Theorem tfrlem13 2961

Description: Lemma for transfinite recursion. If dom F is in On, then C is acceptable, and thus a subset of F, but dom C is bigger than dom F. This is a contradiction, so dom F must be On.

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
tfrlem13	⊢ dom F = On

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

Proof of Theorem tfrlem13

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	tfrlem8 2956	. . 3 ⊢ Ord dom F
4		ordeirr 2217	. . . 4 ⊢ (Ord dom F → ¬ dom F ∈ dom F)
5		elssuni 1940	. . . . . . 7 ⊢ (C ∈ A → C ⊆ ∪A)
6	5, 2	syl6ssr 1547	. . . . . 6 ⊢ (C ∈ A → C ⊆ F)
7		dmss 2530	. . . . . 6 ⊢ (C ⊆ F → dom C ⊆ dom F)
8		ssel 1502	. . . . . 6 ⊢ (dom C ⊆ dom F → (dom F ∈ dom C → dom F ∈ dom F))
9	6, 7, 8	3syl 21	. . . . 5 ⊢ (C ∈ A → (dom F ∈ dom C → dom F ∈ dom F))
10		tfrlem.3	. . . . . 6 ⊢ C = (F ∪ {⟨dom F, (G ‘(F ↾ dom F))⟩})
11	1, 2, 10	tfrlem12 2960	. . . . 5 ⊢ (dom F ∈ On → C ∈ A)
12		sucidg 2305	. . . . . 6 ⊢ (dom F ∈ On → dom F ∈ suc dom F)
13	1, 2, 10	tfrlem10 2958	. . . . . . 7 ⊢ (dom F ∈ On → C Fn suc dom F)
14		fndm 2723	. . . . . . 7 ⊢ (C Fn suc dom F → dom C = suc dom F)
15	13, 14	syl 12	. . . . . 6 ⊢ (dom F ∈ On → dom C = suc dom F)
16	12, 15	eleqtrrd 1166	. . . . 5 ⊢ (dom F ∈ On → dom F ∈ dom C)
17	9, 11, 16	sylc 62	. . . 4 ⊢ (dom F ∈ On → dom F ∈ dom F)
18	4, 17	nsyl 102	. . 3 ⊢ (Ord dom F → ¬ dom F ∈ On)
19	3, 18	ax-mp 6	. 2 ⊢ ¬ dom F ∈ On
20		ordeleqon 2241	. . 3 ⊢ (Ord dom F ↔ (dom F ∈ On ∨ dom F = On))
21	3, 20	mpbi 164	. 2 ⊢ (dom F ∈ On ∨ dom F = On)
22		orel1 212	. 2 ⊢ (¬ dom F ∈ On → ((dom F ∈ On ∨ dom F = On) → dom F = On))
23	19, 21, 22	mp2 43	1 ⊢ dom F = On

Syntax hints:  ¬ wn 1   → wi 2   ∨ wo 195   ∧ wa 196  {cab 1090   = wceq 1091   ∈ wcel 1092  ∀wral 1201  ∃wrex 1202   ∪ cun 1485   ⊆ wss 1487  {csn 1808  ⟨cop 1810  ∪cuni 1919  Ord word 2198  Oncon0 2199  suc csuc 2201  dom cdm 2410   ↾ cres 2412   Fn wfn 2417   ‘cfv 2422

This theorem is referenced by:  tfr1 2962  tfr2 2963

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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