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Theorem frecsuc 6297
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 4734 . . . . . . . . 9 (𝑓 = 𝑔 → dom 𝑓 = dom 𝑔)
21eqeq1d 2146 . . . . . . . 8 (𝑓 = 𝑔 → (dom 𝑓 = suc 𝑛 ↔ dom 𝑔 = suc 𝑛))
3 fveq1 5413 . . . . . . . . . 10 (𝑓 = 𝑔 → (𝑓𝑛) = (𝑔𝑛))
43fveq2d 5418 . . . . . . . . 9 (𝑓 = 𝑔 → (𝐹‘(𝑓𝑛)) = (𝐹‘(𝑔𝑛)))
54eleq2d 2207 . . . . . . . 8 (𝑓 = 𝑔 → (𝑦 ∈ (𝐹‘(𝑓𝑛)) ↔ 𝑦 ∈ (𝐹‘(𝑔𝑛))))
62, 5anbi12d 464 . . . . . . 7 (𝑓 = 𝑔 → ((dom 𝑓 = suc 𝑛𝑦 ∈ (𝐹‘(𝑓𝑛))) ↔ (dom 𝑔 = suc 𝑛𝑦 ∈ (𝐹‘(𝑔𝑛)))))
76rexbidv 2436 . . . . . 6 (𝑓 = 𝑔 → (∃𝑛 ∈ ω (dom 𝑓 = suc 𝑛𝑦 ∈ (𝐹‘(𝑓𝑛))) ↔ ∃𝑛 ∈ ω (dom 𝑔 = suc 𝑛𝑦 ∈ (𝐹‘(𝑔𝑛)))))
81eqeq1d 2146 . . . . . . 7 (𝑓 = 𝑔 → (dom 𝑓 = ∅ ↔ dom 𝑔 = ∅))
98anbi1d 460 . . . . . 6 (𝑓 = 𝑔 → ((dom 𝑓 = ∅ ∧ 𝑦𝐴) ↔ (dom 𝑔 = ∅ ∧ 𝑦𝐴)))
107, 9orbi12d 782 . . . . 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 2255 . . . 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 4019 . . 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 2200 . . . . . . . 8 (𝑦 = 𝑥 → (𝑦 ∈ (𝐹‘(𝑔𝑛)) ↔ 𝑥 ∈ (𝐹‘(𝑔𝑛))))
1413anbi2d 459 . . . . . . 7 (𝑦 = 𝑥 → ((dom 𝑔 = suc 𝑛𝑦 ∈ (𝐹‘(𝑔𝑛))) ↔ (dom 𝑔 = suc 𝑛𝑥 ∈ (𝐹‘(𝑔𝑛)))))
1514rexbidv 2436 . . . . . 6 (𝑦 = 𝑥 → (∃𝑛 ∈ ω (dom 𝑔 = suc 𝑛𝑦 ∈ (𝐹‘(𝑔𝑛))) ↔ ∃𝑛 ∈ ω (dom 𝑔 = suc 𝑛𝑥 ∈ (𝐹‘(𝑔𝑛)))))
16 eleq1 2200 . . . . . . 7 (𝑦 = 𝑥 → (𝑦𝐴𝑥𝐴))
1716anbi2d 459 . . . . . 6 (𝑦 = 𝑥 → ((dom 𝑔 = ∅ ∧ 𝑦𝐴) ↔ (dom 𝑔 = ∅ ∧ 𝑥𝐴)))
1815, 17orbi12d 782 . . . . 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 2262 . . . 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 4010 . . 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 4319 . . . . . . . . 9 (𝑛 = 𝑚 → suc 𝑛 = suc 𝑚)
2221eqeq2d 2149 . . . . . . . 8 (𝑛 = 𝑚 → (dom 𝑔 = suc 𝑛 ↔ dom 𝑔 = suc 𝑚))
23 fveq2 5414 . . . . . . . . . 10 (𝑛 = 𝑚 → (𝑔𝑛) = (𝑔𝑚))
2423fveq2d 5418 . . . . . . . . 9 (𝑛 = 𝑚 → (𝐹‘(𝑔𝑛)) = (𝐹‘(𝑔𝑚)))
2524eleq2d 2207 . . . . . . . 8 (𝑛 = 𝑚 → (𝑥 ∈ (𝐹‘(𝑔𝑛)) ↔ 𝑥 ∈ (𝐹‘(𝑔𝑚))))
2622, 25anbi12d 464 . . . . . . 7 (𝑛 = 𝑚 → ((dom 𝑔 = suc 𝑛𝑥 ∈ (𝐹‘(𝑔𝑛))) ↔ (dom 𝑔 = suc 𝑚𝑥 ∈ (𝐹‘(𝑔𝑚)))))
2726cbvrexv 2653 . . . . . 6 (∃𝑛 ∈ ω (dom 𝑔 = suc 𝑛𝑥 ∈ (𝐹‘(𝑔𝑛))) ↔ ∃𝑚 ∈ ω (dom 𝑔 = suc 𝑚𝑥 ∈ (𝐹‘(𝑔𝑚))))
2827orbi1i 752 . . . . 5 ((∃𝑛 ∈ ω (dom 𝑔 = suc 𝑛𝑥 ∈ (𝐹‘(𝑔𝑛))) ∨ (dom 𝑔 = ∅ ∧ 𝑥𝐴)) ↔ (∃𝑚 ∈ ω (dom 𝑔 = suc 𝑚𝑥 ∈ (𝐹‘(𝑔𝑚))) ∨ (dom 𝑔 = ∅ ∧ 𝑥𝐴)))
2928abbii 2253 . . . 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 4010 . . 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 2162 . 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 6296 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 697  w3a 962   = wceq 1331  wcel 1480  {cab 2123  wral 2414  wrex 2415  Vcvv 2681  c0 3358  cmpt 3984  suc csuc 4282  ωcom 4499  dom cdm 4534  cfv 5118  freccfrec 6280
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-coll 4038  ax-sep 4041  ax-nul 4049  ax-pow 4093  ax-pr 4126  ax-un 4350  ax-setind 4447  ax-iinf 4497
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2000  df-mo 2001  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ne 2307  df-ral 2419  df-rex 2420  df-reu 2421  df-rab 2423  df-v 2683  df-sbc 2905  df-csb 2999  df-dif 3068  df-un 3070  df-in 3072  df-ss 3079  df-nul 3359  df-pw 3507  df-sn 3528  df-pr 3529  df-op 3531  df-uni 3732  df-int 3767  df-iun 3810  df-br 3925  df-opab 3985  df-mpt 3986  df-tr 4022  df-id 4210  df-iord 4283  df-on 4285  df-ilim 4286  df-suc 4288  df-iom 4500  df-xp 4540  df-rel 4541  df-cnv 4542  df-co 4543  df-dm 4544  df-rn 4545  df-res 4546  df-ima 4547  df-iota 5083  df-fun 5120  df-fn 5121  df-f 5122  df-f1 5123  df-fo 5124  df-f1o 5125  df-fv 5126  df-recs 6195  df-frec 6281
This theorem is referenced by:  frecrdg  6298  frec2uzsucd  10167  frec2uzrdg  10175  frecuzrdgsuc  10180  frecuzrdgg  10182  frecuzrdgsuctlem  10189  seq3val  10224  seqvalcd  10225
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