Theorem frectfr 6458

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 6393 or tfri2d 6394, 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 2766	. . . . . . . 8 ⊢ 𝑔 ∈ V
2	1	a1i 9	. . . . . . 7 ⊢ ((∀𝑧(𝐹‘𝑧) ∈ V ∧ 𝐴 ∈ 𝑉) → 𝑔 ∈ V)
3		simpl 109	. . . . . . 7 ⊢ ((∀𝑧(𝐹‘𝑧) ∈ V ∧ 𝐴 ∈ 𝑉) → ∀𝑧(𝐹‘𝑧) ∈ V)
4		simpr 110	. . . . . . 7 ⊢ ((∀𝑧(𝐹‘𝑧) ∈ V ∧ 𝐴 ∈ 𝑉) → 𝐴 ∈ 𝑉)
5	2, 3, 4	frecabex 6456	. . . . . 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 2571	. . . . 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 5384	. . . . 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 2766	. . . 4 ⊢ 𝑦 ∈ V
11		funfvex 5575	. . . . 5 ⊢ ((Fun 𝐺 ∧ 𝑦 ∈ dom 𝐺) → (𝐺‘𝑦) ∈ V)
12	11	funfni 5358	. . . 4 ⊢ ((𝐺 Fn V ∧ 𝑦 ∈ V) → (𝐺‘𝑦) ∈ V)
13	9, 10, 12	sylancl 413	. . 3 ⊢ ((∀𝑧(𝐹‘𝑧) ∈ V ∧ 𝐴 ∈ 𝑉) → (𝐺‘𝑦) ∈ V)
14	7	funmpt2 5297	. . 3 ⊢ Fun 𝐺
15	13, 14	jctil 312	. 2 ⊢ ((∀𝑧(𝐹‘𝑧) ∈ V ∧ 𝐴 ∈ 𝑉) → (Fun 𝐺 ∧ (𝐺‘𝑦) ∈ V))
16	15	alrimiv 1888	1 ⊢ ((∀𝑧(𝐹‘𝑧) ∈ V ∧ 𝐴 ∈ 𝑉) → ∀𝑦(Fun 𝐺 ∧ (𝐺‘𝑦) ∈ V))

Syntax hints:   → wi 4   ∧ wa 104   ∨ wo 709  ∀wal 1362   = wceq 1364   ∈ wcel 2167  {cab 2182  ∀wral 2475  ∃wrex 2476  Vcvv 2763  ∅c0 3450   ↦ cmpt 4094  suc csuc 4400  ωcom 4626  dom cdm 4663  Fun wfun 5252   Fn wfn 5253  ‘cfv 5258

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-coll 4148  ax-sep 4151  ax-pow 4207  ax-pr 4242  ax-un 4468  ax-iinf 4624

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480  df-rex 2481  df-reu 2482  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-un 3161  df-in 3163  df-ss 3170  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-uni 3840  df-int 3875  df-iun 3918  df-br 4034  df-opab 4095  df-mpt 4096  df-id 4328  df-iom 4627  df-xp 4669  df-rel 4670  df-cnv 4671  df-co 4672  df-dm 4673  df-rn 4674  df-res 4675  df-ima 4676  df-iota 5219  df-fun 5260  df-fn 5261  df-f 5262  df-f1 5263  df-fo 5264  df-f1o 5265  df-fv 5266

This theorem is referenced by:  frecfnom  6459