Theorem phialllem1 4617

Description: Lemma for phiall 4619. Any set of numbers without zero is the Phi of a set. (Contributed by Scott Fenton, 14-Apr-2021.)

Hypothesis

Ref	Expression
phiall.1	⊢ A ∈ V

Assertion

Ref	Expression
phialllem1	⊢ ((A ⊆ Nn ∧ ¬ 0c ∈ A) → ∃x A = Phi x)

Distinct variable group:   x,A

Proof of Theorem phialllem1

Dummy variables y z w are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eleq1 2413	. . . . . . . . . . . . 13 ⊢ (z = 0c → (z ∈ A ↔ 0c ∈ A))
2	1	biimpcd 215	. . . . . . . . . . . 12 ⊢ (z ∈ A → (z = 0c → 0c ∈ A))
3	2	con3d 125	. . . . . . . . . . 11 ⊢ (z ∈ A → (¬ 0c ∈ A → ¬ z = 0c))
4	3	impcom 419	. . . . . . . . . 10 ⊢ ((¬ 0c ∈ A ∧ z ∈ A) → ¬ z = 0c)
5	4	adantll 694	. . . . . . . . 9 ⊢ (((A ⊆ Nn ∧ ¬ 0c ∈ A) ∧ z ∈ A) → ¬ z = 0c)
6		ssel2 3269	. . . . . . . . . . 11 ⊢ ((A ⊆ Nn ∧ z ∈ A) → z ∈ Nn )
7	6	adantlr 695	. . . . . . . . . 10 ⊢ (((A ⊆ Nn ∧ ¬ 0c ∈ A) ∧ z ∈ A) → z ∈ Nn )
8		nnc0suc 4413	. . . . . . . . . 10 ⊢ (z ∈ Nn ↔ (z = 0c ∨ ∃x ∈ Nn z = (x +c 1c)))
9	7, 8	sylib 188	. . . . . . . . 9 ⊢ (((A ⊆ Nn ∧ ¬ 0c ∈ A) ∧ z ∈ A) → (z = 0c ∨ ∃x ∈ Nn z = (x +c 1c)))
10		orel1 371	. . . . . . . . 9 ⊢ (¬ z = 0c → ((z = 0c ∨ ∃x ∈ Nn z = (x +c 1c)) → ∃x ∈ Nn z = (x +c 1c)))
11	5, 9, 10	sylc 56	. . . . . . . 8 ⊢ (((A ⊆ Nn ∧ ¬ 0c ∈ A) ∧ z ∈ A) → ∃x ∈ Nn z = (x +c 1c))
12		anidm 625	. . . . . . . . . 10 ⊢ ((z = (x +c 1c) ∧ z = (x +c 1c)) ↔ z = (x +c 1c))
13		eqeq1 2359	. . . . . . . . . . 11 ⊢ (z = w → (z = (x +c 1c) ↔ w = (x +c 1c)))
14	13	anbi2d 684	. . . . . . . . . 10 ⊢ (z = w → ((z = (x +c 1c) ∧ z = (x +c 1c)) ↔ (z = (x +c 1c) ∧ w = (x +c 1c))))
15	12, 14	syl5bbr 250	. . . . . . . . 9 ⊢ (z = w → (z = (x +c 1c) ↔ (z = (x +c 1c) ∧ w = (x +c 1c))))
16	15	rexbidv 2636	. . . . . . . 8 ⊢ (z = w → (∃x ∈ Nn z = (x +c 1c) ↔ ∃x ∈ Nn (z = (x +c 1c) ∧ w = (x +c 1c))))
17	11, 16	syl5ibcom 211	. . . . . . 7 ⊢ (((A ⊆ Nn ∧ ¬ 0c ∈ A) ∧ z ∈ A) → (z = w → ∃x ∈ Nn (z = (x +c 1c) ∧ w = (x +c 1c))))
18		eqtr3 2372	. . . . . . . 8 ⊢ ((z = (x +c 1c) ∧ w = (x +c 1c)) → z = w)
19	18	rexlimivw 2735	. . . . . . 7 ⊢ (∃x ∈ Nn (z = (x +c 1c) ∧ w = (x +c 1c)) → z = w)
20	17, 19	impbid1 194	. . . . . 6 ⊢ (((A ⊆ Nn ∧ ¬ 0c ∈ A) ∧ z ∈ A) → (z = w ↔ ∃x ∈ Nn (z = (x +c 1c) ∧ w = (x +c 1c))))
21	20	rexbidva 2632	. . . . 5 ⊢ ((A ⊆ Nn ∧ ¬ 0c ∈ A) → (∃z ∈ A z = w ↔ ∃z ∈ A ∃x ∈ Nn (z = (x +c 1c) ∧ w = (x +c 1c))))
22		risset 2662	. . . . 5 ⊢ (w ∈ A ↔ ∃z ∈ A z = w)
23		rexcom 2773	. . . . 5 ⊢ (∃x ∈ Nn ∃z ∈ A (z = (x +c 1c) ∧ w = (x +c 1c)) ↔ ∃z ∈ A ∃x ∈ Nn (z = (x +c 1c) ∧ w = (x +c 1c)))
24	21, 22, 23	3bitr4g 279	. . . 4 ⊢ ((A ⊆ Nn ∧ ¬ 0c ∈ A) → (w ∈ A ↔ ∃x ∈ Nn ∃z ∈ A (z = (x +c 1c) ∧ w = (x +c 1c))))
25	24	eqabdv 2469	. . 3 ⊢ ((A ⊆ Nn ∧ ¬ 0c ∈ A) → A = {w ∣ ∃x ∈ Nn ∃z ∈ A (z = (x +c 1c) ∧ w = (x +c 1c))})
26		df-phi 4566	. . . 4 ⊢ Phi {y ∈ Nn ∣ ∃z ∈ A z = (y +c 1c)} = {w ∣ ∃x ∈ {y ∈ Nn ∣ ∃z ∈ A z = (y +c 1c)}w = if(x ∈ Nn , (x +c 1c), x)}
27		addceq1 4384	. . . . . . . . 9 ⊢ (y = x → (y +c 1c) = (x +c 1c))
28	27	eqeq2d 2364	. . . . . . . 8 ⊢ (y = x → (z = (y +c 1c) ↔ z = (x +c 1c)))
29	28	rexbidv 2636	. . . . . . 7 ⊢ (y = x → (∃z ∈ A z = (y +c 1c) ↔ ∃z ∈ A z = (x +c 1c)))
30	29	rexrab 3001	. . . . . 6 ⊢ (∃x ∈ {y ∈ Nn ∣ ∃z ∈ A z = (y +c 1c)}w = if(x ∈ Nn , (x +c 1c), x) ↔ ∃x ∈ Nn (∃z ∈ A z = (x +c 1c) ∧ w = if(x ∈ Nn , (x +c 1c), x)))
31		iftrue 3669	. . . . . . . . . 10 ⊢ (x ∈ Nn → if(x ∈ Nn , (x +c 1c), x) = (x +c 1c))
32	31	eqeq2d 2364	. . . . . . . . 9 ⊢ (x ∈ Nn → (w = if(x ∈ Nn , (x +c 1c), x) ↔ w = (x +c 1c)))
33	32	anbi2d 684	. . . . . . . 8 ⊢ (x ∈ Nn → ((∃z ∈ A z = (x +c 1c) ∧ w = if(x ∈ Nn , (x +c 1c), x)) ↔ (∃z ∈ A z = (x +c 1c) ∧ w = (x +c 1c))))
34		r19.41v 2765	. . . . . . . 8 ⊢ (∃z ∈ A (z = (x +c 1c) ∧ w = (x +c 1c)) ↔ (∃z ∈ A z = (x +c 1c) ∧ w = (x +c 1c)))
35	33, 34	syl6bbr 254	. . . . . . 7 ⊢ (x ∈ Nn → ((∃z ∈ A z = (x +c 1c) ∧ w = if(x ∈ Nn , (x +c 1c), x)) ↔ ∃z ∈ A (z = (x +c 1c) ∧ w = (x +c 1c))))
36	35	rexbiia 2648	. . . . . 6 ⊢ (∃x ∈ Nn (∃z ∈ A z = (x +c 1c) ∧ w = if(x ∈ Nn , (x +c 1c), x)) ↔ ∃x ∈ Nn ∃z ∈ A (z = (x +c 1c) ∧ w = (x +c 1c)))
37	30, 36	bitri 240	. . . . 5 ⊢ (∃x ∈ {y ∈ Nn ∣ ∃z ∈ A z = (y +c 1c)}w = if(x ∈ Nn , (x +c 1c), x) ↔ ∃x ∈ Nn ∃z ∈ A (z = (x +c 1c) ∧ w = (x +c 1c)))
38	37	abbii 2466	. . . 4 ⊢ {w ∣ ∃x ∈ {y ∈ Nn ∣ ∃z ∈ A z = (y +c 1c)}w = if(x ∈ Nn , (x +c 1c), x)} = {w ∣ ∃x ∈ Nn ∃z ∈ A (z = (x +c 1c) ∧ w = (x +c 1c))}
39	26, 38	eqtri 2373	. . 3 ⊢ Phi {y ∈ Nn ∣ ∃z ∈ A z = (y +c 1c)} = {w ∣ ∃x ∈ Nn ∃z ∈ A (z = (x +c 1c) ∧ w = (x +c 1c))}
40	25, 39	syl6eqr 2403	. 2 ⊢ ((A ⊆ Nn ∧ ¬ 0c ∈ A) → A = Phi {y ∈ Nn ∣ ∃z ∈ A z = (y +c 1c)})
41		dfrab2 3531	. . . 4 ⊢ {y ∈ Nn ∣ ∃z ∈ A z = (y +c 1c)} = ({y ∣ ∃z ∈ A z = (y +c 1c)} ∩ Nn )
42		vex 2863	. . . . . . . . 9 ⊢ y ∈ V
43	42	elimak 4260	. . . . . . . 8 ⊢ (y ∈ (◡kImagek(( Ins3k ∼ (( Ins3k Sk ∩ Ins2k Sk ) “k 𝒫1𝒫11c) ∖ (( Ins2k Ins2k Sk ⊕ ( Ins2k Ins3k Sk ∪ Ins3k SIk SIk Sk )) “k 𝒫1𝒫1𝒫1𝒫11c)) “k 𝒫1𝒫11c) “k A) ↔ ∃z ∈ A ⟪z, y⟫ ∈ ◡kImagek(( Ins3k ∼ (( Ins3k Sk ∩ Ins2k Sk ) “k 𝒫1𝒫11c) ∖ (( Ins2k Ins2k Sk ⊕ ( Ins2k Ins3k Sk ∪ Ins3k SIk SIk Sk )) “k 𝒫1𝒫1𝒫1𝒫11c)) “k 𝒫1𝒫11c))
44		vex 2863	. . . . . . . . . . 11 ⊢ z ∈ V
45	42, 44	opkelimagek 4273	. . . . . . . . . 10 ⊢ (⟪y, z⟫ ∈ Imagek(( Ins3k ∼ (( Ins3k Sk ∩ Ins2k Sk ) “k 𝒫1𝒫11c) ∖ (( Ins2k Ins2k Sk ⊕ ( Ins2k Ins3k Sk ∪ Ins3k SIk SIk Sk )) “k 𝒫1𝒫1𝒫1𝒫11c)) “k 𝒫1𝒫11c) ↔ z = ((( Ins3k ∼ (( Ins3k Sk ∩ Ins2k Sk ) “k 𝒫1𝒫11c) ∖ (( Ins2k Ins2k Sk ⊕ ( Ins2k Ins3k Sk ∪ Ins3k SIk SIk Sk )) “k 𝒫1𝒫1𝒫1𝒫11c)) “k 𝒫1𝒫11c) “k y))
46	44, 42	opkelcnvk 4251	. . . . . . . . . 10 ⊢ (⟪z, y⟫ ∈ ◡kImagek(( Ins3k ∼ (( Ins3k Sk ∩ Ins2k Sk ) “k 𝒫1𝒫11c) ∖ (( Ins2k Ins2k Sk ⊕ ( Ins2k Ins3k Sk ∪ Ins3k SIk SIk Sk )) “k 𝒫1𝒫1𝒫1𝒫11c)) “k 𝒫1𝒫11c) ↔ ⟪y, z⟫ ∈ Imagek(( Ins3k ∼ (( Ins3k Sk ∩ Ins2k Sk ) “k 𝒫1𝒫11c) ∖ (( Ins2k Ins2k Sk ⊕ ( Ins2k Ins3k Sk ∪ Ins3k SIk SIk Sk )) “k 𝒫1𝒫1𝒫1𝒫11c)) “k 𝒫1𝒫11c))
47		dfaddc2 4382	. . . . . . . . . . 11 ⊢ (y +c 1c) = ((( Ins3k ∼ (( Ins3k Sk ∩ Ins2k Sk ) “k 𝒫1𝒫11c) ∖ (( Ins2k Ins2k Sk ⊕ ( Ins2k Ins3k Sk ∪ Ins3k SIk SIk Sk )) “k 𝒫1𝒫1𝒫1𝒫11c)) “k 𝒫1𝒫11c) “k y)
48	47	eqeq2i 2363	. . . . . . . . . 10 ⊢ (z = (y +c 1c) ↔ z = ((( Ins3k ∼ (( Ins3k Sk ∩ Ins2k Sk ) “k 𝒫1𝒫11c) ∖ (( Ins2k Ins2k Sk ⊕ ( Ins2k Ins3k Sk ∪ Ins3k SIk SIk Sk )) “k 𝒫1𝒫1𝒫1𝒫11c)) “k 𝒫1𝒫11c) “k y))
49	45, 46, 48	3bitr4i 268	. . . . . . . . 9 ⊢ (⟪z, y⟫ ∈ ◡kImagek(( Ins3k ∼ (( Ins3k Sk ∩ Ins2k Sk ) “k 𝒫1𝒫11c) ∖ (( Ins2k Ins2k Sk ⊕ ( Ins2k Ins3k Sk ∪ Ins3k SIk SIk Sk )) “k 𝒫1𝒫1𝒫1𝒫11c)) “k 𝒫1𝒫11c) ↔ z = (y +c 1c))
50	49	rexbii 2640	. . . . . . . 8 ⊢ (∃z ∈ A ⟪z, y⟫ ∈ ◡kImagek(( Ins3k ∼ (( Ins3k Sk ∩ Ins2k Sk ) “k 𝒫1𝒫11c) ∖ (( Ins2k Ins2k Sk ⊕ ( Ins2k Ins3k Sk ∪ Ins3k SIk SIk Sk )) “k 𝒫1𝒫1𝒫1𝒫11c)) “k 𝒫1𝒫11c) ↔ ∃z ∈ A z = (y +c 1c))
51	43, 50	bitri 240	. . . . . . 7 ⊢ (y ∈ (◡kImagek(( Ins3k ∼ (( Ins3k Sk ∩ Ins2k Sk ) “k 𝒫1𝒫11c) ∖ (( Ins2k Ins2k Sk ⊕ ( Ins2k Ins3k Sk ∪ Ins3k SIk SIk Sk )) “k 𝒫1𝒫1𝒫1𝒫11c)) “k 𝒫1𝒫11c) “k A) ↔ ∃z ∈ A z = (y +c 1c))
52	51	eqabi 2465	. . . . . 6 ⊢ (◡kImagek(( Ins3k ∼ (( Ins3k Sk ∩ Ins2k Sk ) “k 𝒫1𝒫11c) ∖ (( Ins2k Ins2k Sk ⊕ ( Ins2k Ins3k Sk ∪ Ins3k SIk SIk Sk )) “k 𝒫1𝒫1𝒫1𝒫11c)) “k 𝒫1𝒫11c) “k A) = {y ∣ ∃z ∈ A z = (y +c 1c)}
53		addcexlem 4383	. . . . . . . . . 10 ⊢ ( Ins3k ∼ (( Ins3k Sk ∩ Ins2k Sk ) “k 𝒫1𝒫11c) ∖ (( Ins2k Ins2k Sk ⊕ ( Ins2k Ins3k Sk ∪ Ins3k SIk SIk Sk )) “k 𝒫1𝒫1𝒫1𝒫11c)) ∈ V
54		1cex 4143	. . . . . . . . . . . 12 ⊢ 1c ∈ V
55	54	pw1ex 4304	. . . . . . . . . . 11 ⊢ 𝒫11c ∈ V
56	55	pw1ex 4304	. . . . . . . . . 10 ⊢ 𝒫1𝒫11c ∈ V
57	53, 56	imakex 4301	. . . . . . . . 9 ⊢ (( Ins3k ∼ (( Ins3k Sk ∩ Ins2k Sk ) “k 𝒫1𝒫11c) ∖ (( Ins2k Ins2k Sk ⊕ ( Ins2k Ins3k Sk ∪ Ins3k SIk SIk Sk )) “k 𝒫1𝒫1𝒫1𝒫11c)) “k 𝒫1𝒫11c) ∈ V
58	57	imagekex 4313	. . . . . . . 8 ⊢ Imagek(( Ins3k ∼ (( Ins3k Sk ∩ Ins2k Sk ) “k 𝒫1𝒫11c) ∖ (( Ins2k Ins2k Sk ⊕ ( Ins2k Ins3k Sk ∪ Ins3k SIk SIk Sk )) “k 𝒫1𝒫1𝒫1𝒫11c)) “k 𝒫1𝒫11c) ∈ V
59	58	cnvkex 4288	. . . . . . 7 ⊢ ◡kImagek(( Ins3k ∼ (( Ins3k Sk ∩ Ins2k Sk ) “k 𝒫1𝒫11c) ∖ (( Ins2k Ins2k Sk ⊕ ( Ins2k Ins3k Sk ∪ Ins3k SIk SIk Sk )) “k 𝒫1𝒫1𝒫1𝒫11c)) “k 𝒫1𝒫11c) ∈ V
60		phiall.1	. . . . . . 7 ⊢ A ∈ V
61	59, 60	imakex 4301	. . . . . 6 ⊢ (◡kImagek(( Ins3k ∼ (( Ins3k Sk ∩ Ins2k Sk ) “k 𝒫1𝒫11c) ∖ (( Ins2k Ins2k Sk ⊕ ( Ins2k Ins3k Sk ∪ Ins3k SIk SIk Sk )) “k 𝒫1𝒫1𝒫1𝒫11c)) “k 𝒫1𝒫11c) “k A) ∈ V
62	52, 61	eqeltrri 2424	. . . . 5 ⊢ {y ∣ ∃z ∈ A z = (y +c 1c)} ∈ V
63		nncex 4397	. . . . 5 ⊢ Nn ∈ V
64	62, 63	inex 4106	. . . 4 ⊢ ({y ∣ ∃z ∈ A z = (y +c 1c)} ∩ Nn ) ∈ V
65	41, 64	eqeltri 2423	. . 3 ⊢ {y ∈ Nn ∣ ∃z ∈ A z = (y +c 1c)} ∈ V
66		phieq 4571	. . . 4 ⊢ (x = {y ∈ Nn ∣ ∃z ∈ A z = (y +c 1c)} → Phi x = Phi {y ∈ Nn ∣ ∃z ∈ A z = (y +c 1c)})
67	66	eqeq2d 2364	. . 3 ⊢ (x = {y ∈ Nn ∣ ∃z ∈ A z = (y +c 1c)} → (A = Phi x ↔ A = Phi {y ∈ Nn ∣ ∃z ∈ A z = (y +c 1c)}))
68	65, 67	spcev 2947	. 2 ⊢ (A = Phi {y ∈ Nn ∣ ∃z ∈ A z = (y +c 1c)} → ∃x A = Phi x)
69	40, 68	syl 15	1 ⊢ ((A ⊆ Nn ∧ ¬ 0c ∈ A) → ∃x A = Phi x)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∨ wo 357   ∧ wa 358  ∃wex 1541   = wceq 1642   ∈ wcel 1710  {cab 2339  ∃wrex 2616  {crab 2619  Vcvv 2860   ∼ ccompl 3206   ∖ cdif 3207   ∪ cun 3208   ∩ cin 3209   ⊕ csymdif 3210   ⊆ wss 3258   ifcif 3663  ⟪copk 4058  1_cc1c 4135  𝒫₁cpw1 4136  ^◡_kccnvk 4176   Ins2_k cins2k 4177   Ins3_k cins3k 4178   “_k cimak 4180   SI_k csik 4182  Image_kcimagek 4183   S_k cssetk 4184   Nn cnnc 4374  0_cc0c 4375   +_c cplc 4376   Phi cphi 4563

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334  ax-nin 4079  ax-xp 4080  ax-cnv 4081  ax-1c 4082  ax-sset 4083  ax-si 4084  ax-ins2 4085  ax-ins3 4086  ax-typlower 4087  ax-sn 4088

This proof depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-ne 2519  df-ral 2620  df-rex 2621  df-rab 2624  df-v 2862  df-sbc 3048  df-nin 3212  df-compl 3213  df-in 3214  df-un 3215  df-dif 3216  df-symdif 3217  df-ss 3260  df-nul 3552  df-if 3664  df-pw 3725  df-sn 3742  df-pr 3743  df-uni 3893  df-int 3928  df-opk 4059  df-1c 4137  df-pw1 4138  df-uni1 4139  df-xpk 4186  df-cnvk 4187  df-ins2k 4188  df-ins3k 4189  df-imak 4190  df-cok 4191  df-p6 4192  df-sik 4193  df-ssetk 4194  df-imagek 4195  df-idk 4196  df-0c 4378  df-addc 4379  df-nnc 4380  df-phi 4566

This theorem is used by:  phialllem2  4618