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Theorem frecsuc 5991
 Description: The successor value resulting from finite recursive definition generation. (Contributed by Jim Kingdon, 15-Aug-2019.)
Assertion
Ref Expression
frecsuc ((∀𝑧(𝐹𝑧) ∈ V ∧ 𝐴𝑉𝐵 ∈ ω) → (frec(𝐹, 𝐴)‘suc 𝐵) = (𝐹‘(frec(𝐹, 𝐴)‘𝐵)))
Distinct variable groups:   𝑧,𝐴   𝑧,𝐵   𝑧,𝐹
Allowed substitution hint:   𝑉(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 suceq 4139 . . . . . . . . . 10 (𝑛 = 𝑚 → suc 𝑛 = suc 𝑚)
21eqeq2d 2051 . . . . . . . . 9 (𝑛 = 𝑚 → (dom 𝑓 = suc 𝑛 ↔ dom 𝑓 = suc 𝑚))
3 fveq2 5178 . . . . . . . . . . 11 (𝑛 = 𝑚 → (𝑓𝑛) = (𝑓𝑚))
43fveq2d 5182 . . . . . . . . . 10 (𝑛 = 𝑚 → (𝐹‘(𝑓𝑛)) = (𝐹‘(𝑓𝑚)))
54eleq2d 2107 . . . . . . . . 9 (𝑛 = 𝑚 → (𝑥 ∈ (𝐹‘(𝑓𝑛)) ↔ 𝑥 ∈ (𝐹‘(𝑓𝑚))))
62, 5anbi12d 442 . . . . . . . 8 (𝑛 = 𝑚 → ((dom 𝑓 = suc 𝑛𝑥 ∈ (𝐹‘(𝑓𝑛))) ↔ (dom 𝑓 = suc 𝑚𝑥 ∈ (𝐹‘(𝑓𝑚)))))
76cbvrexv 2534 . . . . . . 7 (∃𝑛 ∈ ω (dom 𝑓 = suc 𝑛𝑥 ∈ (𝐹‘(𝑓𝑛))) ↔ ∃𝑚 ∈ ω (dom 𝑓 = suc 𝑚𝑥 ∈ (𝐹‘(𝑓𝑚))))
87orbi1i 680 . . . . . 6 ((∃𝑛 ∈ ω (dom 𝑓 = suc 𝑛𝑥 ∈ (𝐹‘(𝑓𝑛))) ∨ (dom 𝑓 = ∅ ∧ 𝑥𝐴)) ↔ (∃𝑚 ∈ ω (dom 𝑓 = suc 𝑚𝑥 ∈ (𝐹‘(𝑓𝑚))) ∨ (dom 𝑓 = ∅ ∧ 𝑥𝐴)))
98abbii 2153 . . . . 5 {𝑥 ∣ (∃𝑛 ∈ ω (dom 𝑓 = suc 𝑛𝑥 ∈ (𝐹‘(𝑓𝑛))) ∨ (dom 𝑓 = ∅ ∧ 𝑥𝐴))} = {𝑥 ∣ (∃𝑚 ∈ ω (dom 𝑓 = suc 𝑚𝑥 ∈ (𝐹‘(𝑓𝑚))) ∨ (dom 𝑓 = ∅ ∧ 𝑥𝐴))}
10 eleq1 2100 . . . . . . . . 9 (𝑥 = 𝑦 → (𝑥 ∈ (𝐹‘(𝑓𝑚)) ↔ 𝑦 ∈ (𝐹‘(𝑓𝑚))))
1110anbi2d 437 . . . . . . . 8 (𝑥 = 𝑦 → ((dom 𝑓 = suc 𝑚𝑥 ∈ (𝐹‘(𝑓𝑚))) ↔ (dom 𝑓 = suc 𝑚𝑦 ∈ (𝐹‘(𝑓𝑚)))))
1211rexbidv 2327 . . . . . . 7 (𝑥 = 𝑦 → (∃𝑚 ∈ ω (dom 𝑓 = suc 𝑚𝑥 ∈ (𝐹‘(𝑓𝑚))) ↔ ∃𝑚 ∈ ω (dom 𝑓 = suc 𝑚𝑦 ∈ (𝐹‘(𝑓𝑚)))))
13 eleq1 2100 . . . . . . . 8 (𝑥 = 𝑦 → (𝑥𝐴𝑦𝐴))
1413anbi2d 437 . . . . . . 7 (𝑥 = 𝑦 → ((dom 𝑓 = ∅ ∧ 𝑥𝐴) ↔ (dom 𝑓 = ∅ ∧ 𝑦𝐴)))
1512, 14orbi12d 707 . . . . . 6 (𝑥 = 𝑦 → ((∃𝑚 ∈ ω (dom 𝑓 = suc 𝑚𝑥 ∈ (𝐹‘(𝑓𝑚))) ∨ (dom 𝑓 = ∅ ∧ 𝑥𝐴)) ↔ (∃𝑚 ∈ ω (dom 𝑓 = suc 𝑚𝑦 ∈ (𝐹‘(𝑓𝑚))) ∨ (dom 𝑓 = ∅ ∧ 𝑦𝐴))))
1615cbvabv 2161 . . . . 5 {𝑥 ∣ (∃𝑚 ∈ ω (dom 𝑓 = suc 𝑚𝑥 ∈ (𝐹‘(𝑓𝑚))) ∨ (dom 𝑓 = ∅ ∧ 𝑥𝐴))} = {𝑦 ∣ (∃𝑚 ∈ ω (dom 𝑓 = suc 𝑚𝑦 ∈ (𝐹‘(𝑓𝑚))) ∨ (dom 𝑓 = ∅ ∧ 𝑦𝐴))}
179, 16eqtri 2060 . . . 4 {𝑥 ∣ (∃𝑛 ∈ ω (dom 𝑓 = suc 𝑛𝑥 ∈ (𝐹‘(𝑓𝑛))) ∨ (dom 𝑓 = ∅ ∧ 𝑥𝐴))} = {𝑦 ∣ (∃𝑚 ∈ ω (dom 𝑓 = suc 𝑚𝑦 ∈ (𝐹‘(𝑓𝑚))) ∨ (dom 𝑓 = ∅ ∧ 𝑦𝐴))}
1817mpteq2i 3844 . . 3 (𝑓 ∈ V ↦ {𝑥 ∣ (∃𝑛 ∈ ω (dom 𝑓 = suc 𝑛𝑥 ∈ (𝐹‘(𝑓𝑛))) ∨ (dom 𝑓 = ∅ ∧ 𝑥𝐴))}) = (𝑓 ∈ V ↦ {𝑦 ∣ (∃𝑚 ∈ ω (dom 𝑓 = suc 𝑚𝑦 ∈ (𝐹‘(𝑓𝑚))) ∨ (dom 𝑓 = ∅ ∧ 𝑦𝐴))})
19 dmeq 4535 . . . . . . . . 9 (𝑓 = 𝑔 → dom 𝑓 = dom 𝑔)
2019eqeq1d 2048 . . . . . . . 8 (𝑓 = 𝑔 → (dom 𝑓 = suc 𝑚 ↔ dom 𝑔 = suc 𝑚))
21 fveq1 5177 . . . . . . . . . 10 (𝑓 = 𝑔 → (𝑓𝑚) = (𝑔𝑚))
2221fveq2d 5182 . . . . . . . . 9 (𝑓 = 𝑔 → (𝐹‘(𝑓𝑚)) = (𝐹‘(𝑔𝑚)))
2322eleq2d 2107 . . . . . . . 8 (𝑓 = 𝑔 → (𝑦 ∈ (𝐹‘(𝑓𝑚)) ↔ 𝑦 ∈ (𝐹‘(𝑔𝑚))))
2420, 23anbi12d 442 . . . . . . 7 (𝑓 = 𝑔 → ((dom 𝑓 = suc 𝑚𝑦 ∈ (𝐹‘(𝑓𝑚))) ↔ (dom 𝑔 = suc 𝑚𝑦 ∈ (𝐹‘(𝑔𝑚)))))
2524rexbidv 2327 . . . . . 6 (𝑓 = 𝑔 → (∃𝑚 ∈ ω (dom 𝑓 = suc 𝑚𝑦 ∈ (𝐹‘(𝑓𝑚))) ↔ ∃𝑚 ∈ ω (dom 𝑔 = suc 𝑚𝑦 ∈ (𝐹‘(𝑔𝑚)))))
2619eqeq1d 2048 . . . . . . 7 (𝑓 = 𝑔 → (dom 𝑓 = ∅ ↔ dom 𝑔 = ∅))
2726anbi1d 438 . . . . . 6 (𝑓 = 𝑔 → ((dom 𝑓 = ∅ ∧ 𝑦𝐴) ↔ (dom 𝑔 = ∅ ∧ 𝑦𝐴)))
2825, 27orbi12d 707 . . . . 5 (𝑓 = 𝑔 → ((∃𝑚 ∈ ω (dom 𝑓 = suc 𝑚𝑦 ∈ (𝐹‘(𝑓𝑚))) ∨ (dom 𝑓 = ∅ ∧ 𝑦𝐴)) ↔ (∃𝑚 ∈ ω (dom 𝑔 = suc 𝑚𝑦 ∈ (𝐹‘(𝑔𝑚))) ∨ (dom 𝑔 = ∅ ∧ 𝑦𝐴))))
2928abbidv 2155 . . . 4 (𝑓 = 𝑔 → {𝑦 ∣ (∃𝑚 ∈ ω (dom 𝑓 = suc 𝑚𝑦 ∈ (𝐹‘(𝑓𝑚))) ∨ (dom 𝑓 = ∅ ∧ 𝑦𝐴))} = {𝑦 ∣ (∃𝑚 ∈ ω (dom 𝑔 = suc 𝑚𝑦 ∈ (𝐹‘(𝑔𝑚))) ∨ (dom 𝑔 = ∅ ∧ 𝑦𝐴))})
3029cbvmptv 3852 . . 3 (𝑓 ∈ V ↦ {𝑦 ∣ (∃𝑚 ∈ ω (dom 𝑓 = suc 𝑚𝑦 ∈ (𝐹‘(𝑓𝑚))) ∨ (dom 𝑓 = ∅ ∧ 𝑦𝐴))}) = (𝑔 ∈ V ↦ {𝑦 ∣ (∃𝑚 ∈ ω (dom 𝑔 = suc 𝑚𝑦 ∈ (𝐹‘(𝑔𝑚))) ∨ (dom 𝑔 = ∅ ∧ 𝑦𝐴))})
3118, 30eqtri 2060 . 2 (𝑓 ∈ V ↦ {𝑥 ∣ (∃𝑛 ∈ ω (dom 𝑓 = suc 𝑛𝑥 ∈ (𝐹‘(𝑓𝑛))) ∨ (dom 𝑓 = ∅ ∧ 𝑥𝐴))}) = (𝑔 ∈ V ↦ {𝑦 ∣ (∃𝑚 ∈ ω (dom 𝑔 = suc 𝑚𝑦 ∈ (𝐹‘(𝑔𝑚))) ∨ (dom 𝑔 = ∅ ∧ 𝑦𝐴))})
3231frecsuclem3 5990 1 ((∀𝑧(𝐹𝑧) ∈ V ∧ 𝐴𝑉𝐵 ∈ ω) → (frec(𝐹, 𝐴)‘suc 𝐵) = (𝐹‘(frec(𝐹, 𝐴)‘𝐵)))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97   ∨ wo 629   ∧ w3a 885  ∀wal 1241   = wceq 1243   ∈ wcel 1393  {cab 2026  ∃wrex 2307  Vcvv 2557  ∅c0 3224   ↦ cmpt 3818  suc csuc 4102  ωcom 4313  dom cdm 4345  ‘cfv 4902  freccfrec 5977 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-13 1404  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-coll 3872  ax-sep 3875  ax-nul 3883  ax-pow 3927  ax-pr 3944  ax-un 4170  ax-setind 4262  ax-iinf 4311 This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-fal 1249  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ne 2206  df-ral 2311  df-rex 2312  df-reu 2313  df-rab 2315  df-v 2559  df-sbc 2765  df-csb 2853  df-dif 2920  df-un 2922  df-in 2924  df-ss 2931  df-nul 3225  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-int 3616  df-iun 3659  df-br 3765  df-opab 3819  df-mpt 3820  df-tr 3855  df-id 4030  df-iord 4103  df-on 4105  df-suc 4108  df-iom 4314  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-rn 4356  df-res 4357  df-ima 4358  df-iota 4867  df-fun 4904  df-fn 4905  df-f 4906  df-f1 4907  df-fo 4908  df-f1o 4909  df-fv 4910  df-recs 5920  df-frec 5978 This theorem is referenced by:  frecrdg  5992  freccl  5993  frec2uzzd  9186  frec2uzsucd  9187  frec2uzrdg  9195  frecuzrdgsuc  9201
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