Theorem tfrlem7 2955

Description: Lemma for transfinite recursion. The union of all acceptable functions is a function.

Hypotheses

Ref	Expression
tfrlem.1	⊢ A = {f∣∃x ∈ On (f Fn x ∧ ∀y ∈ x (f ‘y) = (G ‘(f ↾ y)))}
tfrlem.2	⊢ F = ∪A

Assertion

Ref	Expression
tfrlem7	⊢ Fun F

Distinct variable group(s):   x,y,f,A   x,F,y,f   x,G,y,f

Proof of Theorem tfrlem7

Step	Hyp	Ref	Expression
1		tfrlem.1	. . . 4 ⊢ A = {f∣∃x ∈ On (f Fn x ∧ ∀y ∈ x (f ‘y) = (G ‘(f ↾ y)))}
2		tfrlem.2	. . . 4 ⊢ F = ∪A
3	1, 2	tfrlem6 2954	. . 3 ⊢ Rel F
4	2	eleq2i 1153	. . . . . . . . 9 ⊢ (⟨x, u⟩ ∈ F ↔ ⟨x, u⟩ ∈ ∪A)
5		eluni 1922	. . . . . . . . 9 ⊢ (⟨x, u⟩ ∈ ∪A ↔ ∃g(⟨x, u⟩ ∈ g ∧ g ∈ A))
6	4, 5	bitr 151	. . . . . . . 8 ⊢ (⟨x, u⟩ ∈ F ↔ ∃g(⟨x, u⟩ ∈ g ∧ g ∈ A))
7	2	eleq2i 1153	. . . . . . . . 9 ⊢ (⟨x, v⟩ ∈ F ↔ ⟨x, v⟩ ∈ ∪A)
8		eluni 1922	. . . . . . . . 9 ⊢ (⟨x, v⟩ ∈ ∪A ↔ ∃h(⟨x, v⟩ ∈ h ∧ h ∈ A))
9	7, 8	bitr 151	. . . . . . . 8 ⊢ (⟨x, v⟩ ∈ F ↔ ∃h(⟨x, v⟩ ∈ h ∧ h ∈ A))
10	6, 9	anbi12i 369	. . . . . . 7 ⊢ ((⟨x, u⟩ ∈ F ∧ ⟨x, v⟩ ∈ F) ↔ (∃g(⟨x, u⟩ ∈ g ∧ g ∈ A) ∧ ∃h(⟨x, v⟩ ∈ h ∧ h ∈ A)))
11		eeanv 980	. . . . . . 7 ⊢ (∃g∃h((⟨x, u⟩ ∈ g ∧ g ∈ A) ∧ (⟨x, v⟩ ∈ h ∧ h ∈ A)) ↔ (∃g(⟨x, u⟩ ∈ g ∧ g ∈ A) ∧ ∃h(⟨x, v⟩ ∈ h ∧ h ∈ A)))
12	10, 11	bitr4 154	. . . . . 6 ⊢ ((⟨x, u⟩ ∈ F ∧ ⟨x, v⟩ ∈ F) ↔ ∃g∃h((⟨x, u⟩ ∈ g ∧ g ∈ A) ∧ (⟨x, v⟩ ∈ h ∧ h ∈ A)))
13		an4 388	. . . . . . . . 9 ⊢ (((⟨x, u⟩ ∈ g ∧ g ∈ A) ∧ (⟨x, v⟩ ∈ h ∧ h ∈ A)) ↔ ((⟨x, u⟩ ∈ g ∧ ⟨x, v⟩ ∈ h) ∧ (g ∈ A ∧ h ∈ A)))
14		ancom 333	. . . . . . . . 9 ⊢ (((⟨x, u⟩ ∈ g ∧ ⟨x, v⟩ ∈ h) ∧ (g ∈ A ∧ h ∈ A)) ↔ ((g ∈ A ∧ h ∈ A) ∧ (⟨x, u⟩ ∈ g ∧ ⟨x, v⟩ ∈ h)))
15	13, 14	bitr 151	. . . . . . . 8 ⊢ (((⟨x, u⟩ ∈ g ∧ g ∈ A) ∧ (⟨x, v⟩ ∈ h ∧ h ∈ A)) ↔ ((g ∈ A ∧ h ∈ A) ∧ (⟨x, u⟩ ∈ g ∧ ⟨x, v⟩ ∈ h)))
16	1, 2	tfrlem5 2953	. . . . . . . . 9 ⊢ ((g ∈ A ∧ h ∈ A) → ((⟨x, u⟩ ∈ g ∧ ⟨x, v⟩ ∈ h) → u = v))
17	16	imp 277	. . . . . . . 8 ⊢ (((g ∈ A ∧ h ∈ A) ∧ (⟨x, u⟩ ∈ g ∧ ⟨x, v⟩ ∈ h)) → u = v)
18	15, 17	sylbi 174	. . . . . . 7 ⊢ (((⟨x, u⟩ ∈ g ∧ g ∈ A) ∧ (⟨x, v⟩ ∈ h ∧ h ∈ A)) → u = v)
19	18	19.23aivv 953	. . . . . 6 ⊢ (∃g∃h((⟨x, u⟩ ∈ g ∧ g ∈ A) ∧ (⟨x, v⟩ ∈ h ∧ h ∈ A)) → u = v)
20	12, 19	sylbi 174	. . . . 5 ⊢ ((⟨x, u⟩ ∈ F ∧ ⟨x, v⟩ ∈ F) → u = v)
21	20	ax-gen 677	. . . 4 ⊢ ∀v((⟨x, u⟩ ∈ F ∧ ⟨x, v⟩ ∈ F) → u = v)
22	21	gen2 681	. . 3 ⊢ ∀x∀u∀v((⟨x, u⟩ ∈ F ∧ ⟨x, v⟩ ∈ F) → u = v)
23	3, 22	pm3.2i 234	. 2 ⊢ (Rel F ∧ ∀x∀u∀v((⟨x, u⟩ ∈ F ∧ ⟨x, v⟩ ∈ F) → u = v))
24		dffun4 2676	. 2 ⊢ (Fun F ↔ (Rel F ∧ ∀x∀u∀v((⟨x, u⟩ ∈ F ∧ ⟨x, v⟩ ∈ F) → u = v)))
25	23, 24	mpbir 165	1 ⊢ Fun F

Syntax hints:   → wi 2   ∧ wa 196  ∀wal 672  ∃wex 678   = weq 797  {cab 1090   = wceq 1091   ∈ wcel 1092  ∀wral 1201  ∃wrex 1202  ⟨cop 1810  ∪cuni 1919  Oncon0 2199   ↾ cres 2412  Rel wrel 2415  Fun wfun 2416   Fn wfn 2417   ‘cfv 2422

This theorem is referenced by:  tfrlem9 2957  tfrlem10 2958  tfr1 2962  numthlem 3598  zornlem1 3603  zornlem2 3604  zornlem5 3607  zornlem7 3609

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-3or 582  df-3an 583  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-sbc 1441  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-tp 1814  df-op 1815  df-uni 1920  df-tr 2042  df-br 2063  df-opab 2098  df-eprel 2122  df-id 2125  df-po 2128  df-so 2138  df-fr 2169  df-we 2186  df-ord 2202  df-on 2203  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  df-fv 2438

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