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Theorem fpr1 8302
Description: Law of well-founded recursion over a partial order, part one. Establish the functionality and domain of the recursive function generator. Note that by requiring a partial order we can avoid using the axiom of infinity. (Contributed by Scott Fenton, 11-Sep-2023.)
Hypothesis
Ref Expression
fprr.1 𝐹 = frecs(𝑅, 𝐴, 𝐺)
Assertion
Ref Expression
fpr1 ((𝑅 Fr 𝐴𝑅 Po 𝐴𝑅 Se 𝐴) → 𝐹 Fn 𝐴)

Proof of Theorem fpr1
Dummy variables 𝑥 𝑦 𝑧 𝑢 𝑣 𝑎 𝑏 𝑐 𝑓 𝑔 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2728 . . . 4 {𝑓 ∣ ∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥𝐴 ∧ ∀𝑦𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝑦𝐺(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))))} = {𝑓 ∣ ∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥𝐴 ∧ ∀𝑦𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝑦𝐺(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))))}
21frrlem1 8285 . . 3 {𝑓 ∣ ∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥𝐴 ∧ ∀𝑦𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝑦𝐺(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))))} = {𝑎 ∣ ∃𝑏(𝑎 Fn 𝑏 ∧ (𝑏𝐴 ∧ ∀𝑐𝑏 Pred(𝑅, 𝐴, 𝑐) ⊆ 𝑏) ∧ ∀𝑐𝑏 (𝑎𝑐) = (𝑐𝐺(𝑎 ↾ Pred(𝑅, 𝐴, 𝑐))))}
3 fprr.1 . . 3 𝐹 = frecs(𝑅, 𝐴, 𝐺)
42, 3fprlem1 8299 . . 3 (((𝑅 Fr 𝐴𝑅 Po 𝐴𝑅 Se 𝐴) ∧ (𝑔 ∈ {𝑓 ∣ ∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥𝐴 ∧ ∀𝑦𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝑦𝐺(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))))} ∧ ∈ {𝑓 ∣ ∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥𝐴 ∧ ∀𝑦𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝑦𝐺(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))))})) → ((𝑏𝑔𝑢𝑏𝑣) → 𝑢 = 𝑣))
52, 3, 4frrlem9 8293 . 2 ((𝑅 Fr 𝐴𝑅 Po 𝐴𝑅 Se 𝐴) → Fun 𝐹)
6 eqid 2728 . . 3 ((𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)) ∪ {⟨𝑧, (𝑧𝐺(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)))⟩}) = ((𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)) ∪ {⟨𝑧, (𝑧𝐺(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)))⟩})
7 simp1 1134 . . 3 ((𝑅 Fr 𝐴𝑅 Po 𝐴𝑅 Se 𝐴) → 𝑅 Fr 𝐴)
8 ssidd 4001 . . 3 (((𝑅 Fr 𝐴𝑅 Po 𝐴𝑅 Se 𝐴) ∧ 𝑧𝐴) → Pred(𝑅, 𝐴, 𝑧) ⊆ Pred(𝑅, 𝐴, 𝑧))
9 fprlem2 8300 . . 3 (((𝑅 Fr 𝐴𝑅 Po 𝐴𝑅 Se 𝐴) ∧ 𝑧𝐴) → ∀𝑦 ∈ Pred (𝑅, 𝐴, 𝑧)Pred(𝑅, 𝐴, 𝑦) ⊆ Pred(𝑅, 𝐴, 𝑧))
10 setlikespec 6325 . . . . 5 ((𝑧𝐴𝑅 Se 𝐴) → Pred(𝑅, 𝐴, 𝑧) ∈ V)
1110ancoms 458 . . . 4 ((𝑅 Se 𝐴𝑧𝐴) → Pred(𝑅, 𝐴, 𝑧) ∈ V)
12113ad2antl3 1185 . . 3 (((𝑅 Fr 𝐴𝑅 Po 𝐴𝑅 Se 𝐴) ∧ 𝑧𝐴) → Pred(𝑅, 𝐴, 𝑧) ∈ V)
13 predss 6307 . . . 4 Pred(𝑅, 𝐴, 𝑧) ⊆ 𝐴
1413a1i 11 . . 3 (((𝑅 Fr 𝐴𝑅 Po 𝐴𝑅 Se 𝐴) ∧ 𝑧𝐴) → Pred(𝑅, 𝐴, 𝑧) ⊆ 𝐴)
15 difssd 4128 . . . . 5 (((𝑅 Fr 𝐴𝑅 Po 𝐴𝑅 Se 𝐴) ∧ (𝐴 ∖ dom 𝐹) ≠ ∅) → (𝐴 ∖ dom 𝐹) ⊆ 𝐴)
16 simpr 484 . . . . 5 (((𝑅 Fr 𝐴𝑅 Po 𝐴𝑅 Se 𝐴) ∧ (𝐴 ∖ dom 𝐹) ≠ ∅) → (𝐴 ∖ dom 𝐹) ≠ ∅)
1715, 16jca 511 . . . 4 (((𝑅 Fr 𝐴𝑅 Po 𝐴𝑅 Se 𝐴) ∧ (𝐴 ∖ dom 𝐹) ≠ ∅) → ((𝐴 ∖ dom 𝐹) ⊆ 𝐴 ∧ (𝐴 ∖ dom 𝐹) ≠ ∅))
18 frpomin2 6341 . . . 4 (((𝑅 Fr 𝐴𝑅 Po 𝐴𝑅 Se 𝐴) ∧ ((𝐴 ∖ dom 𝐹) ⊆ 𝐴 ∧ (𝐴 ∖ dom 𝐹) ≠ ∅)) → ∃𝑧 ∈ (𝐴 ∖ dom 𝐹)Pred(𝑅, (𝐴 ∖ dom 𝐹), 𝑧) = ∅)
1917, 18syldan 590 . . 3 (((𝑅 Fr 𝐴𝑅 Po 𝐴𝑅 Se 𝐴) ∧ (𝐴 ∖ dom 𝐹) ≠ ∅) → ∃𝑧 ∈ (𝐴 ∖ dom 𝐹)Pred(𝑅, (𝐴 ∖ dom 𝐹), 𝑧) = ∅)
202, 3, 4, 6, 7, 8, 9, 12, 14, 19frrlem14 8298 . 2 ((𝑅 Fr 𝐴𝑅 Po 𝐴𝑅 Se 𝐴) → dom 𝐹 = 𝐴)
21 df-fn 6545 . 2 (𝐹 Fn 𝐴 ↔ (Fun 𝐹 ∧ dom 𝐹 = 𝐴))
225, 20, 21sylanbrc 582 1 ((𝑅 Fr 𝐴𝑅 Po 𝐴𝑅 Se 𝐴) → 𝐹 Fn 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  w3a 1085   = wceq 1534  wex 1774  wcel 2099  {cab 2705  wne 2936  wral 3057  wrex 3066  Vcvv 3470  cdif 3942  cun 3943  wss 3945  c0 4318  {csn 4624  cop 4630   Po wpo 5582   Fr wfr 5624   Se wse 5625  dom cdm 5672  cres 5674  Predcpred 6298  Fun wfun 6536   Fn wfn 6537  cfv 6542  (class class class)co 7414  frecscfrecs 8279
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-rep 5279  ax-sep 5293  ax-nul 5300  ax-pr 5423  ax-un 7734
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2937  df-ral 3058  df-rex 3067  df-reu 3373  df-rab 3429  df-v 3472  df-sbc 3776  df-csb 3891  df-dif 3948  df-un 3950  df-in 3952  df-ss 3962  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-iun 4993  df-br 5143  df-opab 5205  df-mpt 5226  df-id 5570  df-po 5584  df-fr 5627  df-se 5628  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-pred 6299  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-ov 7417  df-frecs 8280
This theorem is referenced by:  fpr2  8303  fpr3  8304  wfr1  8349  on2recsfn  8681  norecfn  27856  norec2fn  27866
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