Theorem zfrep6 2744

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 zfaus 1480 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 1075.

Assertion

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

Distinct variable group(s):   φ,z,w   x,y,z,w

Proof of Theorem zfrep6

Step	Hyp	Ref	Expression
1		ax-17 925	. . 3 ⊢ (v ∈ ran {⟨x, y⟩∣(x ∈ z ∧ φ)} → ∀w v ∈ ran {⟨x, y⟩∣(x ∈ z ∧ φ)})
2		ax-17 925	. . 3 ⊢ (∀x ∈ z ∃y ∈ ran {⟨x, y⟩∣(x ∈ z ∧ φ)}φ → ∀w∀x ∈ z ∃y ∈ ran {⟨x, y⟩∣(x ∈ z ∧ φ)}φ)
3		hbopab1 2112	. . . . . 6 ⊢ (w ∈ {⟨x, y⟩∣(x ∈ z ∧ φ)} → ∀x w ∈ {⟨x, y⟩∣(x ∈ z ∧ φ)})
4	3	hbrn 2564	. . . . 5 ⊢ (w ∈ ran {⟨x, y⟩∣(x ∈ z ∧ φ)} → ∀x w ∈ ran {⟨x, y⟩∣(x ∈ z ∧ φ)})
5	4	hbeleq 1173	. . . 4 ⊢ (w = ran {⟨x, y⟩∣(x ∈ z ∧ φ)} → ∀x w = ran {⟨x, y⟩∣(x ∈ z ∧ φ)})
6		ax-17 925	. . . . 5 ⊢ (v ∈ w → ∀y v ∈ w)
7		hbopab2 2113	. . . . . 6 ⊢ (v ∈ {⟨x, y⟩∣(x ∈ z ∧ φ)} → ∀y v ∈ {⟨x, y⟩∣(x ∈ z ∧ φ)})
8	7	hbrn 2564	. . . . 5 ⊢ (v ∈ ran {⟨x, y⟩∣(x ∈ z ∧ φ)} → ∀y v ∈ ran {⟨x, y⟩∣(x ∈ z ∧ φ)})
9	6, 8	rexeqf 1322	. . . 4 ⊢ (w = ran {⟨x, y⟩∣(x ∈ z ∧ φ)} → (∃y ∈ w φ ↔ ∃y ∈ ran {⟨x, y⟩∣(x ∈ z ∧ φ)}φ))
10	5, 9	birald 1217	. . 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 1395	. 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 2743	. . 3 ⊢ (dom {⟨x, y⟩∣(x ∈ z ∧ φ)} ∈ V → (Fun {⟨x, y⟩∣(x ∈ z ∧ φ)} → ran {⟨x, y⟩∣(x ∈ z ∧ φ)} ∈ V))
13		visset 1350	. . . 4 ⊢ z ∈ V
14		euex 1021	. . . . . . . 8 ⊢ (∃!yφ → ∃yφ)
15	14	r19.20si 1254	. . . . . . 7 ⊢ (∀x ∈ z ∃!yφ → ∀x ∈ z ∃yφ)
16		rabid2 1309	. . . . . . 7 ⊢ (z = {x ∈ z∣∃yφ} ↔ ∀x ∈ z ∃yφ)
17	15, 16	sylibr 175	. . . . . 6 ⊢ (∀x ∈ z ∃!yφ → z = {x ∈ z∣∃yφ})
18		19.42v 966	. . . . . . . 8 ⊢ (∃y(x ∈ z ∧ φ) ↔ (x ∈ z ∧ ∃yφ))
19	18	biabi 1181	. . . . . . 7 ⊢ {x∣∃y(x ∈ z ∧ φ)} = {x∣(x ∈ z ∧ ∃yφ)}
20		dmopab 2539	. . . . . . 7 ⊢ dom {⟨x, y⟩∣(x ∈ z ∧ φ)} = {x∣∃y(x ∈ z ∧ φ)}
21		df-rab 1208	. . . . . . 7 ⊢ {x ∈ z∣∃yφ} = {x∣(x ∈ z ∧ ∃yφ)}
22	19, 20, 21	3eqtr4 1126	. . . . . 6 ⊢ dom {⟨x, y⟩∣(x ∈ z ∧ φ)} = {x ∈ z∣∃yφ}
23	17, 22	syl6reqr 1143	. . . . 5 ⊢ (∀x ∈ z ∃!yφ → dom {⟨x, y⟩∣(x ∈ z ∧ φ)} = z)
24	23	eleq1d 1155	. . . 4 ⊢ (∀x ∈ z ∃!yφ → (dom {⟨x, y⟩∣(x ∈ z ∧ φ)} ∈ V ↔ z ∈ V))
25	13, 24	mpbiri 169	. . 3 ⊢ (∀x ∈ z ∃!yφ → dom {⟨x, y⟩∣(x ∈ z ∧ φ)} ∈ V)
26		eumo 1037	. . . . . . 7 ⊢ (∃!yφ → ∃*yφ)
27	26	syl3 18	. . . . . 6 ⊢ ((x ∈ z → ∃!yφ) → (x ∈ z → ∃*yφ))
28		moanimv 1052	. . . . . 6 ⊢ (∃*y(x ∈ z ∧ φ) ↔ (x ∈ z → ∃*yφ))
29	27, 28	sylibr 175	. . . . 5 ⊢ ((x ∈ z → ∃!yφ) → ∃*y(x ∈ z ∧ φ))
30	29	19.20i 691	. . . 4 ⊢ (∀x(x ∈ z → ∃!yφ) → ∀x∃*y(x ∈ z ∧ φ))
31		df-ral 1205	. . . 4 ⊢ (∀x ∈ z ∃!yφ ↔ ∀x(x ∈ z → ∃!yφ))
32		funopab 2694	. . . 4 ⊢ (Fun {⟨x, y⟩∣(x ∈ z ∧ φ)} ↔ ∀x∃*y(x ∈ z ∧ φ))
33	30, 31, 32	3imtr4 192	. . 3 ⊢ (∀x ∈ z ∃!yφ → Fun {⟨x, y⟩∣(x ∈ z ∧ φ)})
34	12, 25, 33	sylc 62	. 2 ⊢ (∀x ∈ z ∃!yφ → ran {⟨x, y⟩∣(x ∈ z ∧ φ)} ∈ V)
35		hbra1 1237	. . 3 ⊢ (∀x ∈ z ∃!yφ → ∀x∀x ∈ z ∃!yφ)
36	23	eleq2d 1156	. . . 4 ⊢ (∀x ∈ z ∃!yφ → (x ∈ dom {⟨x, y⟩∣(x ∈ z ∧ φ)} ↔ x ∈ z))
37		opabid 2099	. . . . . . . . 9 ⊢ (⟨x, y⟩ ∈ {⟨x, y⟩∣(x ∈ z ∧ φ)} ↔ (x ∈ z ∧ φ))
38		visset 1350	. . . . . . . . . 10 ⊢ x ∈ V
39		visset 1350	. . . . . . . . . 10 ⊢ y ∈ V
40	38, 39	opelrn 2560	. . . . . . . . 9 ⊢ (⟨x, y⟩ ∈ {⟨x, y⟩∣(x ∈ z ∧ φ)} → y ∈ ran {⟨x, y⟩∣(x ∈ z ∧ φ)})
41	37, 40	sylbir 176	. . . . . . . 8 ⊢ ((x ∈ z ∧ φ) → y ∈ ran {⟨x, y⟩∣(x ∈ z ∧ φ)})
42	41	exp 291	. . . . . . 7 ⊢ (x ∈ z → (φ → y ∈ ran {⟨x, y⟩∣(x ∈ z ∧ φ)}))
43	42	impac 304	. . . . . 6 ⊢ ((x ∈ z ∧ φ) → (y ∈ ran {⟨x, y⟩∣(x ∈ z ∧ φ)} ∧ φ))
44	43	19.22i 723	. . . . 5 ⊢ (∃y(x ∈ z ∧ φ) → ∃y(y ∈ ran {⟨x, y⟩∣(x ∈ z ∧ φ)} ∧ φ))
45	20	cleqabi 1176	. . . . 5 ⊢ (x ∈ dom {⟨x, y⟩∣(x ∈ z ∧ φ)} ↔ ∃y(x ∈ z ∧ φ))
46		df-rex 1206	. . . . 5 ⊢ (∃y ∈ ran {⟨x, y⟩∣(x ∈ z ∧ φ)}φ ↔ ∃y(y ∈ ran {⟨x, y⟩∣(x ∈ z ∧ φ)} ∧ φ))
47	44, 45, 46	3imtr4 192	. . . 4 ⊢ (x ∈ dom {⟨x, y⟩∣(x ∈ z ∧ φ)} → ∃y ∈ ran {⟨x, y⟩∣(x ∈ z ∧ φ)}φ)
48	36, 47	syl6bir 188	. . 3 ⊢ (∀x ∈ z ∃!yφ → (x ∈ z → ∃y ∈ ran {⟨x, y⟩∣(x ∈ z ∧ φ)}φ))
49	35, 48	r19.21ai 1258	. 2 ⊢ (∀x ∈ z ∃!yφ → ∀x ∈ z ∃y ∈ ran {⟨x, y⟩∣(x ∈ z ∧ φ)}φ)
50	11, 34, 49	sylc 62	1 ⊢ (∀x ∈ z ∃!yφ → ∃w∀x ∈ z ∃y ∈ w φ)

Syntax hints:   → wi 2   ∧ wa 196  ∀wal 672  ∃wex 678   ∈ wel 803  ∃!weu 1007  ∃*wmo 1008  {cab 1090   = wceq 1091   ∈ wcel 1092  ∀wral 1201  ∃wrex 1202  {crab 1204  Vcvv 1348  ⟨cop 1810  {copab 2055  dom cdm 2410  ran crn 2411  Fun wfun 2416

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-un 1076  ax-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-eu 1009  df-mo 1010  df-clab 1093  df-cleq 1097  df-clel 1099  df-ral 1205  df-rex 1206  df-rab 1208  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-pw 1799  df-sn 1811  df-pr 1812  df-op 1815  df-uni 1920  df-br 2063  df-opab 2098  df-id 2125  df-xp 2424  df-rel 2425  df-cnv 2426  df-co 2427  df-dm 2428  df-rn 2429  df-res 2430  df-ima 2431  df-fun 2432  df-fn 2433

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