Theorem rdgruledefgg 5902

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 2560	. 2 ⊢ (A ∈ 𝑉 → A ∈ V)
2		funmpt 4881	. . . 4 ⊢ Fun (g ∈ V ↦ (A ∪ ∪ x ∈ dom g(𝐹‘(g‘x))))
3		vex 2554	. . . . 5 ⊢ f ∈ V
4		vex 2554	. . . . . . . . . . . . 13 ⊢ g ∈ V
5		vex 2554	. . . . . . . . . . . . 13 ⊢ x ∈ V
6	4, 5	fvex 5138	. . . . . . . . . . . 12 ⊢ (g‘x) ∈ V
7		funfvex 5135	. . . . . . . . . . . . 13 ⊢ ((Fun 𝐹 ∧ (g‘x) ∈ dom 𝐹) → (𝐹‘(g‘x)) ∈ V)
8	7	funfni 4942	. . . . . . . . . . . 12 ⊢ ((𝐹 Fn V ∧ (g‘x) ∈ V) → (𝐹‘(g‘x)) ∈ V)
9	6, 8	mpan2 401	. . . . . . . . . . 11 ⊢ (𝐹 Fn V → (𝐹‘(g‘x)) ∈ V)
10	9	ralrimivw 2387	. . . . . . . . . 10 ⊢ (𝐹 Fn V → ∀x ∈ dom g(𝐹‘(g‘x)) ∈ V)
11	4	dmex 4541	. . . . . . . . . . 11 ⊢ dom g ∈ V
12		iunexg 5688	. . . . . . . . . . 11 ⊢ ((dom g ∈ V ∧ ∀x ∈ dom g(𝐹‘(g‘x)) ∈ V) → ∪ x ∈ dom g(𝐹‘(g‘x)) ∈ V)
13	11, 12	mpan 400	. . . . . . . . . 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 4144	. . . . . . . . 9 ⊢ ((A ∈ V ∧ ∪ x ∈ dom g(𝐹‘(g‘x)) ∈ V) → (A ∪ ∪ x ∈ dom g(𝐹‘(g‘x))) ∈ V)
16	14, 15	sylan2 270	. . . . . . . 8 ⊢ ((A ∈ V ∧ 𝐹 Fn V) → (A ∪ ∪ x ∈ dom g(𝐹‘(g‘x))) ∈ V)
17	16	ancoms 255	. . . . . . 7 ⊢ ((𝐹 Fn V ∧ A ∈ V) → (A ∪ ∪ x ∈ dom g(𝐹‘(g‘x))) ∈ V)
18	17	ralrimivw 2387	. . . . . 6 ⊢ ((𝐹 Fn V ∧ A ∈ V) → ∀g ∈ V (A ∪ ∪ x ∈ dom g(𝐹‘(g‘x))) ∈ V)
19		dmmptg 4761	. . . . . 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	syl5eleqr 2124	. . . 4 ⊢ ((𝐹 Fn V ∧ A ∈ V) → f ∈ dom (g ∈ V ↦ (A ∪ ∪ x ∈ dom g(𝐹‘(g‘x)))))
22		funfvex 5135	. . . 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 393	. . 3 ⊢ ((𝐹 Fn V ∧ A ∈ V) → ((g ∈ V ↦ (A ∪ ∪ x ∈ dom g(𝐹‘(g‘x))))‘f) ∈ V)
24	23, 2	jctil 295	. 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 270	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 97   = wceq 1242   ∈ wcel 1390  ∀wral 2300  Vcvv 2551   ∪ cun 2909  ∪ ciun 3648   ↦ cmpt 3809  dom cdm 4288  Fun wfun 4839   Fn wfn 4840  ‘cfv 4845

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-13 1401  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-coll 3863  ax-sep 3866  ax-pow 3918  ax-pr 3935  ax-un 4136

This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-reu 2307  df-rab 2309  df-v 2553  df-sbc 2759  df-csb 2847  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-iun 3650  df-br 3756  df-opab 3810  df-mpt 3811  df-id 4021  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-rn 4299  df-res 4300  df-ima 4301  df-iota 4810  df-fun 4847  df-fn 4848  df-f 4849  df-f1 4850  df-fo 4851  df-f1o 4852  df-fv 4853

This theorem is referenced by:  rdgruledefg  5903  rdgexggg  5904  rdgifnon  5906  rdgivallem  5908