Theorem rdgruledefgg 6272

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 2697	. 2 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ V)
2		funmpt 5161	. . . 4 ⊢ Fun (𝑔 ∈ V ↦ (𝐴 ∪ ∪ 𝑥 ∈ dom 𝑔(𝐹‘(𝑔‘𝑥))))
3		vex 2689	. . . . 5 ⊢ 𝑓 ∈ V
4		vex 2689	. . . . . . . . . . . . 13 ⊢ 𝑔 ∈ V
5		vex 2689	. . . . . . . . . . . . 13 ⊢ 𝑥 ∈ V
6	4, 5	fvex 5441	. . . . . . . . . . . 12 ⊢ (𝑔‘𝑥) ∈ V
7		funfvex 5438	. . . . . . . . . . . . 13 ⊢ ((Fun 𝐹 ∧ (𝑔‘𝑥) ∈ dom 𝐹) → (𝐹‘(𝑔‘𝑥)) ∈ V)
8	7	funfni 5223	. . . . . . . . . . . 12 ⊢ ((𝐹 Fn V ∧ (𝑔‘𝑥) ∈ V) → (𝐹‘(𝑔‘𝑥)) ∈ V)
9	6, 8	mpan2 421	. . . . . . . . . . 11 ⊢ (𝐹 Fn V → (𝐹‘(𝑔‘𝑥)) ∈ V)
10	9	ralrimivw 2506	. . . . . . . . . 10 ⊢ (𝐹 Fn V → ∀𝑥 ∈ dom 𝑔(𝐹‘(𝑔‘𝑥)) ∈ V)
11	4	dmex 4805	. . . . . . . . . . 11 ⊢ dom 𝑔 ∈ V
12		iunexg 6017	. . . . . . . . . . 11 ⊢ ((dom 𝑔 ∈ V ∧ ∀𝑥 ∈ dom 𝑔(𝐹‘(𝑔‘𝑥)) ∈ V) → ∪ 𝑥 ∈ dom 𝑔(𝐹‘(𝑔‘𝑥)) ∈ V)
13	11, 12	mpan 420	. . . . . . . . . 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 4364	. . . . . . . . 9 ⊢ ((𝐴 ∈ V ∧ ∪ 𝑥 ∈ dom 𝑔(𝐹‘(𝑔‘𝑥)) ∈ V) → (𝐴 ∪ ∪ 𝑥 ∈ dom 𝑔(𝐹‘(𝑔‘𝑥))) ∈ V)
16	14, 15	sylan2 284	. . . . . . . 8 ⊢ ((𝐴 ∈ V ∧ 𝐹 Fn V) → (𝐴 ∪ ∪ 𝑥 ∈ dom 𝑔(𝐹‘(𝑔‘𝑥))) ∈ V)
17	16	ancoms 266	. . . . . . 7 ⊢ ((𝐹 Fn V ∧ 𝐴 ∈ V) → (𝐴 ∪ ∪ 𝑥 ∈ dom 𝑔(𝐹‘(𝑔‘𝑥))) ∈ V)
18	17	ralrimivw 2506	. . . . . 6 ⊢ ((𝐹 Fn V ∧ 𝐴 ∈ V) → ∀𝑔 ∈ V (𝐴 ∪ ∪ 𝑥 ∈ dom 𝑔(𝐹‘(𝑔‘𝑥))) ∈ V)
19		dmmptg 5036	. . . . . 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 2229	. . . 4 ⊢ ((𝐹 Fn V ∧ 𝐴 ∈ V) → 𝑓 ∈ dom (𝑔 ∈ V ↦ (𝐴 ∪ ∪ 𝑥 ∈ dom 𝑔(𝐹‘(𝑔‘𝑥)))))
22		funfvex 5438	. . . 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 410	. . 3 ⊢ ((𝐹 Fn V ∧ 𝐴 ∈ V) → ((𝑔 ∈ V ↦ (𝐴 ∪ ∪ 𝑥 ∈ dom 𝑔(𝐹‘(𝑔‘𝑥))))‘𝑓) ∈ V)
24	23, 2	jctil 310	. 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 284	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 103   = wceq 1331   ∈ wcel 1480  ∀wral 2416  Vcvv 2686   ∪ cun 3069  ∪ ciun 3813   ↦ cmpt 3989  dom cdm 4539  Fun wfun 5117   Fn wfn 5118  ‘cfv 5123

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-coll 4043  ax-sep 4046  ax-pow 4098  ax-pr 4131  ax-un 4355

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422  df-reu 2423  df-rab 2425  df-v 2688  df-sbc 2910  df-csb 3004  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-iun 3815  df-br 3930  df-opab 3990  df-mpt 3991  df-id 4215  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-f1 5128  df-fo 5129  df-f1o 5130  df-fv 5131

This theorem is referenced by:  rdgruledefg  6273  rdgexggg  6274  rdgifnon  6276  rdgivallem  6278