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Theorem frecsuc 6050
 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 4557 . . . . . . . . 9 (𝑓 = 𝑔 → dom 𝑓 = dom 𝑔)
21eqeq1d 2090 . . . . . . . 8 (𝑓 = 𝑔 → (dom 𝑓 = suc 𝑛 ↔ dom 𝑔 = suc 𝑛))
3 fveq1 5202 . . . . . . . . . 10 (𝑓 = 𝑔 → (𝑓𝑛) = (𝑔𝑛))
43fveq2d 5207 . . . . . . . . 9 (𝑓 = 𝑔 → (𝐹‘(𝑓𝑛)) = (𝐹‘(𝑔𝑛)))
54eleq2d 2149 . . . . . . . 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 2370 . . . . . 6 (𝑓 = 𝑔 → (∃𝑛 ∈ ω (dom 𝑓 = suc 𝑛𝑦 ∈ (𝐹‘(𝑓𝑛))) ↔ ∃𝑛 ∈ ω (dom 𝑔 = suc 𝑛𝑦 ∈ (𝐹‘(𝑔𝑛)))))
81eqeq1d 2090 . . . . . . 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 2197 . . . 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 3875 . . 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 2142 . . . . . . . 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 2370 . . . . . 6 (𝑦 = 𝑥 → (∃𝑛 ∈ ω (dom 𝑔 = suc 𝑛𝑦 ∈ (𝐹‘(𝑔𝑛))) ↔ ∃𝑛 ∈ ω (dom 𝑔 = suc 𝑛𝑥 ∈ (𝐹‘(𝑔𝑛)))))
16 eleq1 2142 . . . . . . 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 2203 . . . 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 3867 . . 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 4159 . . . . . . . . 9 (𝑛 = 𝑚 → suc 𝑛 = suc 𝑚)
2221eqeq2d 2093 . . . . . . . 8 (𝑛 = 𝑚 → (dom 𝑔 = suc 𝑛 ↔ dom 𝑔 = suc 𝑚))
23 fveq2 5203 . . . . . . . . . 10 (𝑛 = 𝑚 → (𝑔𝑛) = (𝑔𝑚))
2423fveq2d 5207 . . . . . . . . 9 (𝑛 = 𝑚 → (𝐹‘(𝑔𝑛)) = (𝐹‘(𝑔𝑚)))
2524eleq2d 2149 . . . . . . . 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 2579 . . . . . 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 2195 . . . 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 3867 . . 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 2106 . 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 6049 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 2068  ∀wral 2349  ∃wrex 2350  Vcvv 2602  ∅c0 3252   ↦ cmpt 3841  suc csuc 4122  ωcom 4333  dom cdm 4365  ‘cfv 4926  freccfrec 6033 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 2064  ax-coll 3895  ax-sep 3898  ax-nul 3906  ax-pow 3950  ax-pr 3966  ax-un 4190  ax-setind 4282  ax-iinf 4331 This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1687  df-eu 1945  df-mo 1946  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ne 2247  df-ral 2354  df-rex 2355  df-reu 2356  df-rab 2358  df-v 2604  df-sbc 2817  df-csb 2910  df-dif 2976  df-un 2978  df-in 2980  df-ss 2987  df-nul 3253  df-pw 3386  df-sn 3406  df-pr 3407  df-op 3409  df-uni 3604  df-int 3639  df-iun 3682  df-br 3788  df-opab 3842  df-mpt 3843  df-tr 3878  df-id 4050  df-iord 4123  df-on 4125  df-ilim 4126  df-suc 4128  df-iom 4334  df-xp 4371  df-rel 4372  df-cnv 4373  df-co 4374  df-dm 4375  df-rn 4376  df-res 4377  df-ima 4378  df-iota 4891  df-fun 4928  df-fn 4929  df-f 4930  df-f1 4931  df-fo 4932  df-f1o 4933  df-fv 4934  df-recs 5948  df-frec 6034 This theorem is referenced by:  frecrdg  6051  frec2uzsucd  9472  frec2uzrdg  9480  frecuzrdgsuc  9485  frecuzrdgg  9487  frecuzrdgsuctlem  9494  iseqvalt  9521
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