Theorem frectfr 5896

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 5867 or tfri2d 5868, 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	⊢ ((𝐹 Fn V ∧ A ∈ 𝑉) → ∀y(Fun 𝐺 ∧ (𝐺‘y) ∈ V))

Distinct variable groups:   g,𝑚,x,y,A   g,𝐹,𝑚,x,y   g,𝑉,𝑚,y

Allowed substitution hints:   𝐺(x,y,g,𝑚)   𝑉(x)

Proof of Theorem frectfr

Step	Hyp	Ref	Expression
1		vex 2534	. . . . . . . 8 ⊢ g ∈ V
2	1	a1i 9	. . . . . . 7 ⊢ ((𝐹 Fn V ∧ A ∈ 𝑉) → g ∈ V)
3		ax-ia1 99	. . . . . . 7 ⊢ ((𝐹 Fn V ∧ A ∈ 𝑉) → 𝐹 Fn V)
4		ax-ia2 100	. . . . . . 7 ⊢ ((𝐹 Fn V ∧ A ∈ 𝑉) → A ∈ 𝑉)
5	2, 3, 4	frecabex 5895	. . . . . 6 ⊢ ((𝐹 Fn V ∧ A ∈ 𝑉) → {x ∣ (∃𝑚 ∈ 𝜔 (dom g = suc 𝑚 ∧ x ∈ (𝐹‘(g‘𝑚))) ∨ (dom g = ∅ ∧ x ∈ A))} ∈ V)
6	5	ralrimivw 2367	. . . . 5 ⊢ ((𝐹 Fn 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 4947	. . . . 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 ⊢ ((𝐹 Fn V ∧ A ∈ 𝑉) → 𝐺 Fn V)
10		vex 2534	. . . 4 ⊢ y ∈ V
11		funfvex 5113	. . . . 5 ⊢ ((Fun 𝐺 ∧ y ∈ dom 𝐺) → (𝐺‘y) ∈ V)
12	11	funfni 4921	. . . 4 ⊢ ((𝐺 Fn V ∧ y ∈ V) → (𝐺‘y) ∈ V)
13	9, 10, 12	sylancl 394	. . 3 ⊢ ((𝐹 Fn V ∧ A ∈ 𝑉) → (𝐺‘y) ∈ V)
14	7	funmpt2 4861	. . 3 ⊢ Fun 𝐺
15	13, 14	jctil 295	. 2 ⊢ ((𝐹 Fn V ∧ A ∈ 𝑉) → (Fun 𝐺 ∧ (𝐺‘y) ∈ V))
16	15	alrimiv 1732	1 ⊢ ((𝐹 Fn V ∧ A ∈ 𝑉) → ∀y(Fun 𝐺 ∧ (𝐺‘y) ∈ V))

Syntax hints:   → wi 4   ∧ wa 97   ∨ wo 616  ∀wal 1224   = wceq 1226   ∈ wcel 1370  {cab 2004  ∀wral 2280  ∃wrex 2281  Vcvv 2531  ∅c0 3197   ↦ cmpt 3788  suc csuc 4047  𝜔com 4236  dom cdm 4268  Fun wfun 4819   Fn wfn 4820  ‘cfv 4825

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 617  ax-5 1312  ax-7 1313  ax-gen 1314  ax-ie1 1359  ax-ie2 1360  ax-8 1372  ax-10 1373  ax-11 1374  ax-i12 1375  ax-bnd 1376  ax-4 1377  ax-13 1381  ax-14 1382  ax-17 1396  ax-i9 1400  ax-ial 1405  ax-i5r 1406  ax-ext 2000  ax-coll 3842  ax-sep 3845  ax-pow 3897  ax-pr 3914  ax-un 4116  ax-iinf 4234

This theorem depends on definitions:  df-bi 110  df-3an 873  df-tru 1229  df-nf 1326  df-sb 1624  df-eu 1881  df-mo 1882  df-clab 2005  df-cleq 2011  df-clel 2014  df-nfc 2145  df-ral 2285  df-rex 2286  df-reu 2287  df-rab 2289  df-v 2533  df-sbc 2738  df-csb 2826  df-un 2895  df-in 2897  df-ss 2904  df-pw 3332  df-sn 3352  df-pr 3353  df-op 3355  df-uni 3551  df-int 3586  df-iun 3629  df-br 3735  df-opab 3789  df-mpt 3790  df-id 4000  df-iom 4237  df-xp 4274  df-rel 4275  df-cnv 4276  df-co 4277  df-dm 4278  df-rn 4279  df-res 4280  df-ima 4281  df-iota 4790  df-fun 4827  df-fn 4828  df-f 4829  df-f1 4830  df-fo 4831  df-f1o 4832  df-fv 4833

This theorem is referenced by:  frecfnom  5897  frec0g  5898  frecsuclem1  5899  frecsuclemdm  5900  frecsuclem3  5902