Theorem zorn2lem2 4799

Description: Lemma for zorn2 4806.

Hypotheses

Ref	Expression
zorn2lem.1	⊢ A ∈ V
zorn2lem.2	⊢ B = {f∣∃h ∈ On (f Fn h ⋀ ∀t ∈ h (f ‘t) = (G ‘(f ↾ t)))}
zorn2lem.3	⊢ F = ∪B
zorn2lem.4	⊢ C = {z ∈ A∣∀g ∈ ran f gRz}
zorn2lem.5	⊢ D = {z ∈ A∣∀g ∈ (F “ x)gRz}
zorn2lem.6	⊢ G = {⟨f, t⟩∣t = ∪{v ∈ C∣∀u ∈ C ¬ uwv}}

Assertion

Ref	Expression
zorn2lem2	⊢ ((x ∈ On ⋀ (w We A ⋀ D ≠ ∅)) → (y ∈ x → (F ‘y)R(F ‘x)))

Distinct variable groups:   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 zorn2lem2

Step	Hyp	Ref	Expression
1		onsst 2998	. . . . 5 ⊢ (x ∈ On → x ⊆ On)
2		zorn2lem.2	. . . . . . 7 ⊢ B = {f∣∃h ∈ On (f Fn h ⋀ ∀t ∈ h (f ‘t) = (G ‘(f ↾ t)))}
3		zorn2lem.3	. . . . . . 7 ⊢ F = ∪B
4	2, 3	tfr1 3930	. . . . . 6 ⊢ F Fn On
5		fndm 3593	. . . . . 6 ⊢ (F Fn On → dom F = On)
6	4, 5	ax-mp 7	. . . . 5 ⊢ dom F = On
7	1, 6	syl6ssr 2111	. . . 4 ⊢ (x ∈ On → x ⊆ dom F)
8	2, 3	tfrlem7 3923	. . . . 5 ⊢ Fun F
9		funfvima2 3859	. . . . 5 ⊢ ((Fun F ⋀ x ⊆ dom F) → (y ∈ x → (F ‘y) ∈ (F “ x)))
10	8, 9	mpan 697	. . . 4 ⊢ (x ⊆ dom F → (y ∈ x → (F ‘y) ∈ (F “ x)))
11	7, 10	syl 10	. . 3 ⊢ (x ∈ On → (y ∈ x → (F ‘y) ∈ (F “ x)))
12	11	adantr 391	. 2 ⊢ ((x ∈ On ⋀ (w We A ⋀ D ≠ ∅)) → (y ∈ x → (F ‘y) ∈ (F “ x)))
13		zorn2lem.1	. . . 4 ⊢ A ∈ V
14		zorn2lem.4	. . . 4 ⊢ C = {z ∈ A∣∀g ∈ ran f gRz}
15		zorn2lem.5	. . . 4 ⊢ D = {z ∈ A∣∀g ∈ (F “ x)gRz}
16		zorn2lem.6	. . . 4 ⊢ G = {⟨f, t⟩∣t = ∪{v ∈ C∣∀u ∈ C ¬ uwv}}
17	13, 2, 3, 14, 15, 16	zorn2lem1 4798	. . 3 ⊢ ((x ∈ On ⋀ (w We A ⋀ D ≠ ∅)) → (F ‘x) ∈ D)
18		breq2 2628	. . . . . 6 ⊢ (z = (F ‘x) → (gRz ↔ gR(F ‘x)))
19	18	ralbidv 1666	. . . . 5 ⊢ (z = (F ‘x) → (∀g ∈ (F “ x)gRz ↔ ∀g ∈ (F “ x)gR(F ‘x)))
20	19, 15	elrab2 1910	. . . 4 ⊢ ((F ‘x) ∈ D ↔ ((F ‘x) ∈ A ⋀ ∀g ∈ (F “ x)gR(F ‘x)))
21	20	pm3.27bi 326	. . 3 ⊢ ((F ‘x) ∈ D → ∀g ∈ (F “ x)gR(F ‘x))
22		breq1 2627	. . . 4 ⊢ (g = (F ‘y) → (gR(F ‘x) ↔ (F ‘y)R(F ‘x)))
23	22	rcla4cv 1877	. . 3 ⊢ (∀g ∈ (F “ x)gR(F ‘x) → ((F ‘y) ∈ (F “ x) → (F ‘y)R(F ‘x)))
24	17, 21, 23	3syl 20	. 2 ⊢ ((x ∈ On ⋀ (w We A ⋀ D ≠ ∅)) → ((F ‘y) ∈ (F “ x) → (F ‘y)R(F ‘x)))
25	12, 24	syld 27	1 ⊢ ((x ∈ On ⋀ (w We A ⋀ D ≠ ∅)) → (y ∈ x → (F ‘y)R(F ‘x)))

Syntax hints:  ¬ wn 2   → wi 3   ⋀ wa 223   = wceq 958   ∈ wcel 960  {cab 1466   ≠ wne 1588  ∀wral 1648  ∃wrex 1649  {crab 1651  Vcvv 1814   ⊆ wss 2050  ∅c0 2283  ∪cuni 2507   class class class wbr 2624  {copab 2671   We wwe 2922  Oncon0 2954  dom cdm 3176  ran crn 3177   ↾ cres 3178   “ cima 3179  Fun wfun 3182   Fn wfn 3183   ‘cfv 3188

This theorem is referenced by:  zorn2lem3 4800  zorn2lem6 4803

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-9 967  ax-10 968  ax-11 969  ax-12 970  ax-13 971  ax-14 972  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462  ax-rep 2698  ax-sep 2708  ax-nul 2715  ax-pow 2748  ax-pr 2785  ax-un 2872

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 778  df-3an 779  df-ex 983  df-sb 1174  df-eu 1384  df-mo 1385  df-clab 1467  df-cleq 1472  df-clel 1475  df-ne 1590  df-ral 1652  df-rex 1653  df-reu 1654  df-rab 1655  df-v 1815  df-sbc 1945  df-dif 2052  df-un 2053  df-in 2054  df-ss 2056  df-nul 2284  df-pw 2406  df-sn 2416  df-pr 2417  df-tp 2419  df-op 2420  df-uni 2508  df-iun 2572  df-br 2625  df-opab 2672  df-tr 2686  df-eprel 2838  df-id 2841  df-po 2846  df-so 2856  df-fr 2923  df-we 2940  df-ord 2957  df-on 2958  df-suc 2960  df-xp 3190  df-rel 3191  df-cnv 3192  df-co 3193  df-dm 3194  df-rn 3195  df-res 3196  df-ima 3197  df-fun 3198  df-fn 3199  df-fv 3204

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