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Theorem frecsuc 6186
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 4649 . . . . . . . . 9 (𝑓 = 𝑔 → dom 𝑓 = dom 𝑔)
21eqeq1d 2097 . . . . . . . 8 (𝑓 = 𝑔 → (dom 𝑓 = suc 𝑛 ↔ dom 𝑔 = suc 𝑛))
3 fveq1 5317 . . . . . . . . . 10 (𝑓 = 𝑔 → (𝑓𝑛) = (𝑔𝑛))
43fveq2d 5322 . . . . . . . . 9 (𝑓 = 𝑔 → (𝐹‘(𝑓𝑛)) = (𝐹‘(𝑔𝑛)))
54eleq2d 2158 . . . . . . . 8 (𝑓 = 𝑔 → (𝑦 ∈ (𝐹‘(𝑓𝑛)) ↔ 𝑦 ∈ (𝐹‘(𝑔𝑛))))
62, 5anbi12d 458 . . . . . . 7 (𝑓 = 𝑔 → ((dom 𝑓 = suc 𝑛𝑦 ∈ (𝐹‘(𝑓𝑛))) ↔ (dom 𝑔 = suc 𝑛𝑦 ∈ (𝐹‘(𝑔𝑛)))))
76rexbidv 2382 . . . . . 6 (𝑓 = 𝑔 → (∃𝑛 ∈ ω (dom 𝑓 = suc 𝑛𝑦 ∈ (𝐹‘(𝑓𝑛))) ↔ ∃𝑛 ∈ ω (dom 𝑔 = suc 𝑛𝑦 ∈ (𝐹‘(𝑔𝑛)))))
81eqeq1d 2097 . . . . . . 7 (𝑓 = 𝑔 → (dom 𝑓 = ∅ ↔ dom 𝑔 = ∅))
98anbi1d 454 . . . . . 6 (𝑓 = 𝑔 → ((dom 𝑓 = ∅ ∧ 𝑦𝐴) ↔ (dom 𝑔 = ∅ ∧ 𝑦𝐴)))
107, 9orbi12d 743 . . . . 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 2206 . . . 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 3940 . . 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 2151 . . . . . . . 8 (𝑦 = 𝑥 → (𝑦 ∈ (𝐹‘(𝑔𝑛)) ↔ 𝑥 ∈ (𝐹‘(𝑔𝑛))))
1413anbi2d 453 . . . . . . 7 (𝑦 = 𝑥 → ((dom 𝑔 = suc 𝑛𝑦 ∈ (𝐹‘(𝑔𝑛))) ↔ (dom 𝑔 = suc 𝑛𝑥 ∈ (𝐹‘(𝑔𝑛)))))
1514rexbidv 2382 . . . . . 6 (𝑦 = 𝑥 → (∃𝑛 ∈ ω (dom 𝑔 = suc 𝑛𝑦 ∈ (𝐹‘(𝑔𝑛))) ↔ ∃𝑛 ∈ ω (dom 𝑔 = suc 𝑛𝑥 ∈ (𝐹‘(𝑔𝑛)))))
16 eleq1 2151 . . . . . . 7 (𝑦 = 𝑥 → (𝑦𝐴𝑥𝐴))
1716anbi2d 453 . . . . . 6 (𝑦 = 𝑥 → ((dom 𝑔 = ∅ ∧ 𝑦𝐴) ↔ (dom 𝑔 = ∅ ∧ 𝑥𝐴)))
1815, 17orbi12d 743 . . . . 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 2212 . . . 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 3931 . . 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 4238 . . . . . . . . 9 (𝑛 = 𝑚 → suc 𝑛 = suc 𝑚)
2221eqeq2d 2100 . . . . . . . 8 (𝑛 = 𝑚 → (dom 𝑔 = suc 𝑛 ↔ dom 𝑔 = suc 𝑚))
23 fveq2 5318 . . . . . . . . . 10 (𝑛 = 𝑚 → (𝑔𝑛) = (𝑔𝑚))
2423fveq2d 5322 . . . . . . . . 9 (𝑛 = 𝑚 → (𝐹‘(𝑔𝑛)) = (𝐹‘(𝑔𝑚)))
2524eleq2d 2158 . . . . . . . 8 (𝑛 = 𝑚 → (𝑥 ∈ (𝐹‘(𝑔𝑛)) ↔ 𝑥 ∈ (𝐹‘(𝑔𝑚))))
2622, 25anbi12d 458 . . . . . . 7 (𝑛 = 𝑚 → ((dom 𝑔 = suc 𝑛𝑥 ∈ (𝐹‘(𝑔𝑛))) ↔ (dom 𝑔 = suc 𝑚𝑥 ∈ (𝐹‘(𝑔𝑚)))))
2726cbvrexv 2592 . . . . . 6 (∃𝑛 ∈ ω (dom 𝑔 = suc 𝑛𝑥 ∈ (𝐹‘(𝑔𝑛))) ↔ ∃𝑚 ∈ ω (dom 𝑔 = suc 𝑚𝑥 ∈ (𝐹‘(𝑔𝑚))))
2827orbi1i 716 . . . . 5 ((∃𝑛 ∈ ω (dom 𝑔 = suc 𝑛𝑥 ∈ (𝐹‘(𝑔𝑛))) ∨ (dom 𝑔 = ∅ ∧ 𝑥𝐴)) ↔ (∃𝑚 ∈ ω (dom 𝑔 = suc 𝑚𝑥 ∈ (𝐹‘(𝑔𝑚))) ∨ (dom 𝑔 = ∅ ∧ 𝑥𝐴)))
2928abbii 2204 . . . 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 3931 . . 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 2113 . 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 6185 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 103  wo 665  w3a 925   = wceq 1290  wcel 1439  {cab 2075  wral 2360  wrex 2361  Vcvv 2620  c0 3287  cmpt 3905  suc csuc 4201  ωcom 4418  dom cdm 4451  cfv 5028  freccfrec 6169
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 580  ax-in2 581  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-13 1450  ax-14 1451  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071  ax-coll 3960  ax-sep 3963  ax-nul 3971  ax-pow 4015  ax-pr 4045  ax-un 4269  ax-setind 4366  ax-iinf 4416
This theorem depends on definitions:  df-bi 116  df-3an 927  df-tru 1293  df-fal 1296  df-nf 1396  df-sb 1694  df-eu 1952  df-mo 1953  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-ne 2257  df-ral 2365  df-rex 2366  df-reu 2367  df-rab 2369  df-v 2622  df-sbc 2842  df-csb 2935  df-dif 3002  df-un 3004  df-in 3006  df-ss 3013  df-nul 3288  df-pw 3435  df-sn 3456  df-pr 3457  df-op 3459  df-uni 3660  df-int 3695  df-iun 3738  df-br 3852  df-opab 3906  df-mpt 3907  df-tr 3943  df-id 4129  df-iord 4202  df-on 4204  df-ilim 4205  df-suc 4207  df-iom 4419  df-xp 4457  df-rel 4458  df-cnv 4459  df-co 4460  df-dm 4461  df-rn 4462  df-res 4463  df-ima 4464  df-iota 4993  df-fun 5030  df-fn 5031  df-f 5032  df-f1 5033  df-fo 5034  df-f1o 5035  df-fv 5036  df-recs 6084  df-frec 6170
This theorem is referenced by:  frecrdg  6187  frec2uzsucd  9862  frec2uzrdg  9870  frecuzrdgsuc  9875  frecuzrdgg  9877  frecuzrdgsuctlem  9884  iseqvalt  9927  seq3val  9928
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