Theorem rdgruledefgg 5882

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 2543	. 2 ⊢ (A ∈ 𝑉 → A ∈ V)
2		funmpt 4864	. . . 4 ⊢ Fun (g ∈ V ↦ (A ∪ ∪ x ∈ dom g(𝐹‘(g‘x))))
3		vex 2538	. . . . 5 ⊢ f ∈ V
4		vex 2538	. . . . . . . . . . . . 13 ⊢ g ∈ V
5		vex 2538	. . . . . . . . . . . . 13 ⊢ x ∈ V
6	4, 5	fvex 5120	. . . . . . . . . . . 12 ⊢ (g‘x) ∈ V
7		funfvex 5117	. . . . . . . . . . . . 13 ⊢ ((Fun 𝐹 ∧ (g‘x) ∈ dom 𝐹) → (𝐹‘(g‘x)) ∈ V)
8	7	funfni 4925	. . . . . . . . . . . 12 ⊢ ((𝐹 Fn V ∧ (g‘x) ∈ V) → (𝐹‘(g‘x)) ∈ V)
9	6, 8	mpan2 403	. . . . . . . . . . 11 ⊢ (𝐹 Fn V → (𝐹‘(g‘x)) ∈ V)
10	9	ralrimivw 2371	. . . . . . . . . 10 ⊢ (𝐹 Fn V → ∀x ∈ dom g(𝐹‘(g‘x)) ∈ V)
11	4	dmex 4525	. . . . . . . . . . 11 ⊢ dom g ∈ V
12		iunexg 5669	. . . . . . . . . . 11 ⊢ ((dom g ∈ V ∧ ∀x ∈ dom g(𝐹‘(g‘x)) ∈ V) → ∪ x ∈ dom g(𝐹‘(g‘x)) ∈ V)
13	11, 12	mpan 402	. . . . . . . . . 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 4128	. . . . . . . . 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 2371	. . . . . 6 ⊢ ((𝐹 Fn V ∧ A ∈ V) → ∀g ∈ V (A ∪ ∪ x ∈ dom g(𝐹‘(g‘x))) ∈ V)
19		dmmptg 4745	. . . . . 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 2109	. . . 4 ⊢ ((𝐹 Fn V ∧ A ∈ V) → f ∈ dom (g ∈ V ↦ (A ∪ ∪ x ∈ dom g(𝐹‘(g‘x)))))
22		funfvex 5117	. . . 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 395	. . 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 1228   ∈ wcel 1374  ∀wral 2284  Vcvv 2535   ∪ cun 2892  ∪ ciun 3631   ↦ cmpt 3792  dom cdm 4272  Fun wfun 4823   Fn wfn 4824  ‘cfv 4829

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 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-13 1385  ax-14 1386  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004  ax-coll 3846  ax-sep 3849  ax-pow 3901  ax-pr 3918  ax-un 4120

This theorem depends on definitions:  df-bi 110  df-3an 875  df-tru 1231  df-nf 1330  df-sb 1628  df-eu 1885  df-mo 1886  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-ral 2289  df-rex 2290  df-reu 2291  df-rab 2293  df-v 2537  df-sbc 2742  df-csb 2830  df-un 2899  df-in 2901  df-ss 2908  df-pw 3336  df-sn 3356  df-pr 3357  df-op 3359  df-uni 3555  df-iun 3633  df-br 3739  df-opab 3793  df-mpt 3794  df-id 4004  df-xp 4278  df-rel 4279  df-cnv 4280  df-co 4281  df-dm 4282  df-rn 4283  df-res 4284  df-ima 4285  df-iota 4794  df-fun 4831  df-fn 4832  df-f 4833  df-f1 4834  df-fo 4835  df-f1o 4836  df-fv 4837

This theorem is referenced by:  rdgruledefg  5883  rdgexggg  5884  rdgi0g  5886  rdgifnon  5887  rdgivallem  5888  rdgruledef  5894