Theorem rdgprc0 32649

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 6126	. . . 4 ⊢ ∅ ∈ On
2		rdgval 7915	. . . 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 5745	. . . 4 ⊢ (rec(𝐹, 𝐼) ↾ ∅) = ∅
5	4	fveq2i 6548	. . 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 2821	. 2 ⊢ (rec(𝐹, 𝐼)‘∅) = ((𝑔 ∈ V ↦ if(𝑔 = ∅, 𝐼, if(Lim dom 𝑔, ∪ ran 𝑔, (𝐹‘(𝑔‘∪ dom 𝑔)))))‘∅)
7		eqeq1 2801	. . . . . . . 8 ⊢ (𝑔 = ∅ → (𝑔 = ∅ ↔ ∅ = ∅))
8		dmeq 5665	. . . . . . . . . 10 ⊢ (𝑔 = ∅ → dom 𝑔 = dom ∅)
9		limeq 6085	. . . . . . . . . 10 ⊢ (dom 𝑔 = dom ∅ → (Lim dom 𝑔 ↔ Lim dom ∅))
10	8, 9	syl 17	. . . . . . . . 9 ⊢ (𝑔 = ∅ → (Lim dom 𝑔 ↔ Lim dom ∅))
11		rneq 5695	. . . . . . . . . 10 ⊢ (𝑔 = ∅ → ran 𝑔 = ran ∅)
12	11	unieqd 4761	. . . . . . . . 9 ⊢ (𝑔 = ∅ → ∪ ran 𝑔 = ∪ ran ∅)
13		id 22	. . . . . . . . . . 11 ⊢ (𝑔 = ∅ → 𝑔 = ∅)
14	8	unieqd 4761	. . . . . . . . . . 11 ⊢ (𝑔 = ∅ → ∪ dom 𝑔 = ∪ dom ∅)
15	13, 14	fveq12d 6552	. . . . . . . . . 10 ⊢ (𝑔 = ∅ → (𝑔‘∪ dom 𝑔) = (∅‘∪ dom ∅))
16	15	fveq2d 6549	. . . . . . . . 9 ⊢ (𝑔 = ∅ → (𝐹‘(𝑔‘∪ dom 𝑔)) = (𝐹‘(∅‘∪ dom ∅)))
17	10, 12, 16	ifbieq12d 4414	. . . . . . . 8 ⊢ (𝑔 = ∅ → if(Lim dom 𝑔, ∪ ran 𝑔, (𝐹‘(𝑔‘∪ dom 𝑔))) = if(Lim dom ∅, ∪ ran ∅, (𝐹‘(∅‘∪ dom ∅))))
18	7, 17	ifbieq2d 4412	. . . . . . 7 ⊢ (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 2869	. . . . . 6 ⊢ (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 2797	. . . . . . 7 ⊢ (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 5976	. . . . . 6 ⊢ 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 3624	. . . . 5 ⊢ (∅ ∈ 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 2797	. . . . . . . . 9 ⊢ ∅ = ∅
24	23	iftruei 4394	. . . . . . . 8 ⊢ if(∅ = ∅, 𝐼, if(Lim dom ∅, ∪ ran ∅, (𝐹‘(∅‘∪ dom ∅)))) = 𝐼
25	24	eleq1i 2875	. . . . . . 7 ⊢ (if(∅ = ∅, 𝐼, if(Lim dom ∅, ∪ ran ∅, (𝐹‘(∅‘∪ dom ∅)))) ∈ V ↔ 𝐼 ∈ V)
26	25	biimpi 217	. . . . . 6 ⊢ (if(∅ = ∅, 𝐼, if(Lim dom ∅, ∪ ran ∅, (𝐹‘(∅‘∪ dom ∅)))) ∈ V → 𝐼 ∈ V)
27	26	adantl 482	. . . . 5 ⊢ ((∅ ∈ V ∧ if(∅ = ∅, 𝐼, if(Lim dom ∅, ∪ ran ∅, (𝐹‘(∅‘∪ dom ∅)))) ∈ V) → 𝐼 ∈ V)
28	22, 27	sylbi 218	. . . 4 ⊢ (∅ ∈ dom (𝑔 ∈ V ↦ if(𝑔 = ∅, 𝐼, if(Lim dom 𝑔, ∪ ran 𝑔, (𝐹‘(𝑔‘∪ dom 𝑔))))) → 𝐼 ∈ V)
29	28	con3i 157	. . 3 ⊢ (¬ 𝐼 ∈ V → ¬ ∅ ∈ dom (𝑔 ∈ V ↦ if(𝑔 = ∅, 𝐼, if(Lim dom 𝑔, ∪ ran 𝑔, (𝐹‘(𝑔‘∪ dom 𝑔))))))
30		ndmfv 6575	. . 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)))))‘∅) = ∅)
31	29, 30	syl 17	. 2 ⊢ (¬ 𝐼 ∈ V → ((𝑔 ∈ V ↦ if(𝑔 = ∅, 𝐼, if(Lim dom 𝑔, ∪ ran 𝑔, (𝐹‘(𝑔‘∪ dom 𝑔)))))‘∅) = ∅)
32	6, 31	syl5eq 2845	1 ⊢ (¬ 𝐼 ∈ V → (rec(𝐹, 𝐼)‘∅) = ∅)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 207   ∧ wa 396   = wceq 1525   ∈ wcel 2083  Vcvv 3440  ∅c0 4217  ifcif 4387  ∪ cuni 4751   ↦ cmpt 5047  dom cdm 5450  ran crn 5451   ↾ cres 5452  Oncon0 6073  Lim wlim 6074  ‘cfv 6232  reccrdg 7904

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1781  ax-4 1795  ax-5 1892  ax-6 1951  ax-7 1996  ax-8 2085  ax-9 2093  ax-10 2114  ax-11 2128  ax-12 2143  ax-13 2346  ax-ext 2771  ax-rep 5088  ax-sep 5101  ax-nul 5108  ax-pow 5164  ax-pr 5228  ax-un 7326

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3or 1081  df-3an 1082  df-tru 1528  df-ex 1766  df-nf 1770  df-sb 2045  df-mo 2578  df-eu 2614  df-clab 2778  df-cleq 2790  df-clel 2865  df-nfc 2937  df-ne 2987  df-ral 3112  df-rex 3113  df-reu 3114  df-rab 3116  df-v 3442  df-sbc 3712  df-csb 3818  df-dif 3868  df-un 3870  df-in 3872  df-ss 3880  df-pss 3882  df-nul 4218  df-if 4388  df-pw 4461  df-sn 4479  df-pr 4481  df-tp 4483  df-op 4485  df-uni 4752  df-iun 4833  df-br 4969  df-opab 5031  df-mpt 5048  df-tr 5071  df-id 5355  df-eprel 5360  df-po 5369  df-so 5370  df-fr 5409  df-we 5411  df-xp 5456  df-rel 5457  df-cnv 5458  df-co 5459  df-dm 5460  df-rn 5461  df-res 5462  df-ima 5463  df-pred 6030  df-ord 6076  df-on 6077  df-lim 6078  df-suc 6079  df-iota 6196  df-fun 6234  df-fn 6235  df-f 6236  df-f1 6237  df-fo 6238  df-f1o 6239  df-fv 6240  df-wrecs 7805  df-recs 7867  df-rdg 7905

This theorem is referenced by:  rdgprc  32650