Theorem rdgruledefgg 6428

Description: The recursion rule for the recursive definition generator is defined everywhere. (Contributed by Jim Kingdon, 4-Jul-2019.)

Assertion

Ref	Expression
rdgruledefgg	⊢ ((𝐹 Fn V ∧ 𝐴 ∈ 𝑉) → (Fun (𝑔 ∈ V ↦ (𝐴 ∪ ∪ 𝑥 ∈ dom 𝑔(𝐹‘(𝑔‘𝑥)))) ∧ ((𝑔 ∈ V ↦ (𝐴 ∪ ∪ 𝑥 ∈ dom 𝑔(𝐹‘(𝑔‘𝑥))))‘𝑓) ∈ V))

Distinct variable groups:   𝐴,𝑔   𝑥,𝑔,𝐹

Allowed substitution hints:   𝐴(𝑥,𝑓)   𝐹(𝑓)   𝑉(𝑥,𝑓,𝑔)

Proof of Theorem rdgruledefgg

Step	Hyp	Ref	Expression
1		elex 2771	. 2 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ V)
2		funmpt 5292	. . . 4 ⊢ Fun (𝑔 ∈ V ↦ (𝐴 ∪ ∪ 𝑥 ∈ dom 𝑔(𝐹‘(𝑔‘𝑥))))
3		vex 2763	. . . . 5 ⊢ 𝑓 ∈ V
4		vex 2763	. . . . . . . . . . . . 13 ⊢ 𝑔 ∈ V
5		vex 2763	. . . . . . . . . . . . 13 ⊢ 𝑥 ∈ V
6	4, 5	fvex 5574	. . . . . . . . . . . 12 ⊢ (𝑔‘𝑥) ∈ V
7		funfvex 5571	. . . . . . . . . . . . 13 ⊢ ((Fun 𝐹 ∧ (𝑔‘𝑥) ∈ dom 𝐹) → (𝐹‘(𝑔‘𝑥)) ∈ V)
8	7	funfni 5354	. . . . . . . . . . . 12 ⊢ ((𝐹 Fn V ∧ (𝑔‘𝑥) ∈ V) → (𝐹‘(𝑔‘𝑥)) ∈ V)
9	6, 8	mpan2 425	. . . . . . . . . . 11 ⊢ (𝐹 Fn V → (𝐹‘(𝑔‘𝑥)) ∈ V)
10	9	ralrimivw 2568	. . . . . . . . . 10 ⊢ (𝐹 Fn V → ∀𝑥 ∈ dom 𝑔(𝐹‘(𝑔‘𝑥)) ∈ V)
11	4	dmex 4928	. . . . . . . . . . 11 ⊢ dom 𝑔 ∈ V
12		iunexg 6171	. . . . . . . . . . 11 ⊢ ((dom 𝑔 ∈ V ∧ ∀𝑥 ∈ dom 𝑔(𝐹‘(𝑔‘𝑥)) ∈ V) → ∪ 𝑥 ∈ dom 𝑔(𝐹‘(𝑔‘𝑥)) ∈ V)
13	11, 12	mpan 424	. . . . . . . . . 10 ⊢ (∀𝑥 ∈ dom 𝑔(𝐹‘(𝑔‘𝑥)) ∈ V → ∪ 𝑥 ∈ dom 𝑔(𝐹‘(𝑔‘𝑥)) ∈ V)
14	10, 13	syl 14	. . . . . . . . 9 ⊢ (𝐹 Fn V → ∪ 𝑥 ∈ dom 𝑔(𝐹‘(𝑔‘𝑥)) ∈ V)
15		unexg 4474	. . . . . . . . 9 ⊢ ((𝐴 ∈ V ∧ ∪ 𝑥 ∈ dom 𝑔(𝐹‘(𝑔‘𝑥)) ∈ V) → (𝐴 ∪ ∪ 𝑥 ∈ dom 𝑔(𝐹‘(𝑔‘𝑥))) ∈ V)
16	14, 15	sylan2 286	. . . . . . . 8 ⊢ ((𝐴 ∈ V ∧ 𝐹 Fn V) → (𝐴 ∪ ∪ 𝑥 ∈ dom 𝑔(𝐹‘(𝑔‘𝑥))) ∈ V)
17	16	ancoms 268	. . . . . . 7 ⊢ ((𝐹 Fn V ∧ 𝐴 ∈ V) → (𝐴 ∪ ∪ 𝑥 ∈ dom 𝑔(𝐹‘(𝑔‘𝑥))) ∈ V)
18	17	ralrimivw 2568	. . . . . 6 ⊢ ((𝐹 Fn V ∧ 𝐴 ∈ V) → ∀𝑔 ∈ V (𝐴 ∪ ∪ 𝑥 ∈ dom 𝑔(𝐹‘(𝑔‘𝑥))) ∈ V)
19		dmmptg 5163	. . . . . 6 ⊢ (∀𝑔 ∈ V (𝐴 ∪ ∪ 𝑥 ∈ dom 𝑔(𝐹‘(𝑔‘𝑥))) ∈ V → dom (𝑔 ∈ V ↦ (𝐴 ∪ ∪ 𝑥 ∈ dom 𝑔(𝐹‘(𝑔‘𝑥)))) = V)
20	18, 19	syl 14	. . . . 5 ⊢ ((𝐹 Fn V ∧ 𝐴 ∈ V) → dom (𝑔 ∈ V ↦ (𝐴 ∪ ∪ 𝑥 ∈ dom 𝑔(𝐹‘(𝑔‘𝑥)))) = V)
21	3, 20	eleqtrrid 2283	. . . 4 ⊢ ((𝐹 Fn V ∧ 𝐴 ∈ V) → 𝑓 ∈ dom (𝑔 ∈ V ↦ (𝐴 ∪ ∪ 𝑥 ∈ dom 𝑔(𝐹‘(𝑔‘𝑥)))))
22		funfvex 5571	. . . 4 ⊢ ((Fun (𝑔 ∈ V ↦ (𝐴 ∪ ∪ 𝑥 ∈ dom 𝑔(𝐹‘(𝑔‘𝑥)))) ∧ 𝑓 ∈ dom (𝑔 ∈ V ↦ (𝐴 ∪ ∪ 𝑥 ∈ dom 𝑔(𝐹‘(𝑔‘𝑥))))) → ((𝑔 ∈ V ↦ (𝐴 ∪ ∪ 𝑥 ∈ dom 𝑔(𝐹‘(𝑔‘𝑥))))‘𝑓) ∈ V)
23	2, 21, 22	sylancr 414	. . 3 ⊢ ((𝐹 Fn V ∧ 𝐴 ∈ V) → ((𝑔 ∈ V ↦ (𝐴 ∪ ∪ 𝑥 ∈ dom 𝑔(𝐹‘(𝑔‘𝑥))))‘𝑓) ∈ V)
24	23, 2	jctil 312	. 2 ⊢ ((𝐹 Fn V ∧ 𝐴 ∈ V) → (Fun (𝑔 ∈ V ↦ (𝐴 ∪ ∪ 𝑥 ∈ dom 𝑔(𝐹‘(𝑔‘𝑥)))) ∧ ((𝑔 ∈ V ↦ (𝐴 ∪ ∪ 𝑥 ∈ dom 𝑔(𝐹‘(𝑔‘𝑥))))‘𝑓) ∈ V))
25	1, 24	sylan2 286	1 ⊢ ((𝐹 Fn V ∧ 𝐴 ∈ 𝑉) → (Fun (𝑔 ∈ V ↦ (𝐴 ∪ ∪ 𝑥 ∈ dom 𝑔(𝐹‘(𝑔‘𝑥)))) ∧ ((𝑔 ∈ V ↦ (𝐴 ∪ ∪ 𝑥 ∈ dom 𝑔(𝐹‘(𝑔‘𝑥))))‘𝑓) ∈ V))

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1364   ∈ wcel 2164  ∀wral 2472  Vcvv 2760   ∪ cun 3151  ∪ ciun 3912   ↦ cmpt 4090  dom cdm 4659  Fun wfun 5248   Fn wfn 5249  ‘cfv 5254

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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-coll 4144  ax-sep 4147  ax-pow 4203  ax-pr 4238  ax-un 4464

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-reu 2479  df-rab 2481  df-v 2762  df-sbc 2986  df-csb 3081  df-un 3157  df-in 3159  df-ss 3166  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-iun 3914  df-br 4030  df-opab 4091  df-mpt 4092  df-id 4324  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-rn 4670  df-res 4671  df-ima 4672  df-iota 5215  df-fun 5256  df-fn 5257  df-f 5258  df-f1 5259  df-fo 5260  df-f1o 5261  df-fv 5262

This theorem is referenced by:  rdgruledefg  6429  rdgexggg  6430  rdgifnon  6432  rdgivallem  6434