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‘(g‘𝑚))) ∨ (dom g = ∅ ∧ x ∈ A))})

Assertion

Ref	Expression
frectfr	⊢ ((∀z(𝐹‘z) ∈ V ∧ A ∈ 𝑉) → ∀y(Fun 𝐺 ∧ (𝐺‘y) ∈ 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

Step	Hyp	Ref	Expression
1		vex 2554	. . . . . . . 8 ⊢ g ∈ V
2	1	a1i 9	. . . . . . 7 ⊢ ((∀z(𝐹‘z) ∈ V ∧ A ∈ 𝑉) → g ∈ V)
3		simpl 102	. . . . . . 7 ⊢ ((∀z(𝐹‘z) ∈ V ∧ A ∈ 𝑉) → ∀z(𝐹‘z) ∈ V)
4		simpr 103	. . . . . . 7 ⊢ ((∀z(𝐹‘z) ∈ V ∧ A ∈ 𝑉) → A ∈ 𝑉)
5	2, 3, 4	frecabex 5923	. . . . . 6 ⊢ ((∀z(𝐹‘z) ∈ V ∧ A ∈ 𝑉) → {x ∣ (∃𝑚 ∈ 𝜔 (dom g = suc 𝑚 ∧ x ∈ (𝐹‘(g‘𝑚))) ∨ (dom g = ∅ ∧ x ∈ A))} ∈ V)
6	5	ralrimivw 2387	. . . . 5 ⊢ ((∀z(𝐹‘z) ∈ V ∧ A ∈ 𝑉) → ∀g ∈ V {x ∣ (∃𝑚 ∈ 𝜔 (dom g = suc 𝑚 ∧ x ∈ (𝐹‘(g‘𝑚))) ∨ (dom g = ∅ ∧ x ∈ A))} ∈ V)
7		frectfr.1	. . . . . 6 ⊢ 𝐺 = (g ∈ V ↦ {x ∣ (∃𝑚 ∈ 𝜔 (dom g = suc 𝑚 ∧ x ∈ (𝐹‘(g‘𝑚))) ∨ (dom g = ∅ ∧ x ∈ A))})
8	7	fnmpt 4968	. . . . 5 ⊢ (∀g ∈ V {x ∣ (∃𝑚 ∈ 𝜔 (dom g = suc 𝑚 ∧ x ∈ (𝐹‘(g‘𝑚))) ∨ (dom g = ∅ ∧ x ∈ A))} ∈ V → 𝐺 Fn V)
9	6, 8	syl 14	. . . 4 ⊢ ((∀z(𝐹‘z) ∈ V ∧ A ∈ 𝑉) → 𝐺 Fn V)
10		vex 2554	. . . 4 ⊢ y ∈ V
11		funfvex 5135	. . . . 5 ⊢ ((Fun 𝐺 ∧ y ∈ dom 𝐺) → (𝐺‘y) ∈ V)
12	11	funfni 4942	. . . 4 ⊢ ((𝐺 Fn V ∧ y ∈ V) → (𝐺‘y) ∈ V)
13	9, 10, 12	sylancl 392	. . 3 ⊢ ((∀z(𝐹‘z) ∈ V ∧ A ∈ 𝑉) → (𝐺‘y) ∈ V)
14	7	funmpt2 4882	. . 3 ⊢ Fun 𝐺
15	13, 14	jctil 295	. 2 ⊢ ((∀z(𝐹‘z) ∈ V ∧ A ∈ 𝑉) → (Fun 𝐺 ∧ (𝐺‘y) ∈ V))
16	15	alrimiv 1751	1 ⊢ ((∀z(𝐹‘z) ∈ V ∧ A ∈ 𝑉) → ∀y(Fun 𝐺 ∧ (𝐺‘y) ∈ V))

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-bndl 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