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Theorem frecuzrdgfunlem 10421
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 10417 . . . . 5 (𝜑𝑅:ω⟶((ℤ𝐶) × 𝑆))
7 frn 5376 . . . . 5 (𝑅:ω⟶((ℤ𝐶) × 𝑆) → ran 𝑅 ⊆ ((ℤ𝐶) × 𝑆))
86, 7syl 14 . . . 4 (𝜑 → ran 𝑅 ⊆ ((ℤ𝐶) × 𝑆))
9 xpss 4736 . . . 4 ((ℤ𝐶) × 𝑆) ⊆ (V × V)
108, 9sstrdi 3169 . . 3 (𝜑 → ran 𝑅 ⊆ (V × V))
11 df-rel 4635 . . 3 (Rel ran 𝑅 ↔ ran 𝑅 ⊆ (V × V))
1210, 11sylibr 134 . 2 (𝜑 → Rel ran 𝑅)
13 frecuzrdgfunlem.g . . . . . . . . . 10 𝐺 = frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 𝐶)
141, 13frec2uzf1od 10408 . . . . . . . . 9 (𝜑𝐺:ω–1-1-onto→(ℤ𝐶))
15 f1ocnvdm 5784 . . . . . . . . 9 ((𝐺:ω–1-1-onto→(ℤ𝐶) ∧ 𝑣 ∈ (ℤ𝐶)) → (𝐺𝑣) ∈ ω)
1614, 15sylan 283 . . . . . . . 8 ((𝜑𝑣 ∈ (ℤ𝐶)) → (𝐺𝑣) ∈ ω)
176ffvelcdmda 5653 . . . . . . . 8 ((𝜑 ∧ (𝐺𝑣) ∈ ω) → (𝑅‘(𝐺𝑣)) ∈ ((ℤ𝐶) × 𝑆))
1816, 17syldan 282 . . . . . . 7 ((𝜑𝑣 ∈ (ℤ𝐶)) → (𝑅‘(𝐺𝑣)) ∈ ((ℤ𝐶) × 𝑆))
19 xp2nd 6169 . . . . . . 7 ((𝑅‘(𝐺𝑣)) ∈ ((ℤ𝐶) × 𝑆) → (2nd ‘(𝑅‘(𝐺𝑣))) ∈ 𝑆)
2018, 19syl 14 . . . . . 6 ((𝜑𝑣 ∈ (ℤ𝐶)) → (2nd ‘(𝑅‘(𝐺𝑣))) ∈ 𝑆)
21 ffn 5367 . . . . . . . . . 10 (𝑅:ω⟶((ℤ𝐶) × 𝑆) → 𝑅 Fn ω)
22 fvelrnb 5565 . . . . . . . . . 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⟩))
246ffvelcdmda 5653 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑤 ∈ ω) → (𝑅𝑤) ∈ ((ℤ𝐶) × 𝑆))
25 1st2nd2 6178 . . . . . . . . . . . . . . . . . . 19 ((𝑅𝑤) ∈ ((ℤ𝐶) × 𝑆) → (𝑅𝑤) = ⟨(1st ‘(𝑅𝑤)), (2nd ‘(𝑅𝑤))⟩)
2624, 25syl 14 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑤 ∈ ω) → (𝑅𝑤) = ⟨(1st ‘(𝑅𝑤)), (2nd ‘(𝑅𝑤))⟩)
271adantr 276 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑤 ∈ ω) → 𝐶 ∈ ℤ)
282adantr 276 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑤 ∈ ω) → 𝐴𝑆)
293adantr 276 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑤 ∈ ω) → 𝑆𝑇)
304adantlr 477 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑤 ∈ ω) ∧ (𝑥 ∈ (ℤ𝐶) ∧ 𝑦𝑆)) → (𝑥𝐹𝑦) ∈ 𝑆)
31 simpr 110 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑤 ∈ ω) → 𝑤 ∈ ω)
3227, 28, 29, 30, 5, 31, 13frecuzrdgg 10418 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑤 ∈ ω) → (1st ‘(𝑅𝑤)) = (𝐺𝑤))
3332opeq1d 3786 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑤 ∈ ω) → ⟨(1st ‘(𝑅𝑤)), (2nd ‘(𝑅𝑤))⟩ = ⟨(𝐺𝑤), (2nd ‘(𝑅𝑤))⟩)
3426, 33eqtrd 2210 . . . . . . . . . . . . . . . . 17 ((𝜑𝑤 ∈ ω) → (𝑅𝑤) = ⟨(𝐺𝑤), (2nd ‘(𝑅𝑤))⟩)
3534eqeq1d 2186 . . . . . . . . . . . . . . . 16 ((𝜑𝑤 ∈ ω) → ((𝑅𝑤) = ⟨𝑣, 𝑧⟩ ↔ ⟨(𝐺𝑤), (2nd ‘(𝑅𝑤))⟩ = ⟨𝑣, 𝑧⟩))
36 vex 2742 . . . . . . . . . . . . . . . . . 18 𝑣 ∈ V
37 vex 2742 . . . . . . . . . . . . . . . . . 18 𝑧 ∈ V
3836, 37opth2 4242 . . . . . . . . . . . . . . . . 17 (⟨(𝐺𝑤), (2nd ‘(𝑅𝑤))⟩ = ⟨𝑣, 𝑧⟩ ↔ ((𝐺𝑤) = 𝑣 ∧ (2nd ‘(𝑅𝑤)) = 𝑧))
3938simplbi 274 . . . . . . . . . . . . . . . 16 (⟨(𝐺𝑤), (2nd ‘(𝑅𝑤))⟩ = ⟨𝑣, 𝑧⟩ → (𝐺𝑤) = 𝑣)
4035, 39biimtrdi 163 . . . . . . . . . . . . . . 15 ((𝜑𝑤 ∈ ω) → ((𝑅𝑤) = ⟨𝑣, 𝑧⟩ → (𝐺𝑤) = 𝑣))
41 f1ocnvfv 5782 . . . . . . . . . . . . . . . 16 ((𝐺:ω–1-1-onto→(ℤ𝐶) ∧ 𝑤 ∈ ω) → ((𝐺𝑤) = 𝑣 → (𝐺𝑣) = 𝑤))
4214, 41sylan 283 . . . . . . . . . . . . . . 15 ((𝜑𝑤 ∈ ω) → ((𝐺𝑤) = 𝑣 → (𝐺𝑣) = 𝑤))
4340, 42syld 45 . . . . . . . . . . . . . 14 ((𝜑𝑤 ∈ ω) → ((𝑅𝑤) = ⟨𝑣, 𝑧⟩ → (𝐺𝑣) = 𝑤))
44 fveq2 5517 . . . . . . . . . . . . . . 15 ((𝐺𝑣) = 𝑤 → (𝑅‘(𝐺𝑣)) = (𝑅𝑤))
4544fveq2d 5521 . . . . . . . . . . . . . 14 ((𝐺𝑣) = 𝑤 → (2nd ‘(𝑅‘(𝐺𝑣))) = (2nd ‘(𝑅𝑤)))
4643, 45syl6 33 . . . . . . . . . . . . 13 ((𝜑𝑤 ∈ ω) → ((𝑅𝑤) = ⟨𝑣, 𝑧⟩ → (2nd ‘(𝑅‘(𝐺𝑣))) = (2nd ‘(𝑅𝑤))))
4746imp 124 . . . . . . . . . . . 12 (((𝜑𝑤 ∈ ω) ∧ (𝑅𝑤) = ⟨𝑣, 𝑧⟩) → (2nd ‘(𝑅‘(𝐺𝑣))) = (2nd ‘(𝑅𝑤)))
4836, 37op2ndd 6152 . . . . . . . . . . . . 13 ((𝑅𝑤) = ⟨𝑣, 𝑧⟩ → (2nd ‘(𝑅𝑤)) = 𝑧)
4948adantl 277 . . . . . . . . . . . 12 (((𝜑𝑤 ∈ ω) ∧ (𝑅𝑤) = ⟨𝑣, 𝑧⟩) → (2nd ‘(𝑅𝑤)) = 𝑧)
5047, 49eqtr2d 2211 . . . . . . . . . . 11 (((𝜑𝑤 ∈ ω) ∧ (𝑅𝑤) = ⟨𝑣, 𝑧⟩) → 𝑧 = (2nd ‘(𝑅‘(𝐺𝑣))))
5150ex 115 . . . . . . . . . 10 ((𝜑𝑤 ∈ ω) → ((𝑅𝑤) = ⟨𝑣, 𝑧⟩ → 𝑧 = (2nd ‘(𝑅‘(𝐺𝑣)))))
5251rexlimdva 2594 . . . . . . . . 9 (𝜑 → (∃𝑤 ∈ ω (𝑅𝑤) = ⟨𝑣, 𝑧⟩ → 𝑧 = (2nd ‘(𝑅‘(𝐺𝑣)))))
5323, 52sylbid 150 . . . . . . . 8 (𝜑 → (⟨𝑣, 𝑧⟩ ∈ ran 𝑅𝑧 = (2nd ‘(𝑅‘(𝐺𝑣)))))
5453alrimiv 1874 . . . . . . 7 (𝜑 → ∀𝑧(⟨𝑣, 𝑧⟩ ∈ ran 𝑅𝑧 = (2nd ‘(𝑅‘(𝐺𝑣)))))
5554adantr 276 . . . . . 6 ((𝜑𝑣 ∈ (ℤ𝐶)) → ∀𝑧(⟨𝑣, 𝑧⟩ ∈ ran 𝑅𝑧 = (2nd ‘(𝑅‘(𝐺𝑣)))))
56 eqeq2 2187 . . . . . . . . 9 (𝑤 = (2nd ‘(𝑅‘(𝐺𝑣))) → (𝑧 = 𝑤𝑧 = (2nd ‘(𝑅‘(𝐺𝑣)))))
5756imbi2d 230 . . . . . . . 8 (𝑤 = (2nd ‘(𝑅‘(𝐺𝑣))) → ((⟨𝑣, 𝑧⟩ ∈ ran 𝑅𝑧 = 𝑤) ↔ (⟨𝑣, 𝑧⟩ ∈ ran 𝑅𝑧 = (2nd ‘(𝑅‘(𝐺𝑣))))))
5857albidv 1824 . . . . . . 7 (𝑤 = (2nd ‘(𝑅‘(𝐺𝑣))) → (∀𝑧(⟨𝑣, 𝑧⟩ ∈ ran 𝑅𝑧 = 𝑤) ↔ ∀𝑧(⟨𝑣, 𝑧⟩ ∈ ran 𝑅𝑧 = (2nd ‘(𝑅‘(𝐺𝑣))))))
5958spcegv 2827 . . . . . 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 1528 . . . . . 6 𝑤𝑣, 𝑧⟩ ∈ ran 𝑅
6261mo2r 2078 . . . . 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 10420 . . . . . 6 (𝜑 → dom ran 𝑅 = (ℤ𝐶))
6564eleq2d 2247 . . . . 5 (𝜑 → (𝑣 ∈ dom ran 𝑅𝑣 ∈ (ℤ𝐶)))
6665pm5.32i 454 . . . 4 ((𝜑𝑣 ∈ dom ran 𝑅) ↔ (𝜑𝑣 ∈ (ℤ𝐶)))
67 df-br 4006 . . . . 5 (𝑣ran 𝑅 𝑧 ↔ ⟨𝑣, 𝑧⟩ ∈ ran 𝑅)
6867mobii 2063 . . . 4 (∃*𝑧 𝑣ran 𝑅 𝑧 ↔ ∃*𝑧𝑣, 𝑧⟩ ∈ ran 𝑅)
6963, 66, 683imtr4i 201 . . 3 ((𝜑𝑣 ∈ dom ran 𝑅) → ∃*𝑧 𝑣ran 𝑅 𝑧)
7069ralrimiva 2550 . 2 (𝜑 → ∀𝑣 ∈ dom ran 𝑅∃*𝑧 𝑣ran 𝑅 𝑧)
71 dffun7 5245 . 2 (Fun ran 𝑅 ↔ (Rel ran 𝑅 ∧ ∀𝑣 ∈ dom ran 𝑅∃*𝑧 𝑣ran 𝑅 𝑧))
7212, 70, 71sylanbrc 417 1 (𝜑 → Fun ran 𝑅)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  wb 105  wal 1351   = wceq 1353  wex 1492  ∃*wmo 2027  wcel 2148  wral 2455  wrex 2456  Vcvv 2739  wss 3131  cop 3597   class class class wbr 4005  cmpt 4066  ωcom 4591   × cxp 4626  ccnv 4627  dom cdm 4628  ran crn 4629  Rel wrel 4633  Fun wfun 5212   Fn wfn 5213  wf 5214  1-1-ontowf1o 5217  cfv 5218  (class class class)co 5877  cmpo 5879  1st c1st 6141  2nd c2nd 6142  freccfrec 6393  1c1 7814   + caddc 7816  cz 9255  cuz 9530
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4120  ax-sep 4123  ax-nul 4131  ax-pow 4176  ax-pr 4211  ax-un 4435  ax-setind 4538  ax-iinf 4589  ax-cnex 7904  ax-resscn 7905  ax-1cn 7906  ax-1re 7907  ax-icn 7908  ax-addcl 7909  ax-addrcl 7910  ax-mulcl 7911  ax-addcom 7913  ax-addass 7915  ax-distr 7917  ax-i2m1 7918  ax-0lt1 7919  ax-0id 7921  ax-rnegex 7922  ax-cnre 7924  ax-pre-ltirr 7925  ax-pre-ltwlin 7926  ax-pre-lttrn 7927  ax-pre-ltadd 7929
This theorem depends on definitions:  df-bi 117  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2741  df-sbc 2965  df-csb 3060  df-dif 3133  df-un 3135  df-in 3137  df-ss 3144  df-nul 3425  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-int 3847  df-iun 3890  df-br 4006  df-opab 4067  df-mpt 4068  df-tr 4104  df-id 4295  df-iord 4368  df-on 4370  df-ilim 4371  df-suc 4373  df-iom 4592  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-rn 4639  df-res 4640  df-ima 4641  df-iota 5180  df-fun 5220  df-fn 5221  df-f 5222  df-f1 5223  df-fo 5224  df-f1o 5225  df-fv 5226  df-riota 5833  df-ov 5880  df-oprab 5881  df-mpo 5882  df-1st 6143  df-2nd 6144  df-recs 6308  df-frec 6394  df-pnf 7996  df-mnf 7997  df-xr 7998  df-ltxr 7999  df-le 8000  df-sub 8132  df-neg 8133  df-inn 8922  df-n0 9179  df-z 9256  df-uz 9531
This theorem is referenced by:  frecuzrdgfun  10422
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