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Theorem frecsuc 5930
Description: The successor value resulting from finite recursive definition generation. (Contributed by Jim Kingdon, 15-Aug-2019.)
Assertion
Ref Expression
frecsuc ((z(𝐹z) V A 𝑉 B 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 4105 . . . . . . . . . 10 (𝑛 = 𝑚 → suc 𝑛 = suc 𝑚)
21eqeq2d 2048 . . . . . . . . 9 (𝑛 = 𝑚 → (dom f = suc 𝑛 ↔ dom f = suc 𝑚))
3 fveq2 5121 . . . . . . . . . . 11 (𝑛 = 𝑚 → (f𝑛) = (f𝑚))
43fveq2d 5125 . . . . . . . . . 10 (𝑛 = 𝑚 → (𝐹‘(f𝑛)) = (𝐹‘(f𝑚)))
54eleq2d 2104 . . . . . . . . 9 (𝑛 = 𝑚 → (x (𝐹‘(f𝑛)) ↔ x (𝐹‘(f𝑚))))
62, 5anbi12d 442 . . . . . . . 8 (𝑛 = 𝑚 → ((dom f = suc 𝑛 x (𝐹‘(f𝑛))) ↔ (dom f = suc 𝑚 x (𝐹‘(f𝑚)))))
76cbvrexv 2528 . . . . . . 7 (𝑛 𝜔 (dom f = suc 𝑛 x (𝐹‘(f𝑛))) ↔ 𝑚 𝜔 (dom f = suc 𝑚 x (𝐹‘(f𝑚))))
87orbi1i 679 . . . . . 6 ((𝑛 𝜔 (dom f = suc 𝑛 x (𝐹‘(f𝑛))) (dom f = ∅ x A)) ↔ (𝑚 𝜔 (dom f = suc 𝑚 x (𝐹‘(f𝑚))) (dom f = ∅ x A)))
98abbii 2150 . . . . 5 {x ∣ (𝑛 𝜔 (dom f = suc 𝑛 x (𝐹‘(f𝑛))) (dom f = ∅ x A))} = {x ∣ (𝑚 𝜔 (dom f = suc 𝑚 x (𝐹‘(f𝑚))) (dom f = ∅ x A))}
10 eleq1 2097 . . . . . . . . 9 (x = y → (x (𝐹‘(f𝑚)) ↔ y (𝐹‘(f𝑚))))
1110anbi2d 437 . . . . . . . 8 (x = y → ((dom f = suc 𝑚 x (𝐹‘(f𝑚))) ↔ (dom f = suc 𝑚 y (𝐹‘(f𝑚)))))
1211rexbidv 2321 . . . . . . 7 (x = y → (𝑚 𝜔 (dom f = suc 𝑚 x (𝐹‘(f𝑚))) ↔ 𝑚 𝜔 (dom f = suc 𝑚 y (𝐹‘(f𝑚)))))
13 eleq1 2097 . . . . . . . 8 (x = y → (x Ay A))
1413anbi2d 437 . . . . . . 7 (x = y → ((dom f = ∅ x A) ↔ (dom f = ∅ y A)))
1512, 14orbi12d 706 . . . . . 6 (x = y → ((𝑚 𝜔 (dom f = suc 𝑚 x (𝐹‘(f𝑚))) (dom f = ∅ x A)) ↔ (𝑚 𝜔 (dom f = suc 𝑚 y (𝐹‘(f𝑚))) (dom f = ∅ y A))))
1615cbvabv 2158 . . . . 5 {x ∣ (𝑚 𝜔 (dom f = suc 𝑚 x (𝐹‘(f𝑚))) (dom f = ∅ x A))} = {y ∣ (𝑚 𝜔 (dom f = suc 𝑚 y (𝐹‘(f𝑚))) (dom f = ∅ y A))}
179, 16eqtri 2057 . . . 4 {x ∣ (𝑛 𝜔 (dom f = suc 𝑛 x (𝐹‘(f𝑛))) (dom f = ∅ x A))} = {y ∣ (𝑚 𝜔 (dom f = suc 𝑚 y (𝐹‘(f𝑚))) (dom f = ∅ y A))}
1817mpteq2i 3835 . . 3 (f V ↦ {x ∣ (𝑛 𝜔 (dom f = suc 𝑛 x (𝐹‘(f𝑛))) (dom f = ∅ x A))}) = (f V ↦ {y ∣ (𝑚 𝜔 (dom f = suc 𝑚 y (𝐹‘(f𝑚))) (dom f = ∅ y A))})
19 dmeq 4478 . . . . . . . . 9 (f = g → dom f = dom g)
2019eqeq1d 2045 . . . . . . . 8 (f = g → (dom f = suc 𝑚 ↔ dom g = suc 𝑚))
21 fveq1 5120 . . . . . . . . . 10 (f = g → (f𝑚) = (g𝑚))
2221fveq2d 5125 . . . . . . . . 9 (f = g → (𝐹‘(f𝑚)) = (𝐹‘(g𝑚)))
2322eleq2d 2104 . . . . . . . 8 (f = g → (y (𝐹‘(f𝑚)) ↔ y (𝐹‘(g𝑚))))
2420, 23anbi12d 442 . . . . . . 7 (f = g → ((dom f = suc 𝑚 y (𝐹‘(f𝑚))) ↔ (dom g = suc 𝑚 y (𝐹‘(g𝑚)))))
2524rexbidv 2321 . . . . . 6 (f = g → (𝑚 𝜔 (dom f = suc 𝑚 y (𝐹‘(f𝑚))) ↔ 𝑚 𝜔 (dom g = suc 𝑚 y (𝐹‘(g𝑚)))))
2619eqeq1d 2045 . . . . . . 7 (f = g → (dom f = ∅ ↔ dom g = ∅))
2726anbi1d 438 . . . . . 6 (f = g → ((dom f = ∅ y A) ↔ (dom g = ∅ y A)))
2825, 27orbi12d 706 . . . . 5 (f = g → ((𝑚 𝜔 (dom f = suc 𝑚 y (𝐹‘(f𝑚))) (dom f = ∅ y A)) ↔ (𝑚 𝜔 (dom g = suc 𝑚 y (𝐹‘(g𝑚))) (dom g = ∅ y A))))
2928abbidv 2152 . . . 4 (f = g → {y ∣ (𝑚 𝜔 (dom f = suc 𝑚 y (𝐹‘(f𝑚))) (dom f = ∅ y A))} = {y ∣ (𝑚 𝜔 (dom g = suc 𝑚 y (𝐹‘(g𝑚))) (dom g = ∅ y A))})
3029cbvmptv 3843 . . 3 (f V ↦ {y ∣ (𝑚 𝜔 (dom f = suc 𝑚 y (𝐹‘(f𝑚))) (dom f = ∅ y A))}) = (g V ↦ {y ∣ (𝑚 𝜔 (dom g = suc 𝑚 y (𝐹‘(g𝑚))) (dom g = ∅ y A))})
3118, 30eqtri 2057 . 2 (f V ↦ {x ∣ (𝑛 𝜔 (dom f = suc 𝑛 x (𝐹‘(f𝑛))) (dom f = ∅ x A))}) = (g V ↦ {y ∣ (𝑚 𝜔 (dom g = suc 𝑚 y (𝐹‘(g𝑚))) (dom g = ∅ y A))})
3231frecsuclem3 5929 1 ((z(𝐹z) V A 𝑉 B 𝜔) → (frec(𝐹, A)‘suc B) = (𝐹‘(frec(𝐹, A)‘B)))
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97   wo 628   w3a 884  wal 1240   = wceq 1242   wcel 1390  {cab 2023  wrex 2301  Vcvv 2551  c0 3218  cmpt 3809  suc csuc 4068  𝜔com 4256  dom cdm 4288  cfv 4845  freccfrec 5917
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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-13 1401  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-coll 3863  ax-sep 3866  ax-nul 3874  ax-pow 3918  ax-pr 3935  ax-un 4136  ax-setind 4220  ax-iinf 4254
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-fal 1248  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ne 2203  df-ral 2305  df-rex 2306  df-reu 2307  df-rab 2309  df-v 2553  df-sbc 2759  df-csb 2847  df-dif 2914  df-un 2916  df-in 2918  df-ss 2925  df-nul 3219  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-int 3607  df-iun 3650  df-br 3756  df-opab 3810  df-mpt 3811  df-tr 3846  df-id 4021  df-iord 4069  df-on 4071  df-suc 4074  df-iom 4257  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-rn 4299  df-res 4300  df-ima 4301  df-iota 4810  df-fun 4847  df-fn 4848  df-f 4849  df-f1 4850  df-fo 4851  df-f1o 4852  df-fv 4853  df-recs 5861  df-frec 5918
This theorem is referenced by:  frecrdg  5931  freccl  5932  frec2uzzd  8847  frec2uzsucd  8848  frec2uzrdg  8856  frecuzrdgsuc  8862
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