈ dom ran 𝑅) → ∃*𝑧 𝑣ran 𝑅 𝑧)
7069ralrimiva 2446 . 2 (𝜑 → ∀𝑣 ∈ dom ran 𝑅∃*𝑧 𝑣ran 𝑅 𝑧)
71 dffun7 5028 . 2 (Fun ran 𝑅 ↔ (Rel ran 𝑅 ∧ ∀𝑣 ∈ dom ran 𝑅∃*𝑧 𝑣ran 𝑅 𝑧))
7212, 70, 71sylanbrc 408 1 (𝜑 → Fun ran 𝑅)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102  wb 103  wal 1287   = wceq 1289  wex 1426  wcel 1438  ∃*wmo 1949  wral 2359  wrex 2360  Vcvv 2619  wss 2997  cop 3444   class class class wbr 3837  cmpt 3891  ωcom 4395   × cxp 4426  ccnv 4427  dom cdm 4428  ran crn 4429  Rel wrel 4433  Fun wfun 4996   Fn wfn 4997  wf 4998  1-1-ontowf1o 5001  cfv 5002  (class class class)co 5634  cmpt2 5636  1st c1st 5891  2nd c2nd 5892  freccfrec 6137  1c1 7330   + caddc 7332  cz 8720  cuz 8988
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-coll 3946  ax-sep 3949  ax-nul 3957  ax-pow 4001  ax-pr 4027  ax-un 4251  ax-setind 4343  ax-iinf 4393  ax-cnex 7415  ax-resscn 7416  ax-1cn 7417  ax-1re 7418  ax-icn 7419  ax-addcl 7420  ax-addrcl 7421  ax-mulcl 7422  ax-addcom 7424  ax-addass 7426  ax-distr 7428  ax-i2m1 7429  ax-0lt1 7430  ax-0id 7432  ax-rnegex 7433  ax-cnre 7435  ax-pre-ltirr 7436  ax-pre-ltwlin 7437  ax-pre-lttrn 7438  ax-pre-ltadd 7440
This theorem depends on definitions:  df-bi 115  df-3or 925  df-3an 926  df-tru 1292  df-fal 1295  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ne 2256  df-nel 2351  df-ral 2364  df-rex 2365  df-reu 2366  df-rab 2368  df-v 2621  df-sbc 2839  df-csb 2932  df-dif 2999  df-un 3001  df-in 3003  df-ss 3010  df-nul 3285  df-pw 3427  df-sn 3447  df-pr 3448  df-op 3450  df-uni 3649  df-int 3684  df-iun 3727  df-br 3838  df-opab 3892  df-mpt 3893  df-tr 3929  df-id 4111  df-iord 4184  df-on 4186  df-ilim 4187  df-suc 4189  df-iom 4396  df-xp 4434  df-rel 4435  df-cnv 4436  df-co 4437  df-dm 4438  df-rn 4439  df-res 4440  df-ima 4441  df-iota 4967  df-fun 5004  df-fn 5005  df-f 5006  df-f1 5007  df-fo 5008  df-f1o 5009  df-fv 5010  df-riota 5590  df-ov 5637  df-oprab 5638  df-mpt2 5639  df-1st 5893  df-2nd 5894  df-recs 6052  df-frec 6138  df-pnf 7503  df-mnf 7504  df-xr 7505  df-ltxr 7506  df-le 7507  df-sub 7634  df-neg 7635  df-inn 8395  df-n0 8644  df-z 8721  df-uz 8989
This theorem is referenced by:  frecuzrdgfun  9792
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