Theorem rdgprc0 34368

Description: The value of the recursive definition generator at ∅ when the base value is a proper class. (Contributed by Scott Fenton, 26-Mar-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)

Assertion

Ref	Expression
rdgprc0	⊢ (¬ 𝐼 ∈ V → (rec(𝐹, 𝐼)‘∅) = ∅)

Proof of Theorem rdgprc0

Dummy variable 𝑔 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		0elon 6371	. . . 4 ⊢ ∅ ∈ On
2		rdgval 8366	. . . 4 ⊢ (∅ ∈ On → (rec(𝐹, 𝐼)‘∅) = ((𝑔 ∈ V ↦ if(𝑔 = ∅, 𝐼, if(Lim dom 𝑔, ∪ ran 𝑔, (𝐹‘(𝑔‘∪ dom 𝑔)))))‘(rec(𝐹, 𝐼) ↾ ∅)))
3	1, 2	ax-mp 5	. . 3 ⊢ (rec(𝐹, 𝐼)‘∅) = ((𝑔 ∈ V ↦ if(𝑔 = ∅, 𝐼, if(Lim dom 𝑔, ∪ ran 𝑔, (𝐹‘(𝑔‘∪ dom 𝑔)))))‘(rec(𝐹, 𝐼) ↾ ∅))
4		res0 5941	. . . 4 ⊢ (rec(𝐹, 𝐼) ↾ ∅) = ∅
5	4	fveq2i 6845	. . 3 ⊢ ((𝑔 ∈ V ↦ if(𝑔 = ∅, 𝐼, if(Lim dom 𝑔, ∪ ran 𝑔, (𝐹‘(𝑔‘∪ dom 𝑔)))))‘(rec(𝐹, 𝐼) ↾ ∅)) = ((𝑔 ∈ V ↦ if(𝑔 = ∅, 𝐼, if(Lim dom 𝑔, ∪ ran 𝑔, (𝐹‘(𝑔‘∪ dom 𝑔)))))‘∅)
6	3, 5	eqtri 2764	. 2 ⊢ (rec(𝐹, 𝐼)‘∅) = ((𝑔 ∈ V ↦ if(𝑔 = ∅, 𝐼, if(Lim dom 𝑔, ∪ ran 𝑔, (𝐹‘(𝑔‘∪ dom 𝑔)))))‘∅)
7		eqeq1 2740	. . . . . . 7 ⊢ (𝑔 = ∅ → (𝑔 = ∅ ↔ ∅ = ∅))
8		dmeq 5859	. . . . . . . . 9 ⊢ (𝑔 = ∅ → dom 𝑔 = dom ∅)
9		limeq 6329	. . . . . . . . 9 ⊢ (dom 𝑔 = dom ∅ → (Lim dom 𝑔 ↔ Lim dom ∅))
10	8, 9	syl 17	. . . . . . . 8 ⊢ (𝑔 = ∅ → (Lim dom 𝑔 ↔ Lim dom ∅))
11		rneq 5891	. . . . . . . . 9 ⊢ (𝑔 = ∅ → ran 𝑔 = ran ∅)
12	11	unieqd 4879	. . . . . . . 8 ⊢ (𝑔 = ∅ → ∪ ran 𝑔 = ∪ ran ∅)
13		id 22	. . . . . . . . . 10 ⊢ (𝑔 = ∅ → 𝑔 = ∅)
14	8	unieqd 4879	. . . . . . . . . 10 ⊢ (𝑔 = ∅ → ∪ dom 𝑔 = ∪ dom ∅)
15	13, 14	fveq12d 6849	. . . . . . . . 9 ⊢ (𝑔 = ∅ → (𝑔‘∪ dom 𝑔) = (∅‘∪ dom ∅))
16	15	fveq2d 6846	. . . . . . . 8 ⊢ (𝑔 = ∅ → (𝐹‘(𝑔‘∪ dom 𝑔)) = (𝐹‘(∅‘∪ dom ∅)))
17	10, 12, 16	ifbieq12d 4514	. . . . . . 7 ⊢ (𝑔 = ∅ → if(Lim dom 𝑔, ∪ ran 𝑔, (𝐹‘(𝑔‘∪ dom 𝑔))) = if(Lim dom ∅, ∪ ran ∅, (𝐹‘(∅‘∪ dom ∅))))
18	7, 17	ifbieq2d 4512	. . . . . 6 ⊢ (𝑔 = ∅ → if(𝑔 = ∅, 𝐼, if(Lim dom 𝑔, ∪ ran 𝑔, (𝐹‘(𝑔‘∪ dom 𝑔)))) = if(∅ = ∅, 𝐼, if(Lim dom ∅, ∪ ran ∅, (𝐹‘(∅‘∪ dom ∅)))))
19	18	eleq1d 2822	. . . . 5 ⊢ (𝑔 = ∅ → (if(𝑔 = ∅, 𝐼, if(Lim dom 𝑔, ∪ ran 𝑔, (𝐹‘(𝑔‘∪ dom 𝑔)))) ∈ V ↔ if(∅ = ∅, 𝐼, if(Lim dom ∅, ∪ ran ∅, (𝐹‘(∅‘∪ dom ∅)))) ∈ V))
20		eqid 2736	. . . . . 6 ⊢ (𝑔 ∈ V ↦ if(𝑔 = ∅, 𝐼, if(Lim dom 𝑔, ∪ ran 𝑔, (𝐹‘(𝑔‘∪ dom 𝑔))))) = (𝑔 ∈ V ↦ if(𝑔 = ∅, 𝐼, if(Lim dom 𝑔, ∪ ran 𝑔, (𝐹‘(𝑔‘∪ dom 𝑔)))))
21	20	dmmpt 6192	. . . . 5 ⊢ dom (𝑔 ∈ V ↦ if(𝑔 = ∅, 𝐼, if(Lim dom 𝑔, ∪ ran 𝑔, (𝐹‘(𝑔‘∪ dom 𝑔))))) = {𝑔 ∈ V ∣ if(𝑔 = ∅, 𝐼, if(Lim dom 𝑔, ∪ ran 𝑔, (𝐹‘(𝑔‘∪ dom 𝑔)))) ∈ V}
22	19, 21	elrab2 3648	. . . 4 ⊢ (∅ ∈ dom (𝑔 ∈ V ↦ if(𝑔 = ∅, 𝐼, if(Lim dom 𝑔, ∪ ran 𝑔, (𝐹‘(𝑔‘∪ dom 𝑔))))) ↔ (∅ ∈ V ∧ if(∅ = ∅, 𝐼, if(Lim dom ∅, ∪ ran ∅, (𝐹‘(∅‘∪ dom ∅)))) ∈ V))
23		eqid 2736	. . . . . . 7 ⊢ ∅ = ∅
24	23	iftruei 4493	. . . . . 6 ⊢ if(∅ = ∅, 𝐼, if(Lim dom ∅, ∪ ran ∅, (𝐹‘(∅‘∪ dom ∅)))) = 𝐼
25	24	eleq1i 2828	. . . . 5 ⊢ (if(∅ = ∅, 𝐼, if(Lim dom ∅, ∪ ran ∅, (𝐹‘(∅‘∪ dom ∅)))) ∈ V ↔ 𝐼 ∈ V)
26	25	biimpi 215	. . . 4 ⊢ (if(∅ = ∅, 𝐼, if(Lim dom ∅, ∪ ran ∅, (𝐹‘(∅‘∪ dom ∅)))) ∈ V → 𝐼 ∈ V)
27	22, 26	simplbiim 505	. . 3 ⊢ (∅ ∈ dom (𝑔 ∈ V ↦ if(𝑔 = ∅, 𝐼, if(Lim dom 𝑔, ∪ ran 𝑔, (𝐹‘(𝑔‘∪ dom 𝑔))))) → 𝐼 ∈ V)
28		ndmfv 6877	. . 3 ⊢ (¬ ∅ ∈ dom (𝑔 ∈ V ↦ if(𝑔 = ∅, 𝐼, if(Lim dom 𝑔, ∪ ran 𝑔, (𝐹‘(𝑔‘∪ dom 𝑔))))) → ((𝑔 ∈ V ↦ if(𝑔 = ∅, 𝐼, if(Lim dom 𝑔, ∪ ran 𝑔, (𝐹‘(𝑔‘∪ dom 𝑔)))))‘∅) = ∅)
29	27, 28	nsyl5 159	. 2 ⊢ (¬ 𝐼 ∈ V → ((𝑔 ∈ V ↦ if(𝑔 = ∅, 𝐼, if(Lim dom 𝑔, ∪ ran 𝑔, (𝐹‘(𝑔‘∪ dom 𝑔)))))‘∅) = ∅)
30	6, 29	eqtrid 2788	1 ⊢ (¬ 𝐼 ∈ V → (rec(𝐹, 𝐼)‘∅) = ∅)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   = wceq 1541   ∈ wcel 2106  Vcvv 3445  ∅c0 4282  ifcif 4486  ∪ cuni 4865   ↦ cmpt 5188  dom cdm 5633  ran crn 5634   ↾ cres 5635  Oncon0 6317  Lim wlim 6318  ‘cfv 6496  reccrdg 8355

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pr 5384  ax-un 7672

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-ral 3065  df-rex 3074  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-ov 7360  df-2nd 7922  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356

This theorem is referenced by:  rdgprc  34369