Theorem rdgprc0 35617

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 6430	. . . 4 ⊢ ∅ ∈ On
2		rdgval 8450	. . . 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 5993	. . . 4 ⊢ (rec(𝐹, 𝐼) ↾ ∅) = ∅
5	4	fveq2i 6904	. . 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 2754	. 2 ⊢ (rec(𝐹, 𝐼)‘∅) = ((𝑔 ∈ V ↦ if(𝑔 = ∅, 𝐼, if(Lim dom 𝑔, ∪ ran 𝑔, (𝐹‘(𝑔‘∪ dom 𝑔)))))‘∅)
7		eqeq1 2730	. . . . . . 7 ⊢ (𝑔 = ∅ → (𝑔 = ∅ ↔ ∅ = ∅))
8		dmeq 5910	. . . . . . . . 9 ⊢ (𝑔 = ∅ → dom 𝑔 = dom ∅)
9		limeq 6388	. . . . . . . . 9 ⊢ (dom 𝑔 = dom ∅ → (Lim dom 𝑔 ↔ Lim dom ∅))
10	8, 9	syl 17	. . . . . . . 8 ⊢ (𝑔 = ∅ → (Lim dom 𝑔 ↔ Lim dom ∅))
11		rneq 5942	. . . . . . . . 9 ⊢ (𝑔 = ∅ → ran 𝑔 = ran ∅)
12	11	unieqd 4926	. . . . . . . 8 ⊢ (𝑔 = ∅ → ∪ ran 𝑔 = ∪ ran ∅)
13		id 22	. . . . . . . . . 10 ⊢ (𝑔 = ∅ → 𝑔 = ∅)
14	8	unieqd 4926	. . . . . . . . . 10 ⊢ (𝑔 = ∅ → ∪ dom 𝑔 = ∪ dom ∅)
15	13, 14	fveq12d 6908	. . . . . . . . 9 ⊢ (𝑔 = ∅ → (𝑔‘∪ dom 𝑔) = (∅‘∪ dom ∅))
16	15	fveq2d 6905	. . . . . . . 8 ⊢ (𝑔 = ∅ → (𝐹‘(𝑔‘∪ dom 𝑔)) = (𝐹‘(∅‘∪ dom ∅)))
17	10, 12, 16	ifbieq12d 4561	. . . . . . 7 ⊢ (𝑔 = ∅ → if(Lim dom 𝑔, ∪ ran 𝑔, (𝐹‘(𝑔‘∪ dom 𝑔))) = if(Lim dom ∅, ∪ ran ∅, (𝐹‘(∅‘∪ dom ∅))))
18	7, 17	ifbieq2d 4559	. . . . . 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 2811	. . . . 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 2726	. . . . . 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 6251	. . . . 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 3684	. . . 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 2726	. . . . . . 7 ⊢ ∅ = ∅
24	23	iftruei 4540	. . . . . 6 ⊢ if(∅ = ∅, 𝐼, if(Lim dom ∅, ∪ ran ∅, (𝐹‘(∅‘∪ dom ∅)))) = 𝐼
25	24	eleq1i 2817	. . . . 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 503	. . 3 ⊢ (∅ ∈ dom (𝑔 ∈ V ↦ if(𝑔 = ∅, 𝐼, if(Lim dom 𝑔, ∪ ran 𝑔, (𝐹‘(𝑔‘∪ dom 𝑔))))) → 𝐼 ∈ V)
28		ndmfv 6936	. . 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 2778	1 ⊢ (¬ 𝐼 ∈ V → (rec(𝐹, 𝐼)‘∅) = ∅)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   = wceq 1534   ∈ wcel 2099  Vcvv 3462  ∅c0 4325  ifcif 4533  ∪ cuni 4913   ↦ cmpt 5236  dom cdm 5682  ran crn 5683   ↾ cres 5684  Oncon0 6376  Lim wlim 6377  ‘cfv 6554  reccrdg 8439

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2697  ax-rep 5290  ax-sep 5304  ax-nul 5311  ax-pr 5433  ax-un 7746

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2704  df-cleq 2718  df-clel 2803  df-nfc 2878  df-ne 2931  df-ral 3052  df-rex 3061  df-reu 3365  df-rab 3420  df-v 3464  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3967  df-nul 4326  df-if 4534  df-pw 4609  df-sn 4634  df-pr 4636  df-op 4640  df-uni 4914  df-iun 5003  df-br 5154  df-opab 5216  df-mpt 5237  df-tr 5271  df-id 5580  df-eprel 5586  df-po 5594  df-so 5595  df-fr 5637  df-we 5639  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-pred 6312  df-ord 6379  df-on 6380  df-lim 6381  df-suc 6382  df-iota 6506  df-fun 6556  df-fn 6557  df-f 6558  df-f1 6559  df-fo 6560  df-f1o 6561  df-fv 6562  df-ov 7427  df-2nd 8004  df-frecs 8296  df-wrecs 8327  df-recs 8401  df-rdg 8440

This theorem is referenced by:  rdgprc  35618