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Theorem frectfr 5924
 Description: Lemma to connect transfinite recursion theorems with finite recursion. That is, given the conditions 𝐹 Fn V and A ∈ 𝑉 on frec(𝐹, A), we want to be able to apply tfri1d 5890 or tfri2d 5891, and this lemma lets us satisfy hypotheses of those theorems. (Contributed by Jim Kingdon, 15-Aug-2019.)
Hypothesis
Ref Expression
frectfr.1 𝐺 = (g V ↦ {x ∣ (𝑚 𝜔 (dom g = suc 𝑚 x (𝐹‘(g𝑚))) (dom g = ∅ x A))})
Assertion
Ref Expression
frectfr ((z(𝐹z) V A 𝑉) → y(Fun 𝐺 (𝐺y) V))
Distinct variable groups:   g,𝑚,x,y,A   z,g,𝐹,𝑚,x,y   g,𝑉,𝑚,y
Allowed substitution hints:   A(z)   𝐺(x,y,z,g,𝑚)   𝑉(x,z)

Proof of Theorem frectfr
StepHypRef Expression
1 vex 2554 . . . . . . . 8 g V
21a1i 9 . . . . . . 7 ((z(𝐹z) V A 𝑉) → g V)
3 simpl 102 . . . . . . 7 ((z(𝐹z) V A 𝑉) → z(𝐹z) V)
4 simpr 103 . . . . . . 7 ((z(𝐹z) V A 𝑉) → A 𝑉)
52, 3, 4frecabex 5923 . . . . . 6 ((z(𝐹z) V A 𝑉) → {x ∣ (𝑚 𝜔 (dom g = suc 𝑚 x (𝐹‘(g𝑚))) (dom g = ∅ x A))} V)
65ralrimivw 2387 . . . . 5 ((z(𝐹z) V A 𝑉) → g V {x ∣ (𝑚 𝜔 (dom g = suc 𝑚 x (𝐹‘(g𝑚))) (dom g = ∅ x A))} V)
7 frectfr.1 . . . . . 6 𝐺 = (g V ↦ {x ∣ (𝑚 𝜔 (dom g = suc 𝑚 x (𝐹‘(g𝑚))) (dom g = ∅ x A))})
87fnmpt 4968 . . . . 5 (g V {x ∣ (𝑚 𝜔 (dom g = suc 𝑚 x (𝐹‘(g𝑚))) (dom g = ∅ x A))} V → 𝐺 Fn V)
96, 8syl 14 . . . 4 ((z(𝐹z) V A 𝑉) → 𝐺 Fn V)
10 vex 2554 . . . 4 y V
11 funfvex 5135 . . . . 5 ((Fun 𝐺 y dom 𝐺) → (𝐺y) V)
1211funfni 4942 . . . 4 ((𝐺 Fn V y V) → (𝐺y) V)
139, 10, 12sylancl 392 . . 3 ((z(𝐹z) V A 𝑉) → (𝐺y) V)
147funmpt2 4882 . . 3 Fun 𝐺
1513, 14jctil 295 . 2 ((z(𝐹z) V A 𝑉) → (Fun 𝐺 (𝐺y) V))
1615alrimiv 1751 1 ((z(𝐹z) V A 𝑉) → y(Fun 𝐺 (𝐺y) V))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97   ∨ wo 628  ∀wal 1240   = wceq 1242   ∈ wcel 1390  {cab 2023  ∀wral 2300  ∃wrex 2301  Vcvv 2551  ∅c0 3218   ↦ cmpt 3809  suc csuc 4068  𝜔com 4256  dom cdm 4288  Fun wfun 4839   Fn wfn 4840  ‘cfv 4845 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-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-pow 3918  ax-pr 3935  ax-un 4136  ax-iinf 4254 This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  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-ral 2305  df-rex 2306  df-reu 2307  df-rab 2309  df-v 2553  df-sbc 2759  df-csb 2847  df-un 2916  df-in 2918  df-ss 2925  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-id 4021  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 This theorem is referenced by:  frecfnom  5925  frecsuclem1  5926  frecsuclemdm  5927  frecsuclem3  5929
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