ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  frectfr Structured version   GIF version

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
  Copyright terms: Public domain W3C validator