Theorem tfrlem6 2954

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

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
tfrlem6	⊢ Rel F

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

Proof of Theorem tfrlem6

Step	Hyp	Ref	Expression
1		tfrlem.2	. 2 ⊢ F = ∪A
2		reluni 2493	. . . 4 ⊢ (Rel ∪A ↔ ∀g ∈ A Rel g)
3		tfrlem.1	. . . . . 6 ⊢ A = {f∣∃x ∈ On (f Fn x ∧ ∀y ∈ x (f ‘y) = (G ‘(f ↾ y)))}
4	3, 1	tfrlem4 2952	. . . . 5 ⊢ (g ∈ A → Fun g)
5		funrel 2681	. . . . 5 ⊢ (Fun g → Rel g)
6	4, 5	syl 12	. . . 4 ⊢ (g ∈ A → Rel g)
7	2, 6	mprgbir 1250	. . 3 ⊢ Rel ∪A
8		releq 2477	. . 3 ⊢ (F = ∪A → (Rel F ↔ Rel ∪A))
9	7, 8	mpbiri 169	. 2 ⊢ (F = ∪A → Rel F)
10	1, 9	ax-mp 6	1 ⊢ Rel F

Syntax hints:   ∧ wa 196  {cab 1090   = wceq 1091   ∈ wcel 1092  ∀wral 1201  ∃wrex 1202  ∪cuni 1919  Oncon0 2199   ↾ cres 2412  Rel wrel 2415  Fun wfun 2416   Fn wfn 2417   ‘cfv 2422

This theorem is referenced by:  tfrlem7 2955  zornlem4 3606

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-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  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-v 1349  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-op 1815  df-uni 1920  df-br 2063  df-opab 2098  df-id 2125  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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