Theorem zfrep6 3600

Description: A version of the Axiom of Replacement. Normally φ would have free variables x and y. Axiom 6 of [Kunen] p. 12. The Separation Scheme ax-sep 2693 cannot be derived from this version and must be stated as a separate axiom in an axiom system (such as Kunen's) that uses this version in place of our ax-rep 2683.

Assertion

Ref	Expression
zfrep6	⊢ (∀x ∈ z ∃!yφ → ∃w∀x ∈ z ∃y ∈ w φ)

Distinct variable groups:   φ,w   x,y,z,w

Proof of Theorem zfrep6

Step	Hyp	Ref	Expression
1		ax-17 968	. . 3 ⊢ (v ∈ ran {⟨x, y⟩∣(x ∈ z ⋀ φ)} → ∀w v ∈ ran {⟨x, y⟩∣(x ∈ z ⋀ φ)})
2		ax-17 968	. . 3 ⊢ (∀x ∈ z ∃y ∈ ran {⟨x, y⟩∣(x ∈ z ⋀ φ)}φ → ∀w∀x ∈ z ∃y ∈ ran {⟨x, y⟩∣(x ∈ z ⋀ φ)}φ)
3		hbopab1 2802	. . . . . 6 ⊢ (w ∈ {⟨x, y⟩∣(x ∈ z ⋀ φ)} → ∀x w ∈ {⟨x, y⟩∣(x ∈ z ⋀ φ)})
4	3	hbrn 3337	. . . . 5 ⊢ (w ∈ ran {⟨x, y⟩∣(x ∈ z ⋀ φ)} → ∀x w ∈ ran {⟨x, y⟩∣(x ∈ z ⋀ φ)})
5	4	hbeleq 1559	. . . 4 ⊢ (w = ran {⟨x, y⟩∣(x ∈ z ⋀ φ)} → ∀x w = ran {⟨x, y⟩∣(x ∈ z ⋀ φ)})
6		ax-17 968	. . . . 5 ⊢ (v ∈ w → ∀y v ∈ w)
7		hbopab2 2803	. . . . . 6 ⊢ (v ∈ {⟨x, y⟩∣(x ∈ z ⋀ φ)} → ∀y v ∈ {⟨x, y⟩∣(x ∈ z ⋀ φ)})
8	7	hbrn 3337	. . . . 5 ⊢ (v ∈ ran {⟨x, y⟩∣(x ∈ z ⋀ φ)} → ∀y v ∈ ran {⟨x, y⟩∣(x ∈ z ⋀ φ)})
9	6, 8	rexeq1f 1776	. . . 4 ⊢ (w = ran {⟨x, y⟩∣(x ∈ z ⋀ φ)} → (∃y ∈ w φ ↔ ∃y ∈ ran {⟨x, y⟩∣(x ∈ z ⋀ φ)}φ))
10	5, 9	ralbid 1653	. . 3 ⊢ (w = ran {⟨x, y⟩∣(x ∈ z ⋀ φ)} → (∀x ∈ z ∃y ∈ w φ ↔ ∀x ∈ z ∃y ∈ ran {⟨x, y⟩∣(x ∈ z ⋀ φ)}φ))
11	1, 2, 10	cla4egf 1852	. 2 ⊢ (ran {⟨x, y⟩∣(x ∈ z ⋀ φ)} ∈ V → (∀x ∈ z ∃y ∈ ran {⟨x, y⟩∣(x ∈ z ⋀ φ)}φ → ∃w∀x ∈ z ∃y ∈ w φ))
12		funrnex 3599	. . 3 ⊢ (dom {⟨x, y⟩∣(x ∈ z ⋀ φ)} ∈ V → (Fun {⟨x, y⟩∣(x ∈ z ⋀ φ)} → ran {⟨x, y⟩∣(x ∈ z ⋀ φ)} ∈ V))
13		euex 1387	. . . . . . 7 ⊢ (∃!yφ → ∃yφ)
14	13	r19.20si 1698	. . . . . 6 ⊢ (∀x ∈ z ∃!yφ → ∀x ∈ z ∃yφ)
15		rabid2 1762	. . . . . 6 ⊢ (z = {x ∈ z∣∃yφ} ↔ ∀x ∈ z ∃yφ)
16	14, 15	sylibr 200	. . . . 5 ⊢ (∀x ∈ z ∃!yφ → z = {x ∈ z∣∃yφ})
17		19.42v 1303	. . . . . . 7 ⊢ (∃y(x ∈ z ⋀ φ) ↔ (x ∈ z ⋀ ∃yφ))
18	17	abbii 1567	. . . . . 6 ⊢ {x∣∃y(x ∈ z ⋀ φ)} = {x∣(x ∈ z ⋀ ∃yφ)}
19		dmopab 3309	. . . . . 6 ⊢ dom {⟨x, y⟩∣(x ∈ z ⋀ φ)} = {x∣∃y(x ∈ z ⋀ φ)}
20		df-rab 1644	. . . . . 6 ⊢ {x ∈ z∣∃yφ} = {x∣(x ∈ z ⋀ ∃yφ)}
21	18, 19, 20	3eqtr4 1497	. . . . 5 ⊢ dom {⟨x, y⟩∣(x ∈ z ⋀ φ)} = {x ∈ z∣∃yφ}
22	16, 21	syl6reqr 1518	. . . 4 ⊢ (∀x ∈ z ∃!yφ → dom {⟨x, y⟩∣(x ∈ z ⋀ φ)} = z)
23		visset 1804	. . . 4 ⊢ z ∈ V
24	22, 23	syl6eqel 1548	. . 3 ⊢ (∀x ∈ z ∃!yφ → dom {⟨x, y⟩∣(x ∈ z ⋀ φ)} ∈ V)
25		eumo 1404	. . . . . . 7 ⊢ (∃!yφ → ∃*yφ)
26	25	imim2i 17	. . . . . 6 ⊢ ((x ∈ z → ∃!yφ) → (x ∈ z → ∃*yφ))
27		moanimv 1422	. . . . . 6 ⊢ (∃*y(x ∈ z ⋀ φ) ↔ (x ∈ z → ∃*yφ))
28	26, 27	sylibr 200	. . . . 5 ⊢ ((x ∈ z → ∃!yφ) → ∃*y(x ∈ z ⋀ φ))
29	28	19.20i 989	. . . 4 ⊢ (∀x(x ∈ z → ∃!yφ) → ∀x∃*y(x ∈ z ⋀ φ))
30		df-ral 1641	. . . 4 ⊢ (∀x ∈ z ∃!yφ ↔ ∀x(x ∈ z → ∃!yφ))
31		funopab 3534	. . . 4 ⊢ (Fun {⟨x, y⟩∣(x ∈ z ⋀ φ)} ↔ ∀x∃*y(x ∈ z ⋀ φ))
32	29, 30, 31	3imtr4 219	. . 3 ⊢ (∀x ∈ z ∃!yφ → Fun {⟨x, y⟩∣(x ∈ z ⋀ φ)})
33	12, 24, 32	sylc 68	. 2 ⊢ (∀x ∈ z ∃!yφ → ran {⟨x, y⟩∣(x ∈ z ⋀ φ)} ∈ V)
34		hbra1 1679	. . 3 ⊢ (∀x ∈ z ∃!yφ → ∀x∀x ∈ z ∃!yφ)
35	22	eleq2d 1533	. . . 4 ⊢ (∀x ∈ z ∃!yφ → (x ∈ dom {⟨x, y⟩∣(x ∈ z ⋀ φ)} ↔ x ∈ z))
36		opabid 2799	. . . . . . . . 9 ⊢ (⟨x, y⟩ ∈ {⟨x, y⟩∣(x ∈ z ⋀ φ)} ↔ (x ∈ z ⋀ φ))
37		visset 1804	. . . . . . . . . 10 ⊢ x ∈ V
38		visset 1804	. . . . . . . . . 10 ⊢ y ∈ V
39	37, 38	opelrn 3333	. . . . . . . . 9 ⊢ (⟨x, y⟩ ∈ {⟨x, y⟩∣(x ∈ z ⋀ φ)} → y ∈ ran {⟨x, y⟩∣(x ∈ z ⋀ φ)})
40	36, 39	sylbir 201	. . . . . . . 8 ⊢ ((x ∈ z ⋀ φ) → y ∈ ran {⟨x, y⟩∣(x ∈ z ⋀ φ)})
41	40	ex 373	. . . . . . 7 ⊢ (x ∈ z → (φ → y ∈ ran {⟨x, y⟩∣(x ∈ z ⋀ φ)}))
42	41	impac 387	. . . . . 6 ⊢ ((x ∈ z ⋀ φ) → (y ∈ ran {⟨x, y⟩∣(x ∈ z ⋀ φ)} ⋀ φ))
43	42	19.22i 1036	. . . . 5 ⊢ (∃y(x ∈ z ⋀ φ) → ∃y(y ∈ ran {⟨x, y⟩∣(x ∈ z ⋀ φ)} ⋀ φ))
44	19	abeq2i 1562	. . . . 5 ⊢ (x ∈ dom {⟨x, y⟩∣(x ∈ z ⋀ φ)} ↔ ∃y(x ∈ z ⋀ φ))
45		df-rex 1642	. . . . 5 ⊢ (∃y ∈ ran {⟨x, y⟩∣(x ∈ z ⋀ φ)}φ ↔ ∃y(y ∈ ran {⟨x, y⟩∣(x ∈ z ⋀ φ)} ⋀ φ))
46	43, 44, 45	3imtr4 219	. . . 4 ⊢ (x ∈ dom {⟨x, y⟩∣(x ∈ z ⋀ φ)} → ∃y ∈ ran {⟨x, y⟩∣(x ∈ z ⋀ φ)}φ)
47	35, 46	syl6bir 215	. . 3 ⊢ (∀x ∈ z ∃!yφ → (x ∈ z → ∃y ∈ ran {⟨x, y⟩∣(x ∈ z ⋀ φ)}φ))
48	34, 47	r19.21ai 1704	. 2 ⊢ (∀x ∈ z ∃!yφ → ∀x ∈ z ∃y ∈ ran {⟨x, y⟩∣(x ∈ z ⋀ φ)}φ)
49	11, 33, 48	sylc 68	1 ⊢ (∀x ∈ z ∃!yφ → ∃w∀x ∈ z ∃y ∈ w φ)

Syntax hints:   → wi 3   ⋀ wa 223  ∀wal 951   = wceq 953   ∈ wcel 955  ∃wex 977  ∃!weu 1373  ∃*wmo 1374  {cab 1456  ∀wral 1637  ∃wrex 1638  {crab 1640  Vcvv 1802  ⟨cop 2401  {copab 2656  dom cdm 3160  ran crn 3161  Fun wfun 3166

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-9 962  ax-10 963  ax-11 964  ax-12 965  ax-13 966  ax-14 967  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-10o 1136  ax-16 1206  ax-11o 1213  ax-ext 1452  ax-rep 2683  ax-sep 2693  ax-pow 2732  ax-pr 2769  ax-un 2857

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 978  df-sb 1168  df-eu 1375  df-mo 1376  df-clab 1457  df-cleq 1462  df-clel 1465  df-ne 1579  df-ral 1641  df-rex 1642  df-rab 1644  df-v 1803  df-dif 2039  df-un 2040  df-in 2041  df-ss 2043  df-nul 2271  df-pw 2392  df-sn 2402  df-pr 2403  df-op 2406  df-uni 2494  df-br 2610  df-opab 2657  df-id 2824  df-xp 3174  df-rel 3175  df-cnv 3176  df-co 3177  df-dm 3178  df-rn 3179  df-res 3180  df-ima 3181  df-fun 3182  df-fn 3183

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