Theorem rdgivallem 6546

Description: Value of the recursive definition generator. Lemma for rdgival 6547 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 6535	. . . 4 ⊢ rec(𝐹, 𝐴) = recs((𝑔 ∈ V ↦ (𝐴 ∪ ∪ 𝑥 ∈ dom 𝑔(𝐹‘(𝑔‘𝑥)))))
2		rdgruledefgg 6540	. . . . 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 1922	. . . 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 6501	. . 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 1220	. 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 2232	. . 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 4931	. . . . . 6 ⊢ (𝑔 = (rec(𝐹, 𝐴) ↾ 𝐵) → dom 𝑔 = dom (rec(𝐹, 𝐴) ↾ 𝐵))
8		onss 4591	. . . . . . . . 9 ⊢ (𝐵 ∈ On → 𝐵 ⊆ On)
9	8	3ad2ant3 1046	. . . . . . . 8 ⊢ ((𝐹 Fn V ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ On) → 𝐵 ⊆ On)
10		rdgifnon 6544	. . . . . . . . . 10 ⊢ ((𝐹 Fn V ∧ 𝐴 ∈ 𝑉) → rec(𝐹, 𝐴) Fn On)
11		fndm 5429	. . . . . . . . . 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 1043	. . . . . . . 8 ⊢ ((𝐹 Fn V ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ On) → dom rec(𝐹, 𝐴) = On)
14	9, 13	sseqtrrd 3266	. . . . . . 7 ⊢ ((𝐹 Fn V ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ On) → 𝐵 ⊆ dom rec(𝐹, 𝐴))
15		ssdmres 5035	. . . . . . 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 2286	. . . . 5 ⊢ (((𝐹 Fn V ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ On) ∧ 𝑔 = (rec(𝐹, 𝐴) ↾ 𝐵)) → dom 𝑔 = 𝐵)
18		fveq1 5638	. . . . . . 7 ⊢ (𝑔 = (rec(𝐹, 𝐴) ↾ 𝐵) → (𝑔‘𝑥) = ((rec(𝐹, 𝐴) ↾ 𝐵)‘𝑥))
19	18	fveq2d 5643	. . . . . 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 3994	. . . 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 3361	. . 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 6538	. . . . 5 ⊢ Fun rec(𝐹, 𝐴)
24		resfunexg 5874	. . . . 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 1046	. . 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 2805	. . . . . . . . . 10 ⊢ 𝑥 ∈ V
29		fvexg 5658	. . . . . . . . . 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 2606	. . . . . . . 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 5656	. . . . . . . . . . 11 ⊢ ((Fun 𝐹 ∧ ((rec(𝐹, 𝐴) ↾ 𝐵)‘𝑥) ∈ dom 𝐹) → (𝐹‘((rec(𝐹, 𝐴) ↾ 𝐵)‘𝑥)) ∈ V)
34	33	funfni 5432	. . . . . . . . . 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 2600	. . . . . . . 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 6280	. . . . . 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 1042	. . . 4 ⊢ ((𝐹 Fn V ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ On) → ∪ 𝑥 ∈ 𝐵 (𝐹‘((rec(𝐹, 𝐴) ↾ 𝐵)‘𝑥)) ∈ V)
42		unexg 4540	. . . . . 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 1045	. . . 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 5727	. 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 2264	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 1004   = wceq 1397   ∈ wcel 2202  ∀wral 2510  Vcvv 2802   ∪ cun 3198   ⊆ wss 3200  ∪ ciun 3970   ↦ cmpt 4150  Oncon0 4460  dom cdm 4725   ↾ cres 4727  Fun wfun 5320   Fn wfn 5321  ‘cfv 5326  reccrdg 6534

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 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4204  ax-sep 4207  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-ral 2515  df-rex 2516  df-reu 2517  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-nul 3495  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-iun 3972  df-br 4089  df-opab 4151  df-mpt 4152  df-tr 4188  df-id 4390  df-iord 4463  df-on 4465  df-suc 4468  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-f1 5331  df-fo 5332  df-f1o 5333  df-fv 5334  df-recs 6470  df-irdg 6535

This theorem is referenced by:  rdgival  6547