Theorem rdgruledefgg 6226

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 2668	. 2 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ V)
2		funmpt 5119	. . . 4 ⊢ Fun (𝑔 ∈ V ↦ (𝐴 ∪ ∪ 𝑥 ∈ dom 𝑔(𝐹‘(𝑔‘𝑥))))
3		vex 2660	. . . . 5 ⊢ 𝑓 ∈ V
4		vex 2660	. . . . . . . . . . . . 13 ⊢ 𝑔 ∈ V
5		vex 2660	. . . . . . . . . . . . 13 ⊢ 𝑥 ∈ V
6	4, 5	fvex 5395	. . . . . . . . . . . 12 ⊢ (𝑔‘𝑥) ∈ V
7		funfvex 5392	. . . . . . . . . . . . 13 ⊢ ((Fun 𝐹 ∧ (𝑔‘𝑥) ∈ dom 𝐹) → (𝐹‘(𝑔‘𝑥)) ∈ V)
8	7	funfni 5181	. . . . . . . . . . . 12 ⊢ ((𝐹 Fn V ∧ (𝑔‘𝑥) ∈ V) → (𝐹‘(𝑔‘𝑥)) ∈ V)
9	6, 8	mpan2 419	. . . . . . . . . . 11 ⊢ (𝐹 Fn V → (𝐹‘(𝑔‘𝑥)) ∈ V)
10	9	ralrimivw 2480	. . . . . . . . . 10 ⊢ (𝐹 Fn V → ∀𝑥 ∈ dom 𝑔(𝐹‘(𝑔‘𝑥)) ∈ V)
11	4	dmex 4763	. . . . . . . . . . 11 ⊢ dom 𝑔 ∈ V
12		iunexg 5971	. . . . . . . . . . 11 ⊢ ((dom 𝑔 ∈ V ∧ ∀𝑥 ∈ dom 𝑔(𝐹‘(𝑔‘𝑥)) ∈ V) → ∪ 𝑥 ∈ dom 𝑔(𝐹‘(𝑔‘𝑥)) ∈ V)
13	11, 12	mpan 418	. . . . . . . . . 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 4324	. . . . . . . . 9 ⊢ ((𝐴 ∈ V ∧ ∪ 𝑥 ∈ dom 𝑔(𝐹‘(𝑔‘𝑥)) ∈ V) → (𝐴 ∪ ∪ 𝑥 ∈ dom 𝑔(𝐹‘(𝑔‘𝑥))) ∈ V)
16	14, 15	sylan2 282	. . . . . . . 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 2480	. . . . . 6 ⊢ ((𝐹 Fn V ∧ 𝐴 ∈ V) → ∀𝑔 ∈ V (𝐴 ∪ ∪ 𝑥 ∈ dom 𝑔(𝐹‘(𝑔‘𝑥))) ∈ V)
19		dmmptg 4994	. . . . . 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 2204	. . . 4 ⊢ ((𝐹 Fn V ∧ 𝐴 ∈ V) → 𝑓 ∈ dom (𝑔 ∈ V ↦ (𝐴 ∪ ∪ 𝑥 ∈ dom 𝑔(𝐹‘(𝑔‘𝑥)))))
22		funfvex 5392	. . . 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 408	. . 3 ⊢ ((𝐹 Fn V ∧ 𝐴 ∈ V) → ((𝑔 ∈ V ↦ (𝐴 ∪ ∪ 𝑥 ∈ dom 𝑔(𝐹‘(𝑔‘𝑥))))‘𝑓) ∈ V)
24	23, 2	jctil 308	. 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 282	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 1314   ∈ wcel 1463  ∀wral 2390  Vcvv 2657   ∪ cun 3035  ∪ ciun 3779   ↦ cmpt 3949  dom cdm 4499  Fun wfun 5075   Fn wfn 5076  ‘cfv 5081

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-13 1474  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-coll 4003  ax-sep 4006  ax-pow 4058  ax-pr 4091  ax-un 4315

This theorem depends on definitions:  df-bi 116  df-3an 947  df-tru 1317  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2244  df-ral 2395  df-rex 2396  df-reu 2397  df-rab 2399  df-v 2659  df-sbc 2879  df-csb 2972  df-un 3041  df-in 3043  df-ss 3050  df-pw 3478  df-sn 3499  df-pr 3500  df-op 3502  df-uni 3703  df-iun 3781  df-br 3896  df-opab 3950  df-mpt 3951  df-id 4175  df-xp 4505  df-rel 4506  df-cnv 4507  df-co 4508  df-dm 4509  df-rn 4510  df-res 4511  df-ima 4512  df-iota 5046  df-fun 5083  df-fn 5084  df-f 5085  df-f1 5086  df-fo 5087  df-f1o 5088  df-fv 5089

This theorem is referenced by:  rdgruledefg  6227  rdgexggg  6228  rdgifnon  6230  rdgivallem  6232