Theorem rdgivallem 6625

Description: Value of the recursive definition generator. Lemma for rdgival 6626 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 6614	. . . 4 ⊢ rec(𝐹, 𝐴) = recs((𝑔 ∈ V ↦ (𝐴 ∪ ∪ 𝑥 ∈ dom 𝑔(𝐹‘(𝑔‘𝑥)))))
2		rdgruledefgg 6619	. . . . 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 1923	. . . 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 6580	. . 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 1221	. 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 2235	. . 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 4961	. . . . . 6 ⊢ (𝑔 = (rec(𝐹, 𝐴) ↾ 𝐵) → dom 𝑔 = dom (rec(𝐹, 𝐴) ↾ 𝐵))
8		onss 4620	. . . . . . . . 9 ⊢ (𝐵 ∈ On → 𝐵 ⊆ On)
9	8	3ad2ant3 1047	. . . . . . . 8 ⊢ ((𝐹 Fn V ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ On) → 𝐵 ⊆ On)
10		rdgifnon 6623	. . . . . . . . . 10 ⊢ ((𝐹 Fn V ∧ 𝐴 ∈ 𝑉) → rec(𝐹, 𝐴) Fn On)
11		fndm 5460	. . . . . . . . . 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 1044	. . . . . . . 8 ⊢ ((𝐹 Fn V ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ On) → dom rec(𝐹, 𝐴) = On)
14	9, 13	sseqtrrd 3281	. . . . . . 7 ⊢ ((𝐹 Fn V ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ On) → 𝐵 ⊆ dom rec(𝐹, 𝐴))
15		ssdmres 5065	. . . . . . 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 2289	. . . . 5 ⊢ (((𝐹 Fn V ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ On) ∧ 𝑔 = (rec(𝐹, 𝐴) ↾ 𝐵)) → dom 𝑔 = 𝐵)
18		fveq1 5674	. . . . . . 7 ⊢ (𝑔 = (rec(𝐹, 𝐴) ↾ 𝐵) → (𝑔‘𝑥) = ((rec(𝐹, 𝐴) ↾ 𝐵)‘𝑥))
19	18	fveq2d 5679	. . . . . 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 4020	. . . 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 3377	. . 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 6617	. . . . 5 ⊢ Fun rec(𝐹, 𝐴)
24		resfunexg 5910	. . . . 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 1047	. . 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 2818	. . . . . . . . . 10 ⊢ 𝑥 ∈ V
29		fvexg 5694	. . . . . . . . . 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 2618	. . . . . . . 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 5692	. . . . . . . . . . 11 ⊢ ((Fun 𝐹 ∧ ((rec(𝐹, 𝐴) ↾ 𝐵)‘𝑥) ∈ dom 𝐹) → (𝐹‘((rec(𝐹, 𝐴) ↾ 𝐵)‘𝑥)) ∈ V)
34	33	funfni 5463	. . . . . . . . . 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 2612	. . . . . . . 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 6321	. . . . . 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 1043	. . . 4 ⊢ ((𝐹 Fn V ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ On) → ∪ 𝑥 ∈ 𝐵 (𝐹‘((rec(𝐹, 𝐴) ↾ 𝐵)‘𝑥)) ∈ V)
42		unexg 4569	. . . . . 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 1046	. . . 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 5763	. 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 2267	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 1005   = wceq 1398   ∈ wcel 2205  ∀wral 2522  Vcvv 2815   ∪ cun 3212   ⊆ wss 3214  ∪ ciun 3996   ↦ cmpt 4176  Oncon0 4489  dom cdm 4754   ↾ cres 4756  Fun wfun 5351   Fn wfn 5352  ‘cfv 5357  reccrdg 6613

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-coll 4230  ax-sep 4233  ax-pow 4292  ax-pr 4327  ax-un 4559  ax-setind 4664

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-ral 2527  df-rex 2528  df-reu 2529  df-rab 2531  df-v 2817  df-sbc 3046  df-csb 3142  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-nul 3513  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-iun 3998  df-br 4115  df-opab 4177  df-mpt 4178  df-tr 4214  df-id 4419  df-iord 4492  df-on 4494  df-suc 4497  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-res 4766  df-ima 4767  df-iota 5317  df-fun 5359  df-fn 5360  df-f 5361  df-f1 5362  df-fo 5363  df-f1o 5364  df-fv 5365  df-recs 6549  df-irdg 6614

This theorem is referenced by:  rdgival  6626