Theorem rdgruledefgg 6511

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 2811	. 2 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ V)
2		funmpt 5352	. . . 4 ⊢ Fun (𝑔 ∈ V ↦ (𝐴 ∪ ∪ 𝑥 ∈ dom 𝑔(𝐹‘(𝑔‘𝑥))))
3		vex 2802	. . . . 5 ⊢ 𝑓 ∈ V
4		vex 2802	. . . . . . . . . . . . 13 ⊢ 𝑔 ∈ V
5		vex 2802	. . . . . . . . . . . . 13 ⊢ 𝑥 ∈ V
6	4, 5	fvex 5643	. . . . . . . . . . . 12 ⊢ (𝑔‘𝑥) ∈ V
7		funfvex 5640	. . . . . . . . . . . . 13 ⊢ ((Fun 𝐹 ∧ (𝑔‘𝑥) ∈ dom 𝐹) → (𝐹‘(𝑔‘𝑥)) ∈ V)
8	7	funfni 5419	. . . . . . . . . . . 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 2604	. . . . . . . . . 10 ⊢ (𝐹 Fn V → ∀𝑥 ∈ dom 𝑔(𝐹‘(𝑔‘𝑥)) ∈ V)
11	4	dmex 4987	. . . . . . . . . . 11 ⊢ dom 𝑔 ∈ V
12		iunexg 6254	. . . . . . . . . . 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 4531	. . . . . . . . 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 2604	. . . . . 6 ⊢ ((𝐹 Fn V ∧ 𝐴 ∈ V) → ∀𝑔 ∈ V (𝐴 ∪ ∪ 𝑥 ∈ dom 𝑔(𝐹‘(𝑔‘𝑥))) ∈ V)
19		dmmptg 5222	. . . . . 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 2319	. . . 4 ⊢ ((𝐹 Fn V ∧ 𝐴 ∈ V) → 𝑓 ∈ dom (𝑔 ∈ V ↦ (𝐴 ∪ ∪ 𝑥 ∈ dom 𝑔(𝐹‘(𝑔‘𝑥)))))
22		funfvex 5640	. . . 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 1395   ∈ wcel 2200  ∀wral 2508  Vcvv 2799   ∪ cun 3195  ∪ ciun 3964   ↦ cmpt 4144  dom cdm 4716  Fun wfun 5308   Fn wfn 5309  ‘cfv 5314

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4198  ax-sep 4201  ax-pow 4257  ax-pr 4292  ax-un 4521

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3888  df-iun 3966  df-br 4083  df-opab 4145  df-mpt 4146  df-id 4381  df-xp 4722  df-rel 4723  df-cnv 4724  df-co 4725  df-dm 4726  df-rn 4727  df-res 4728  df-ima 4729  df-iota 5274  df-fun 5316  df-fn 5317  df-f 5318  df-f1 5319  df-fo 5320  df-f1o 5321  df-fv 5322

This theorem is referenced by:  rdgruledefg  6512  rdgexggg  6513  rdgifnon  6515  rdgivallem  6517