Theorem tfrlem12 2960

Description: Lemma for transfinite recursion. Show C is an acceptable function.

Hypotheses

Ref	Expression
tfrlem.1	⊢ A = {f∣∃x ∈ On (f Fn x ∧ ∀y ∈ x (f ‘y) = (G ‘(f ↾ y)))}
tfrlem.2	⊢ F = ∪A
tfrlem.3	⊢ C = (F ∪ {⟨dom F, (G ‘(F ↾ dom F))⟩})

Assertion

Ref	Expression
tfrlem12	⊢ (dom F ∈ On → C ∈ A)

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

Proof of Theorem tfrlem12

Step	Hyp	Ref	Expression
1		fneq2 2719	. . . . 5 ⊢ (x = suc dom F → (C Fn x ↔ C Fn suc dom F))
2		raleq 1324	. . . . 5 ⊢ (x = suc dom F → (∀y ∈ x (C ‘y) = (G ‘(C ↾ y)) ↔ ∀y ∈ suc dom F(C ‘y) = (G ‘(C ↾ y))))
3	1, 2	anbi12d 476	. . . 4 ⊢ (x = suc dom F → ((C Fn x ∧ ∀y ∈ x (C ‘y) = (G ‘(C ↾ y))) ↔ (C Fn suc dom F ∧ ∀y ∈ suc dom F(C ‘y) = (G ‘(C ↾ y)))))
4	3	rcla4ev 1403	. . 3 ⊢ ((suc dom F ∈ On ∧ (C Fn suc dom F ∧ ∀y ∈ suc dom F(C ‘y) = (G ‘(C ↾ y)))) → ∃x ∈ On (C Fn x ∧ ∀y ∈ x (C ‘y) = (G ‘(C ↾ y))))
5		suceloni 2314	. . 3 ⊢ (dom F ∈ On → suc dom F ∈ On)
6		tfrlem.1	. . . . 5 ⊢ A = {f∣∃x ∈ On (f Fn x ∧ ∀y ∈ x (f ‘y) = (G ‘(f ↾ y)))}
7		tfrlem.2	. . . . 5 ⊢ F = ∪A
8		tfrlem.3	. . . . 5 ⊢ C = (F ∪ {⟨dom F, (G ‘(F ↾ dom F))⟩})
9	6, 7, 8	tfrlem10 2958	. . . 4 ⊢ (dom F ∈ On → C Fn suc dom F)
10	6, 7, 8	tfrlem11 2959	. . . . 5 ⊢ (dom F ∈ On → (y ∈ suc dom F → (C ‘y) = (G ‘(C ↾ y))))
11	10	r19.21aiv 1259	. . . 4 ⊢ (dom F ∈ On → ∀y ∈ suc dom F(C ‘y) = (G ‘(C ↾ y)))
12	9, 11	jca 236	. . 3 ⊢ (dom F ∈ On → (C Fn suc dom F ∧ ∀y ∈ suc dom F(C ‘y) = (G ‘(C ↾ y))))
13	4, 5, 12	sylanc 361	. 2 ⊢ (dom F ∈ On → ∃x ∈ On (C Fn x ∧ ∀y ∈ x (C ‘y) = (G ‘(C ↾ y))))
14		fnex 2740	. . . 4 ⊢ (suc dom F ∈ On → (C Fn suc dom F → C ∈ V))
15	14, 5, 9	sylc 62	. . 3 ⊢ (dom F ∈ On → C ∈ V)
16		fneq1 2718	. . . . . 6 ⊢ (f = C → (f Fn x ↔ C Fn x))
17		fveq1 2831	. . . . . . . 8 ⊢ (f = C → (f ‘y) = (C ‘y))
18		reseq1 2575	. . . . . . . . 9 ⊢ (f = C → (f ↾ y) = (C ↾ y))
19	18	fveq2d 2836	. . . . . . . 8 ⊢ (f = C → (G ‘(f ↾ y)) = (G ‘(C ↾ y)))
20	17, 19	cleq12d 1115	. . . . . . 7 ⊢ (f = C → ((f ‘y) = (G ‘(f ↾ y)) ↔ (C ‘y) = (G ‘(C ↾ y))))
21	20	biraldv 1219	. . . . . 6 ⊢ (f = C → (∀y ∈ x (f ‘y) = (G ‘(f ↾ y)) ↔ ∀y ∈ x (C ‘y) = (G ‘(C ↾ y))))
22	16, 21	anbi12d 476	. . . . 5 ⊢ (f = C → ((f Fn x ∧ ∀y ∈ x (f ‘y) = (G ‘(f ↾ y))) ↔ (C Fn x ∧ ∀y ∈ x (C ‘y) = (G ‘(C ↾ y)))))
23	22	birexdv 1220	. . . 4 ⊢ (f = C → (∃x ∈ On (f Fn x ∧ ∀y ∈ x (f ‘y) = (G ‘(f ↾ y))) ↔ ∃x ∈ On (C Fn x ∧ ∀y ∈ x (C ‘y) = (G ‘(C ↾ y)))))
24	23, 6	elab2g 1418	. . 3 ⊢ (C ∈ V → (C ∈ A ↔ ∃x ∈ On (C Fn x ∧ ∀y ∈ x (C ‘y) = (G ‘(C ↾ y)))))
25	15, 24	syl 12	. 2 ⊢ (dom F ∈ On → (C ∈ A ↔ ∃x ∈ On (C Fn x ∧ ∀y ∈ x (C ‘y) = (G ‘(C ↾ y)))))
26	13, 25	mpbird 171	1 ⊢ (dom F ∈ On → C ∈ A)

Syntax hints:   → wi 2   ↔ wb 127   ∧ wa 196  {cab 1090   = wceq 1091   ∈ wcel 1092  ∀wral 1201  ∃wrex 1202  Vcvv 1348   ∪ cun 1485  {csn 1808  ⟨cop 1810  ∪cuni 1919  Oncon0 2199  suc csuc 2201  dom cdm 2410   ↾ cres 2412   Fn wfn 2417   ‘cfv 2422

This theorem is referenced by:  tfrlem13 2961

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-suc 2205  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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