Theorem rdgivallem 6480

Description: Value of the recursive definition generator. Lemma for rdgival 6481 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 6469	. . . 4 ⊢ rec(𝐹, 𝐴) = recs((𝑔 ∈ V ↦ (𝐴 ∪ ∪ 𝑥 ∈ dom 𝑔(𝐹‘(𝑔‘𝑥)))))
2		rdgruledefgg 6474	. . . . 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 1898	. . . 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 6435	. . 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 1197	. 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 2207	. . 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 4887	. . . . . 6 ⊢ (𝑔 = (rec(𝐹, 𝐴) ↾ 𝐵) → dom 𝑔 = dom (rec(𝐹, 𝐴) ↾ 𝐵))
8		onss 4549	. . . . . . . . 9 ⊢ (𝐵 ∈ On → 𝐵 ⊆ On)
9	8	3ad2ant3 1023	. . . . . . . 8 ⊢ ((𝐹 Fn V ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ On) → 𝐵 ⊆ On)
10		rdgifnon 6478	. . . . . . . . . 10 ⊢ ((𝐹 Fn V ∧ 𝐴 ∈ 𝑉) → rec(𝐹, 𝐴) Fn On)
11		fndm 5382	. . . . . . . . . 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 1020	. . . . . . . 8 ⊢ ((𝐹 Fn V ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ On) → dom rec(𝐹, 𝐴) = On)
14	9, 13	sseqtrrd 3236	. . . . . . 7 ⊢ ((𝐹 Fn V ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ On) → 𝐵 ⊆ dom rec(𝐹, 𝐴))
15		ssdmres 4990	. . . . . . 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 2261	. . . . 5 ⊢ (((𝐹 Fn V ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ On) ∧ 𝑔 = (rec(𝐹, 𝐴) ↾ 𝐵)) → dom 𝑔 = 𝐵)
18		fveq1 5588	. . . . . . 7 ⊢ (𝑔 = (rec(𝐹, 𝐴) ↾ 𝐵) → (𝑔‘𝑥) = ((rec(𝐹, 𝐴) ↾ 𝐵)‘𝑥))
19	18	fveq2d 5593	. . . . . 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 3957	. . . 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 3331	. . 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 6472	. . . . 5 ⊢ Fun rec(𝐹, 𝐴)
24		resfunexg 5818	. . . . 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 1023	. . 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 2776	. . . . . . . . . 10 ⊢ 𝑥 ∈ V
29		fvexg 5608	. . . . . . . . . 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 2581	. . . . . . . 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 5606	. . . . . . . . . . 11 ⊢ ((Fun 𝐹 ∧ ((rec(𝐹, 𝐴) ↾ 𝐵)‘𝑥) ∈ dom 𝐹) → (𝐹‘((rec(𝐹, 𝐴) ↾ 𝐵)‘𝑥)) ∈ V)
34	33	funfni 5385	. . . . . . . . . 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 2575	. . . . . . . 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 6217	. . . . . 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 1019	. . . 4 ⊢ ((𝐹 Fn V ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ On) → ∪ 𝑥 ∈ 𝐵 (𝐹‘((rec(𝐹, 𝐴) ↾ 𝐵)‘𝑥)) ∈ V)
42		unexg 4498	. . . . . 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 1022	. . . 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 5673	. 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 2239	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 981   = wceq 1373   ∈ wcel 2177  ∀wral 2485  Vcvv 2773   ∪ cun 3168   ⊆ wss 3170  ∪ ciun 3933   ↦ cmpt 4113  Oncon0 4418  dom cdm 4683   ↾ cres 4685  Fun wfun 5274   Fn wfn 5275  ‘cfv 5280  reccrdg 6468

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 615  ax-in2 616  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2179  ax-14 2180  ax-ext 2188  ax-coll 4167  ax-sep 4170  ax-pow 4226  ax-pr 4261  ax-un 4488  ax-setind 4593

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-ne 2378  df-ral 2490  df-rex 2491  df-reu 2492  df-rab 2494  df-v 2775  df-sbc 3003  df-csb 3098  df-dif 3172  df-un 3174  df-in 3176  df-ss 3183  df-nul 3465  df-pw 3623  df-sn 3644  df-pr 3645  df-op 3647  df-uni 3857  df-iun 3935  df-br 4052  df-opab 4114  df-mpt 4115  df-tr 4151  df-id 4348  df-iord 4421  df-on 4423  df-suc 4426  df-xp 4689  df-rel 4690  df-cnv 4691  df-co 4692  df-dm 4693  df-rn 4694  df-res 4695  df-ima 4696  df-iota 5241  df-fun 5282  df-fn 5283  df-f 5284  df-f1 5285  df-fo 5286  df-f1o 5287  df-fv 5288  df-recs 6404  df-irdg 6469

This theorem is referenced by:  rdgival  6481