Theorem rdgruledefgg 6433

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 2774	. 2 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ V)
2		funmpt 5296	. . . 4 ⊢ Fun (𝑔 ∈ V ↦ (𝐴 ∪ ∪ 𝑥 ∈ dom 𝑔(𝐹‘(𝑔‘𝑥))))
3		vex 2766	. . . . 5 ⊢ 𝑓 ∈ V
4		vex 2766	. . . . . . . . . . . . 13 ⊢ 𝑔 ∈ V
5		vex 2766	. . . . . . . . . . . . 13 ⊢ 𝑥 ∈ V
6	4, 5	fvex 5578	. . . . . . . . . . . 12 ⊢ (𝑔‘𝑥) ∈ V
7		funfvex 5575	. . . . . . . . . . . . 13 ⊢ ((Fun 𝐹 ∧ (𝑔‘𝑥) ∈ dom 𝐹) → (𝐹‘(𝑔‘𝑥)) ∈ V)
8	7	funfni 5358	. . . . . . . . . . . 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 2571	. . . . . . . . . 10 ⊢ (𝐹 Fn V → ∀𝑥 ∈ dom 𝑔(𝐹‘(𝑔‘𝑥)) ∈ V)
11	4	dmex 4932	. . . . . . . . . . 11 ⊢ dom 𝑔 ∈ V
12		iunexg 6176	. . . . . . . . . . 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 4478	. . . . . . . . 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 2571	. . . . . 6 ⊢ ((𝐹 Fn V ∧ 𝐴 ∈ V) → ∀𝑔 ∈ V (𝐴 ∪ ∪ 𝑥 ∈ dom 𝑔(𝐹‘(𝑔‘𝑥))) ∈ V)
19		dmmptg 5167	. . . . . 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 2286	. . . 4 ⊢ ((𝐹 Fn V ∧ 𝐴 ∈ V) → 𝑓 ∈ dom (𝑔 ∈ V ↦ (𝐴 ∪ ∪ 𝑥 ∈ dom 𝑔(𝐹‘(𝑔‘𝑥)))))
22		funfvex 5575	. . . 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 2167  ∀wral 2475  Vcvv 2763   ∪ cun 3155  ∪ ciun 3916   ↦ cmpt 4094  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

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-iun 3918  df-br 4034  df-opab 4095  df-mpt 4096  df-id 4328  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:  rdgruledefg  6434  rdgexggg  6435  rdgifnon  6437  rdgivallem  6439