Theorem rdgivallem 6542

Description: Value of the recursive definition generator. Lemma for rdgival 6543 which simplifies the value further. (Contributed by Jim Kingdon, 13-Jul-2019.) (New usage is discouraged.)

Assertion

Ref	Expression
rdgivallem	⊢ ((𝐹 Fn V ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ On) → (rec(𝐹, 𝐴)‘𝐵) = (𝐴 ∪ ∪ 𝑥 ∈ 𝐵 (𝐹‘((rec(𝐹, 𝐴) ↾ 𝐵)‘𝑥))))

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵   𝑥,𝐹   𝑥,𝑉

Proof of Theorem rdgivallem

Dummy variables 𝑔 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-irdg 6531	. . . 4 ⊢ rec(𝐹, 𝐴) = recs((𝑔 ∈ V ↦ (𝐴 ∪ ∪ 𝑥 ∈ dom 𝑔(𝐹‘(𝑔‘𝑥)))))
2		rdgruledefgg 6536	. . . . 5 ⊢ ((𝐹 Fn V ∧ 𝐴 ∈ 𝑉) → (Fun (𝑔 ∈ V ↦ (𝐴 ∪ ∪ 𝑥 ∈ dom 𝑔(𝐹‘(𝑔‘𝑥)))) ∧ ((𝑔 ∈ V ↦ (𝐴 ∪ ∪ 𝑥 ∈ dom 𝑔(𝐹‘(𝑔‘𝑥))))‘𝑦) ∈ V))
3	2	alrimiv 1920	. . . 4 ⊢ ((𝐹 Fn V ∧ 𝐴 ∈ 𝑉) → ∀𝑦(Fun (𝑔 ∈ V ↦ (𝐴 ∪ ∪ 𝑥 ∈ dom 𝑔(𝐹‘(𝑔‘𝑥)))) ∧ ((𝑔 ∈ V ↦ (𝐴 ∪ ∪ 𝑥 ∈ dom 𝑔(𝐹‘(𝑔‘𝑥))))‘𝑦) ∈ V))
4	1, 3	tfri2d 6497	. . 3 ⊢ (((𝐹 Fn V ∧ 𝐴 ∈ 𝑉) ∧ 𝐵 ∈ On) → (rec(𝐹, 𝐴)‘𝐵) = ((𝑔 ∈ V ↦ (𝐴 ∪ ∪ 𝑥 ∈ dom 𝑔(𝐹‘(𝑔‘𝑥))))‘(rec(𝐹, 𝐴) ↾ 𝐵)))
5	4	3impa 1218	. 2 ⊢ ((𝐹 Fn V ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ On) → (rec(𝐹, 𝐴)‘𝐵) = ((𝑔 ∈ V ↦ (𝐴 ∪ ∪ 𝑥 ∈ dom 𝑔(𝐹‘(𝑔‘𝑥))))‘(rec(𝐹, 𝐴) ↾ 𝐵)))
6		eqidd 2230	. . 3 ⊢ ((𝐹 Fn V ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ On) → (𝑔 ∈ V ↦ (𝐴 ∪ ∪ 𝑥 ∈ dom 𝑔(𝐹‘(𝑔‘𝑥)))) = (𝑔 ∈ V ↦ (𝐴 ∪ ∪ 𝑥 ∈ dom 𝑔(𝐹‘(𝑔‘𝑥)))))
7		dmeq 4929	. . . . . 6 ⊢ (𝑔 = (rec(𝐹, 𝐴) ↾ 𝐵) → dom 𝑔 = dom (rec(𝐹, 𝐴) ↾ 𝐵))
8		onss 4589	. . . . . . . . 9 ⊢ (𝐵 ∈ On → 𝐵 ⊆ On)
9	8	3ad2ant3 1044	. . . . . . . 8 ⊢ ((𝐹 Fn V ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ On) → 𝐵 ⊆ On)
10		rdgifnon 6540	. . . . . . . . . 10 ⊢ ((𝐹 Fn V ∧ 𝐴 ∈ 𝑉) → rec(𝐹, 𝐴) Fn On)
11		fndm 5426	. . . . . . . . . 10 ⊢ (rec(𝐹, 𝐴) Fn On → dom rec(𝐹, 𝐴) = On)
12	10, 11	syl 14	. . . . . . . . 9 ⊢ ((𝐹 Fn V ∧ 𝐴 ∈ 𝑉) → dom rec(𝐹, 𝐴) = On)
13	12	3adant3 1041	. . . . . . . 8 ⊢ ((𝐹 Fn V ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ On) → dom rec(𝐹, 𝐴) = On)
14	9, 13	sseqtrrd 3264	. . . . . . 7 ⊢ ((𝐹 Fn V ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ On) → 𝐵 ⊆ dom rec(𝐹, 𝐴))
15		ssdmres 5033	. . . . . . 7 ⊢ (𝐵 ⊆ dom rec(𝐹, 𝐴) ↔ dom (rec(𝐹, 𝐴) ↾ 𝐵) = 𝐵)
16	14, 15	sylib 122	. . . . . 6 ⊢ ((𝐹 Fn V ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ On) → dom (rec(𝐹, 𝐴) ↾ 𝐵) = 𝐵)
17	7, 16	sylan9eqr 2284	. . . . 5 ⊢ (((𝐹 Fn V ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ On) ∧ 𝑔 = (rec(𝐹, 𝐴) ↾ 𝐵)) → dom 𝑔 = 𝐵)
18		fveq1 5634	. . . . . . 7 ⊢ (𝑔 = (rec(𝐹, 𝐴) ↾ 𝐵) → (𝑔‘𝑥) = ((rec(𝐹, 𝐴) ↾ 𝐵)‘𝑥))
19	18	fveq2d 5639	. . . . . 6 ⊢ (𝑔 = (rec(𝐹, 𝐴) ↾ 𝐵) → (𝐹‘(𝑔‘𝑥)) = (𝐹‘((rec(𝐹, 𝐴) ↾ 𝐵)‘𝑥)))
20	19	adantl 277	. . . . 5 ⊢ (((𝐹 Fn V ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ On) ∧ 𝑔 = (rec(𝐹, 𝐴) ↾ 𝐵)) → (𝐹‘(𝑔‘𝑥)) = (𝐹‘((rec(𝐹, 𝐴) ↾ 𝐵)‘𝑥)))
21	17, 20	iuneq12d 3992	. . . 4 ⊢ (((𝐹 Fn V ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ On) ∧ 𝑔 = (rec(𝐹, 𝐴) ↾ 𝐵)) → ∪ 𝑥 ∈ dom 𝑔(𝐹‘(𝑔‘𝑥)) = ∪ 𝑥 ∈ 𝐵 (𝐹‘((rec(𝐹, 𝐴) ↾ 𝐵)‘𝑥)))
22	21	uneq2d 3359	. . 3 ⊢ (((𝐹 Fn V ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ On) ∧ 𝑔 = (rec(𝐹, 𝐴) ↾ 𝐵)) → (𝐴 ∪ ∪ 𝑥 ∈ dom 𝑔(𝐹‘(𝑔‘𝑥))) = (𝐴 ∪ ∪ 𝑥 ∈ 𝐵 (𝐹‘((rec(𝐹, 𝐴) ↾ 𝐵)‘𝑥))))
23		rdgfun 6534	. . . . 5 ⊢ Fun rec(𝐹, 𝐴)
24		resfunexg 5870	. . . . 5 ⊢ ((Fun rec(𝐹, 𝐴) ∧ 𝐵 ∈ On) → (rec(𝐹, 𝐴) ↾ 𝐵) ∈ V)
25	23, 24	mpan 424	. . . 4 ⊢ (𝐵 ∈ On → (rec(𝐹, 𝐴) ↾ 𝐵) ∈ V)
26	25	3ad2ant3 1044	. . 3 ⊢ ((𝐹 Fn V ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ On) → (rec(𝐹, 𝐴) ↾ 𝐵) ∈ V)
27		simpr 110	. . . . . 6 ⊢ ((𝐹 Fn V ∧ 𝐵 ∈ On) → 𝐵 ∈ On)
28		vex 2803	. . . . . . . . . 10 ⊢ 𝑥 ∈ V
29		fvexg 5654	. . . . . . . . . 10 ⊢ (((rec(𝐹, 𝐴) ↾ 𝐵) ∈ V ∧ 𝑥 ∈ V) → ((rec(𝐹, 𝐴) ↾ 𝐵)‘𝑥) ∈ V)
30	25, 28, 29	sylancl 413	. . . . . . . . 9 ⊢ (𝐵 ∈ On → ((rec(𝐹, 𝐴) ↾ 𝐵)‘𝑥) ∈ V)
31	30	ralrimivw 2604	. . . . . . . 8 ⊢ (𝐵 ∈ On → ∀𝑥 ∈ 𝐵 ((rec(𝐹, 𝐴) ↾ 𝐵)‘𝑥) ∈ V)
32	31	adantl 277	. . . . . . 7 ⊢ ((𝐹 Fn V ∧ 𝐵 ∈ On) → ∀𝑥 ∈ 𝐵 ((rec(𝐹, 𝐴) ↾ 𝐵)‘𝑥) ∈ V)
33		funfvex 5652	. . . . . . . . . . 11 ⊢ ((Fun 𝐹 ∧ ((rec(𝐹, 𝐴) ↾ 𝐵)‘𝑥) ∈ dom 𝐹) → (𝐹‘((rec(𝐹, 𝐴) ↾ 𝐵)‘𝑥)) ∈ V)
34	33	funfni 5429	. . . . . . . . . 10 ⊢ ((𝐹 Fn V ∧ ((rec(𝐹, 𝐴) ↾ 𝐵)‘𝑥) ∈ V) → (𝐹‘((rec(𝐹, 𝐴) ↾ 𝐵)‘𝑥)) ∈ V)
35	34	ex 115	. . . . . . . . 9 ⊢ (𝐹 Fn V → (((rec(𝐹, 𝐴) ↾ 𝐵)‘𝑥) ∈ V → (𝐹‘((rec(𝐹, 𝐴) ↾ 𝐵)‘𝑥)) ∈ V))
36	35	ralimdv 2598	. . . . . . . 8 ⊢ (𝐹 Fn V → (∀𝑥 ∈ 𝐵 ((rec(𝐹, 𝐴) ↾ 𝐵)‘𝑥) ∈ V → ∀𝑥 ∈ 𝐵 (𝐹‘((rec(𝐹, 𝐴) ↾ 𝐵)‘𝑥)) ∈ V))
37	36	adantr 276	. . . . . . 7 ⊢ ((𝐹 Fn V ∧ 𝐵 ∈ On) → (∀𝑥 ∈ 𝐵 ((rec(𝐹, 𝐴) ↾ 𝐵)‘𝑥) ∈ V → ∀𝑥 ∈ 𝐵 (𝐹‘((rec(𝐹, 𝐴) ↾ 𝐵)‘𝑥)) ∈ V))
38	32, 37	mpd 13	. . . . . 6 ⊢ ((𝐹 Fn V ∧ 𝐵 ∈ On) → ∀𝑥 ∈ 𝐵 (𝐹‘((rec(𝐹, 𝐴) ↾ 𝐵)‘𝑥)) ∈ V)
39		iunexg 6276	. . . . . 6 ⊢ ((𝐵 ∈ On ∧ ∀𝑥 ∈ 𝐵 (𝐹‘((rec(𝐹, 𝐴) ↾ 𝐵)‘𝑥)) ∈ V) → ∪ 𝑥 ∈ 𝐵 (𝐹‘((rec(𝐹, 𝐴) ↾ 𝐵)‘𝑥)) ∈ V)
40	27, 38, 39	syl2anc 411	. . . . 5 ⊢ ((𝐹 Fn V ∧ 𝐵 ∈ On) → ∪ 𝑥 ∈ 𝐵 (𝐹‘((rec(𝐹, 𝐴) ↾ 𝐵)‘𝑥)) ∈ V)
41	40	3adant2 1040	. . . 4 ⊢ ((𝐹 Fn V ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ On) → ∪ 𝑥 ∈ 𝐵 (𝐹‘((rec(𝐹, 𝐴) ↾ 𝐵)‘𝑥)) ∈ V)
42		unexg 4538	. . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ ∪ 𝑥 ∈ 𝐵 (𝐹‘((rec(𝐹, 𝐴) ↾ 𝐵)‘𝑥)) ∈ V) → (𝐴 ∪ ∪ 𝑥 ∈ 𝐵 (𝐹‘((rec(𝐹, 𝐴) ↾ 𝐵)‘𝑥))) ∈ V)
43	42	ex 115	. . . . 5 ⊢ (𝐴 ∈ 𝑉 → (∪ 𝑥 ∈ 𝐵 (𝐹‘((rec(𝐹, 𝐴) ↾ 𝐵)‘𝑥)) ∈ V → (𝐴 ∪ ∪ 𝑥 ∈ 𝐵 (𝐹‘((rec(𝐹, 𝐴) ↾ 𝐵)‘𝑥))) ∈ V))
44	43	3ad2ant2 1043	. . . 4 ⊢ ((𝐹 Fn V ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ On) → (∪ 𝑥 ∈ 𝐵 (𝐹‘((rec(𝐹, 𝐴) ↾ 𝐵)‘𝑥)) ∈ V → (𝐴 ∪ ∪ 𝑥 ∈ 𝐵 (𝐹‘((rec(𝐹, 𝐴) ↾ 𝐵)‘𝑥))) ∈ V))
45	41, 44	mpd 13	. . 3 ⊢ ((𝐹 Fn V ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ On) → (𝐴 ∪ ∪ 𝑥 ∈ 𝐵 (𝐹‘((rec(𝐹, 𝐴) ↾ 𝐵)‘𝑥))) ∈ V)
46	6, 22, 26, 45	fvmptd 5723	. 2 ⊢ ((𝐹 Fn V ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ On) → ((𝑔 ∈ V ↦ (𝐴 ∪ ∪ 𝑥 ∈ dom 𝑔(𝐹‘(𝑔‘𝑥))))‘(rec(𝐹, 𝐴) ↾ 𝐵)) = (𝐴 ∪ ∪ 𝑥 ∈ 𝐵 (𝐹‘((rec(𝐹, 𝐴) ↾ 𝐵)‘𝑥))))
47	5, 46	eqtrd 2262	1 ⊢ ((𝐹 Fn V ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ On) → (rec(𝐹, 𝐴)‘𝐵) = (𝐴 ∪ ∪ 𝑥 ∈ 𝐵 (𝐹‘((rec(𝐹, 𝐴) ↾ 𝐵)‘𝑥))))

Syntax hints:   → wi 4   ∧ wa 104   ∧ w3a 1002   = wceq 1395   ∈ wcel 2200  ∀wral 2508  Vcvv 2800   ∪ cun 3196   ⊆ wss 3198  ∪ ciun 3968   ↦ cmpt 4148  Oncon0 4458  dom cdm 4723   ↾ cres 4725  Fun wfun 5318   Fn wfn 5319  ‘cfv 5324  reccrdg 6530

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-in1 617  ax-in2 618  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 4202  ax-sep 4205  ax-pow 4262  ax-pr 4297  ax-un 4528  ax-setind 4633

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-fal 1401  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-ne 2401  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2517  df-v 2802  df-sbc 3030  df-csb 3126  df-dif 3200  df-un 3202  df-in 3204  df-ss 3211  df-nul 3493  df-pw 3652  df-sn 3673  df-pr 3674  df-op 3676  df-uni 3892  df-iun 3970  df-br 4087  df-opab 4149  df-mpt 4150  df-tr 4186  df-id 4388  df-iord 4461  df-on 4463  df-suc 4466  df-xp 4729  df-rel 4730  df-cnv 4731  df-co 4732  df-dm 4733  df-rn 4734  df-res 4735  df-ima 4736  df-iota 5284  df-fun 5326  df-fn 5327  df-f 5328  df-f1 5329  df-fo 5330  df-f1o 5331  df-fv 5332  df-recs 6466  df-irdg 6531

This theorem is referenced by:  rdgival  6543