Theorem zornlem2 3604

Description: Lemma for Zorn's lemma.

Hypotheses

Ref	Expression
zornlem.1	⊢ A ∈ V
zornlem.2	⊢ B = {f∣∃h ∈ On (f Fn h ∧ ∀t ∈ h (f ‘t) = (G ‘(f ↾ t)))}
zornlem.3	⊢ F = ∪B
zornlem.4	⊢ C = {z ∈ A∣∀g ∈ ran fgRz}
zornlem.5	⊢ D = {z ∈ A∣∀g ∈ (F “ x)gRz}
zornlem.6	⊢ G = {⟨f, t⟩∣t = ∪{v ∈ C∣∀u ∈ C ¬ uwv}}

Assertion

Ref	Expression
zornlem2	⊢ ((x ∈ On ∧ (w We A ∧ ¬ D = ∅)) → (y ∈ x → (F ‘y)R(F ‘x)))

Distinct variable group(s):   x,y,w,h,t,z,f,g,u,v,A   B,h,t,f   x,F,y,z,v,u,f,g,h,t   h,G,t,f   t,C   y,D,u,v,f,t   x,R,y,z,w,g,u,v,f,t

Proof of Theorem zornlem2

Step	Hyp	Ref	Expression
1		onsst 2243	. . . . 5 ⊢ (x ∈ On → x ⊆ On)
2		zornlem.2	. . . . . . 7 ⊢ B = {f∣∃h ∈ On (f Fn h ∧ ∀t ∈ h (f ‘t) = (G ‘(f ↾ t)))}
3		zornlem.3	. . . . . . 7 ⊢ F = ∪B
4	2, 3	tfr1 2962	. . . . . 6 ⊢ F Fn On
5		fndm 2723	. . . . . 6 ⊢ (F Fn On → dom F = On)
6	4, 5	ax-mp 6	. . . . 5 ⊢ dom F = On
7	1, 6	syl6ssr 1547	. . . 4 ⊢ (x ∈ On → x ⊆ dom F)
8	2, 3	tfrlem7 2955	. . . . 5 ⊢ Fun F
9		funfvima2 2905	. . . . 5 ⊢ ((Fun F ∧ x ⊆ dom F) → (y ∈ x → (F ‘y) ∈ (F “ x)))
10	8, 9	mpan 518	. . . 4 ⊢ (x ⊆ dom F → (y ∈ x → (F ‘y) ∈ (F “ x)))
11	7, 10	syl 12	. . 3 ⊢ (x ∈ On → (y ∈ x → (F ‘y) ∈ (F “ x)))
12	11	adantr 306	. 2 ⊢ ((x ∈ On ∧ (w We A ∧ ¬ D = ∅)) → (y ∈ x → (F ‘y) ∈ (F “ x)))
13		zornlem.1	. . . 4 ⊢ A ∈ V
14		zornlem.4	. . . 4 ⊢ C = {z ∈ A∣∀g ∈ ran fgRz}
15		zornlem.5	. . . 4 ⊢ D = {z ∈ A∣∀g ∈ (F “ x)gRz}
16		zornlem.6	. . . 4 ⊢ G = {⟨f, t⟩∣t = ∪{v ∈ C∣∀u ∈ C ¬ uwv}}
17	13, 2, 3, 14, 15, 16	zornlem1 3603	. . 3 ⊢ ((x ∈ On ∧ (w We A ∧ ¬ D = ∅)) → (F ‘x) ∈ D)
18	15	eleq2i 1153	. . . . 5 ⊢ ((F ‘x) ∈ D ↔ (F ‘x) ∈ {z ∈ A∣∀g ∈ (F “ x)gRz})
19		breq2 2066	. . . . . . 7 ⊢ (z = (F ‘x) → (gRz ↔ gR(F ‘x)))
20	19	biraldv 1219	. . . . . 6 ⊢ (z = (F ‘x) → (∀g ∈ (F “ x)gRz ↔ ∀g ∈ (F “ x)gR(F ‘x)))
21	20	elrab 1422	. . . . 5 ⊢ ((F ‘x) ∈ {z ∈ A∣∀g ∈ (F “ x)gRz} ↔ ((F ‘x) ∈ A ∧ ∀g ∈ (F “ x)gR(F ‘x)))
22	18, 21	bitr 151	. . . 4 ⊢ ((F ‘x) ∈ D ↔ ((F ‘x) ∈ A ∧ ∀g ∈ (F “ x)gR(F ‘x)))
23	22	pm3.27bd 263	. . 3 ⊢ ((F ‘x) ∈ D → ∀g ∈ (F “ x)gR(F ‘x))
24		breq1 2065	. . . 4 ⊢ (g = (F ‘y) → (gR(F ‘x) ↔ (F ‘y)R(F ‘x)))
25	24	rcla4v 1402	. . 3 ⊢ (∀g ∈ (F “ x)gR(F ‘x) → ((F ‘y) ∈ (F “ x) → (F ‘y)R(F ‘x)))
26	17, 23, 25	3syl 21	. 2 ⊢ ((x ∈ On ∧ (w We A ∧ ¬ D = ∅)) → ((F ‘y) ∈ (F “ x) → (F ‘y)R(F ‘x)))
27	12, 26	syld 27	1 ⊢ ((x ∈ On ∧ (w We A ∧ ¬ D = ∅)) → (y ∈ x → (F ‘y)R(F ‘x)))

Syntax hints:  ¬ wn 1   → wi 2   ∧ wa 196   ∈ wel 803  {cab 1090   = wceq 1091   ∈ wcel 1092  ∀wral 1201  ∃wrex 1202  {crab 1204  Vcvv 1348   ⊆ wss 1487  ∅c0 1707  ∪cuni 1919   class class class wbr 2054  {copab 2055   We wwe 2062  Oncon0 2199  dom cdm 2410  ran crn 2411   ↾ cres 2412   “ cima 2413  Fun wfun 2416   Fn wfn 2417   ‘cfv 2422

This theorem is referenced by:  zornlem3 3605  zornlem6 3608

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-reu 1207  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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