ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  frecsuc GIF version

Theorem frecsuc 6102
Description: The successor value resulting from finite recursive definition generation. (Contributed by Jim Kingdon, 31-Mar-2022.)
Assertion
Ref Expression
frecsuc ((∀𝑧𝑆 (𝐹𝑧) ∈ 𝑆𝐴𝑆𝐵 ∈ ω) → (frec(𝐹, 𝐴)‘suc 𝐵) = (𝐹‘(frec(𝐹, 𝐴)‘𝐵)))
Distinct variable groups:   𝑧,𝐹   𝑧,𝑆
Allowed substitution hints:   𝐴(𝑧)   𝐵(𝑧)

Proof of Theorem frecsuc
Dummy variables 𝑓 𝑔 𝑚 𝑥 𝑦 𝑛 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 dmeq 4592 . . . . . . . . 9 (𝑓 = 𝑔 → dom 𝑓 = dom 𝑔)
21eqeq1d 2091 . . . . . . . 8 (𝑓 = 𝑔 → (dom 𝑓 = suc 𝑛 ↔ dom 𝑔 = suc 𝑛))
3 fveq1 5250 . . . . . . . . . 10 (𝑓 = 𝑔 → (𝑓𝑛) = (𝑔𝑛))
43fveq2d 5255 . . . . . . . . 9 (𝑓 = 𝑔 → (𝐹‘(𝑓𝑛)) = (𝐹‘(𝑔𝑛)))
54eleq2d 2152 . . . . . . . 8 (𝑓 = 𝑔 → (𝑦 ∈ (𝐹‘(𝑓𝑛)) ↔ 𝑦 ∈ (𝐹‘(𝑔𝑛))))
62, 5anbi12d 457 . . . . . . 7 (𝑓 = 𝑔 → ((dom 𝑓 = suc 𝑛𝑦 ∈ (𝐹‘(𝑓𝑛))) ↔ (dom 𝑔 = suc 𝑛𝑦 ∈ (𝐹‘(𝑔𝑛)))))
76rexbidv 2375 . . . . . 6 (𝑓 = 𝑔 → (∃𝑛 ∈ ω (dom 𝑓 = suc 𝑛𝑦 ∈ (𝐹‘(𝑓𝑛))) ↔ ∃𝑛 ∈ ω (dom 𝑔 = suc 𝑛𝑦 ∈ (𝐹‘(𝑔𝑛)))))
81eqeq1d 2091 . . . . . . 7 (𝑓 = 𝑔 → (dom 𝑓 = ∅ ↔ dom 𝑔 = ∅))
98anbi1d 453 . . . . . 6 (𝑓 = 𝑔 → ((dom 𝑓 = ∅ ∧ 𝑦𝐴) ↔ (dom 𝑔 = ∅ ∧ 𝑦𝐴)))
107, 9orbi12d 740 . . . . 5 (𝑓 = 𝑔 → ((∃𝑛 ∈ ω (dom 𝑓 = suc 𝑛𝑦 ∈ (𝐹‘(𝑓𝑛))) ∨ (dom 𝑓 = ∅ ∧ 𝑦𝐴)) ↔ (∃𝑛 ∈ ω (dom 𝑔 = suc 𝑛𝑦 ∈ (𝐹‘(𝑔𝑛))) ∨ (dom 𝑔 = ∅ ∧ 𝑦𝐴))))
1110abbidv 2200 . . . 4 (𝑓 = 𝑔 → {𝑦 ∣ (∃𝑛 ∈ ω (dom 𝑓 = suc 𝑛𝑦 ∈ (𝐹‘(𝑓𝑛))) ∨ (dom 𝑓 = ∅ ∧ 𝑦𝐴))} = {𝑦 ∣ (∃𝑛 ∈ ω (dom 𝑔 = suc 𝑛𝑦 ∈ (𝐹‘(𝑔𝑛))) ∨ (dom 𝑔 = ∅ ∧ 𝑦𝐴))})
1211cbvmptv 3899 . . 3 (𝑓 ∈ V ↦ {𝑦 ∣ (∃𝑛 ∈ ω (dom 𝑓 = suc 𝑛𝑦 ∈ (𝐹‘(𝑓𝑛))) ∨ (dom 𝑓 = ∅ ∧ 𝑦𝐴))}) = (𝑔 ∈ V ↦ {𝑦 ∣ (∃𝑛 ∈ ω (dom 𝑔 = suc 𝑛𝑦 ∈ (𝐹‘(𝑔𝑛))) ∨ (dom 𝑔 = ∅ ∧ 𝑦𝐴))})
13 eleq1 2145 . . . . . . . 8 (𝑦 = 𝑥 → (𝑦 ∈ (𝐹‘(𝑔𝑛)) ↔ 𝑥 ∈ (𝐹‘(𝑔𝑛))))
1413anbi2d 452 . . . . . . 7 (𝑦 = 𝑥 → ((dom 𝑔 = suc 𝑛𝑦 ∈ (𝐹‘(𝑔𝑛))) ↔ (dom 𝑔 = suc 𝑛𝑥 ∈ (𝐹‘(𝑔𝑛)))))
1514rexbidv 2375 . . . . . 6 (𝑦 = 𝑥 → (∃𝑛 ∈ ω (dom 𝑔 = suc 𝑛𝑦 ∈ (𝐹‘(𝑔𝑛))) ↔ ∃𝑛 ∈ ω (dom 𝑔 = suc 𝑛𝑥 ∈ (𝐹‘(𝑔𝑛)))))
16 eleq1 2145 . . . . . . 7 (𝑦 = 𝑥 → (𝑦𝐴𝑥𝐴))
1716anbi2d 452 . . . . . 6 (𝑦 = 𝑥 → ((dom 𝑔 = ∅ ∧ 𝑦𝐴) ↔ (dom 𝑔 = ∅ ∧ 𝑥𝐴)))
1815, 17orbi12d 740 . . . . 5 (𝑦 = 𝑥 → ((∃𝑛 ∈ ω (dom 𝑔 = suc 𝑛𝑦 ∈ (𝐹‘(𝑔𝑛))) ∨ (dom 𝑔 = ∅ ∧ 𝑦𝐴)) ↔ (∃𝑛 ∈ ω (dom 𝑔 = suc 𝑛𝑥 ∈ (𝐹‘(𝑔𝑛))) ∨ (dom 𝑔 = ∅ ∧ 𝑥𝐴))))
1918cbvabv 2206 . . . 4 {𝑦 ∣ (∃𝑛 ∈ ω (dom 𝑔 = suc 𝑛𝑦 ∈ (𝐹‘(𝑔𝑛))) ∨ (dom 𝑔 = ∅ ∧ 𝑦𝐴))} = {𝑥 ∣ (∃𝑛 ∈ ω (dom 𝑔 = suc 𝑛𝑥 ∈ (𝐹‘(𝑔𝑛))) ∨ (dom 𝑔 = ∅ ∧ 𝑥𝐴))}
2019mpteq2i 3891 . . 3 (𝑔 ∈ V ↦ {𝑦 ∣ (∃𝑛 ∈ ω (dom 𝑔 = suc 𝑛𝑦 ∈ (𝐹‘(𝑔𝑛))) ∨ (dom 𝑔 = ∅ ∧ 𝑦𝐴))}) = (𝑔 ∈ V ↦ {𝑥 ∣ (∃𝑛 ∈ ω (dom 𝑔 = suc 𝑛𝑥 ∈ (𝐹‘(𝑔𝑛))) ∨ (dom 𝑔 = ∅ ∧ 𝑥𝐴))})
21 suceq 4192 . . . . . . . . 9 (𝑛 = 𝑚 → suc 𝑛 = suc 𝑚)
2221eqeq2d 2094 . . . . . . . 8 (𝑛 = 𝑚 → (dom 𝑔 = suc 𝑛 ↔ dom 𝑔 = suc 𝑚))
23 fveq2 5251 . . . . . . . . . 10 (𝑛 = 𝑚 → (𝑔𝑛) = (𝑔𝑚))
2423fveq2d 5255 . . . . . . . . 9 (𝑛 = 𝑚 → (𝐹‘(𝑔𝑛)) = (𝐹‘(𝑔𝑚)))
2524eleq2d 2152 . . . . . . . 8 (𝑛 = 𝑚 → (𝑥 ∈ (𝐹‘(𝑔𝑛)) ↔ 𝑥 ∈ (𝐹‘(𝑔𝑚))))
2622, 25anbi12d 457 . . . . . . 7 (𝑛 = 𝑚 → ((dom 𝑔 = suc 𝑛𝑥 ∈ (𝐹‘(𝑔𝑛))) ↔ (dom 𝑔 = suc 𝑚𝑥 ∈ (𝐹‘(𝑔𝑚)))))
2726cbvrexv 2584 . . . . . 6 (∃𝑛 ∈ ω (dom 𝑔 = suc 𝑛𝑥 ∈ (𝐹‘(𝑔𝑛))) ↔ ∃𝑚 ∈ ω (dom 𝑔 = suc 𝑚𝑥 ∈ (𝐹‘(𝑔𝑚))))
2827orbi1i 713 . . . . 5 ((∃𝑛 ∈ ω (dom 𝑔 = suc 𝑛𝑥 ∈ (𝐹‘(𝑔𝑛))) ∨ (dom 𝑔 = ∅ ∧ 𝑥𝐴)) ↔ (∃𝑚 ∈ ω (dom 𝑔 = suc 𝑚𝑥 ∈ (𝐹‘(𝑔𝑚))) ∨ (dom 𝑔 = ∅ ∧ 𝑥𝐴)))
2928abbii 2198 . . . 4 {𝑥 ∣ (∃𝑛 ∈ ω (dom 𝑔 = suc 𝑛𝑥 ∈ (𝐹‘(𝑔𝑛))) ∨ (dom 𝑔 = ∅ ∧ 𝑥𝐴))} = {𝑥 ∣ (∃𝑚 ∈ ω (dom 𝑔 = suc 𝑚𝑥 ∈ (𝐹‘(𝑔𝑚))) ∨ (dom 𝑔 = ∅ ∧ 𝑥𝐴))}
3029mpteq2i 3891 . . 3 (𝑔 ∈ V ↦ {𝑥 ∣ (∃𝑛 ∈ ω (dom 𝑔 = suc 𝑛𝑥 ∈ (𝐹‘(𝑔𝑛))) ∨ (dom 𝑔 = ∅ ∧ 𝑥𝐴))}) = (𝑔 ∈ V ↦ {𝑥 ∣ (∃𝑚 ∈ ω (dom 𝑔 = suc 𝑚𝑥 ∈ (𝐹‘(𝑔𝑚))) ∨ (dom 𝑔 = ∅ ∧ 𝑥𝐴))})
3112, 20, 303eqtri 2107 . 2 (𝑓 ∈ V ↦ {𝑦 ∣ (∃𝑛 ∈ ω (dom 𝑓 = suc 𝑛𝑦 ∈ (𝐹‘(𝑓𝑛))) ∨ (dom 𝑓 = ∅ ∧ 𝑦𝐴))}) = (𝑔 ∈ V ↦ {𝑥 ∣ (∃𝑚 ∈ ω (dom 𝑔 = suc 𝑚𝑥 ∈ (𝐹‘(𝑔𝑚))) ∨ (dom 𝑔 = ∅ ∧ 𝑥𝐴))})
3231frecsuclem 6101 1 ((∀𝑧𝑆 (𝐹𝑧) ∈ 𝑆𝐴𝑆𝐵 ∈ ω) → (frec(𝐹, 𝐴)‘suc 𝐵) = (𝐹‘(frec(𝐹, 𝐴)‘𝐵)))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102  wo 662  w3a 920   = wceq 1285  wcel 1434  {cab 2069  wral 2353  wrex 2354  Vcvv 2612  c0 3269  cmpt 3865  suc csuc 4155  ωcom 4367  dom cdm 4399  cfv 4967  freccfrec 6085
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 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-coll 3919  ax-sep 3922  ax-nul 3930  ax-pow 3974  ax-pr 3999  ax-un 4223  ax-setind 4315  ax-iinf 4365
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1688  df-eu 1946  df-mo 1947  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ne 2250  df-ral 2358  df-rex 2359  df-reu 2360  df-rab 2362  df-v 2614  df-sbc 2827  df-csb 2920  df-dif 2986  df-un 2988  df-in 2990  df-ss 2997  df-nul 3270  df-pw 3408  df-sn 3428  df-pr 3429  df-op 3431  df-uni 3628  df-int 3663  df-iun 3706  df-br 3812  df-opab 3866  df-mpt 3867  df-tr 3902  df-id 4083  df-iord 4156  df-on 4158  df-ilim 4159  df-suc 4161  df-iom 4368  df-xp 4405  df-rel 4406  df-cnv 4407  df-co 4408  df-dm 4409  df-rn 4410  df-res 4411  df-ima 4412  df-iota 4932  df-fun 4969  df-fn 4970  df-f 4971  df-f1 4972  df-fo 4973  df-f1o 4974  df-fv 4975  df-recs 6000  df-frec 6086
This theorem is referenced by:  frecrdg  6103  frec2uzsucd  9695  frec2uzrdg  9703  frecuzrdgsuc  9708  frecuzrdgg  9710  frecuzrdgsuctlem  9717  iseqvalt  9749
  Copyright terms: Public domain W3C validator