Theorem tfrALTlem 13976

Description: Lemma for deriving transfinite recursion from well-founded recursion.

Assertion

Ref	Expression
tfrALTlem	|- {f | E.x e. On (f Fn x /\ A.y e. x (f` y) = (G` (f |` y)))} = {f | E.x(f Fn x /\ (x C_ On /\ A.y e. x Pred( _E , On, y) C_ x) /\ A.y e. x (f` y) = (G` (f |` Pred( _E , On, y))))}

Distinct variable group:   x,y

Proof of Theorem tfrALTlem

Step	Hyp	Ref	Expression
1		df-rex 2110	. . 3 |- (E.x e. On (f Fn x /\ A.y e. x (f` y) = (G` (f |` y))) <-> E.x(x e. On /\ (f Fn x /\ A.y e. x (f` y) = (G` (f |` y)))))
2		onelon 3683	. . . . . . . . . . 11 |- ((x e. On /\ y e. x) -> y e. On)
3		predon 13904	. . . . . . . . . . . . . 14 |- (y e. On -> Pred( _E , On, y) = y)
4		reseq2 4219	. . . . . . . . . . . . . 14 |- (Pred( _E , On, y) = y -> (f |` Pred( _E , On, y)) = (f |` y))
5	3, 4	syl 12	. . . . . . . . . . . . 13 |- (y e. On -> (f |` Pred( _E , On, y)) = (f |` y))
6	5	fveq2d 4685	. . . . . . . . . . . 12 |- (y e. On -> (G` (f |` Pred( _E , On, y))) = (G` (f |` y)))
7	6	eqeq2d 1895	. . . . . . . . . . 11 |- (y e. On -> ((f` y) = (G` (f |` Pred( _E , On, y))) <-> (f` y) = (G` (f |` y))))
8	2, 7	syl 12	. . . . . . . . . 10 |- ((x e. On /\ y e. x) -> ((f` y) = (G` (f |` Pred( _E , On, y))) <-> (f` y) = (G` (f |` y))))
9	8	pm5.74da 646	. . . . . . . . 9 |- (x e. On -> ((y e. x -> (f` y) = (G` (f |` Pred( _E , On, y)))) <-> (y e. x -> (f` y) = (G` (f |` y)))))
10	9	ralbidv2 2125	. . . . . . . 8 |- (x e. On -> (A.y e. x (f` y) = (G` (f |` Pred( _E , On, y))) <-> A.y e. x (f` y) = (G` (f |` y))))
11	10	pm5.32i 707	. . . . . . 7 |- ((x e. On /\ A.y e. x (f` y) = (G` (f |` Pred( _E , On, y)))) <-> (x e. On /\ A.y e. x (f` y) = (G` (f |` y))))
12		df-ord 3660	. . . . . . . . . . . . . 14 |- (Ord x <-> (Tr x /\ _E We x))
13		ordsson 3867	. . . . . . . . . . . . . 14 |- (Ord x -> x C_ On)
14	12, 13	sylbir 218	. . . . . . . . . . . . 13 |- ((Tr x /\ _E We x) -> x C_ On)
15	14	ex 402	. . . . . . . . . . . 12 |- (Tr x -> ( _E We x -> x C_ On))
16		epweon 3864	. . . . . . . . . . . . 13 |- _E We On
17		wess 3645	. . . . . . . . . . . . 13 |- (x C_ On -> ( _E We On -> _E We x))
18	16, 17	mpi 55	. . . . . . . . . . . 12 |- (x C_ On -> _E We x)
19	15, 18	impbid1 575	. . . . . . . . . . 11 |- (Tr x -> ( _E We x <-> x C_ On))
20	19	pm5.32i 707	. . . . . . . . . 10 |- ((Tr x /\ _E We x) <-> (Tr x /\ x C_ On))
21		ancom 482	. . . . . . . . . 10 |- ((x C_ On /\ Tr x) <-> (Tr x /\ x C_ On))
22	20, 12, 21	3bitr4i 200	. . . . . . . . 9 |- (Ord x <-> (x C_ On /\ Tr x))
23		visset 2295	. . . . . . . . . 10 |- x e. _V
24	23	elon 3666	. . . . . . . . 9 |- (x e. On <-> Ord x)
25		ssel2 2616	. . . . . . . . . . . . . 14 |- ((x C_ On /\ y e. x) -> y e. On)
26	25, 3	syl 12	. . . . . . . . . . . . 13 |- ((x C_ On /\ y e. x) -> Pred( _E , On, y) = y)
27	26	sseq1d 2644	. . . . . . . . . . . 12 |- ((x C_ On /\ y e. x) -> (Pred( _E , On, y) C_ x <-> y C_ x))
28	27	ralbidva 2119	. . . . . . . . . . 11 |- (x C_ On -> (A.y e. x Pred( _E , On, y) C_ x <-> A.y e. x y C_ x))
29		dftr3 3415	. . . . . . . . . . 11 |- (Tr x <-> A.y e. x y C_ x)
30	28, 29	syl6bbr 597	. . . . . . . . . 10 |- (x C_ On -> (A.y e. x Pred( _E , On, y) C_ x <-> Tr x))
31	30	pm5.32i 707	. . . . . . . . 9 |- ((x C_ On /\ A.y e. x Pred( _E , On, y) C_ x) <-> (x C_ On /\ Tr x))
32	22, 24, 31	3bitr4i 200	. . . . . . . 8 |- (x e. On <-> (x C_ On /\ A.y e. x Pred( _E , On, y) C_ x))
33	32	anbi1i 539	. . . . . . 7 |- ((x e. On /\ A.y e. x (f` y) = (G` (f |` Pred( _E , On, y)))) <-> ((x C_ On /\ A.y e. x Pred( _E , On, y) C_ x) /\ A.y e. x (f` y) = (G` (f |` Pred( _E , On, y)))))
34	11, 33	bitr3i 192	. . . . . 6 |- ((x e. On /\ A.y e. x (f` y) = (G` (f |` y))) <-> ((x C_ On /\ A.y e. x Pred( _E , On, y) C_ x) /\ A.y e. x (f` y) = (G` (f |` Pred( _E , On, y)))))
35	34	anbi2i 538	. . . . 5 |- ((f Fn x /\ (x e. On /\ A.y e. x (f` y) = (G` (f |` y)))) <-> (f Fn x /\ ((x C_ On /\ A.y e. x Pred( _E , On, y) C_ x) /\ A.y e. x (f` y) = (G` (f |` Pred( _E , On, y))))))
36		an12 542	. . . . 5 |- ((x e. On /\ (f Fn x /\ A.y e. x (f` y) = (G` (f |` y)))) <-> (f Fn x /\ (x e. On /\ A.y e. x (f` y) = (G` (f |` y)))))
37		3anass 862	. . . . 5 |- ((f Fn x /\ (x C_ On /\ A.y e. x Pred( _E , On, y) C_ x) /\ A.y e. x (f` y) = (G` (f |` Pred( _E , On, y)))) <-> (f Fn x /\ ((x C_ On /\ A.y e. x Pred( _E , On, y) C_ x) /\ A.y e. x (f` y) = (G` (f |` Pred( _E , On, y))))))
38	35, 36, 37	3bitr4i 200	. . . 4 |- ((x e. On /\ (f Fn x /\ A.y e. x (f` y) = (G` (f |` y)))) <-> (f Fn x /\ (x C_ On /\ A.y e. x Pred( _E , On, y) C_ x) /\ A.y e. x (f` y) = (G` (f |` Pred( _E , On, y)))))
39	38	exbii 1398	. . 3 |- (E.x(x e. On /\ (f Fn x /\ A.y e. x (f` y) = (G` (f |` y)))) <-> E.x(f Fn x /\ (x C_ On /\ A.y e. x Pred( _E , On, y) C_ x) /\ A.y e. x (f` y) = (G` (f |` Pred( _E , On, y)))))
40	1, 39	bitri 190	. 2 |- (E.x e. On (f Fn x /\ A.y e. x (f` y) = (G` (f |` y))) <-> E.x(f Fn x /\ (x C_ On /\ A.y e. x Pred( _E , On, y) C_ x) /\ A.y e. x (f` y) = (G` (f |` Pred( _E , On, y)))))
41	40	abbii 2006	1 |- {f | E.x e. On (f Fn x /\ A.y e. x (f` y) = (G` (f |` y)))} = {f | E.x(f Fn x /\ (x C_ On /\ A.y e. x Pred( _E , On, y) C_ x) /\ A.y e. x (f` y) = (G` (f |` Pred( _E , On, y))))}

Syntax hints:   <-> wb 163   /\ wa 240   /\ w3a 858   = wceq 1298   e. wcel 1300  E.wex 1326  {cab 1871  A.wral 2105  E.wrex 2106   C_ wss 2593  Tr wtr 3411   _E cep 3581   We wwe 3624  Ord word 3656  Oncon0 3657   |` cres 3988   Fn wfn 3993  ` cfv 3998  Predcpred 13879

This theorem is referenced by:  tfr1ALT 13977  tfr2ALT 13978  tfr3ALT 13979

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-13 1311  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524  ax-un 3790

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3or 859  df-3an 860  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-ral 2109  df-rex 2110  df-v 2294  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-pss 2607  df-nul 2876  df-pw 3035  df-sn 3049  df-pr 3050  df-tp 3052  df-op 3053  df-uni 3178  df-br 3339  df-opab 3396  df-tr 3412  df-eprel 3583  df-po 3591  df-so 3604  df-fr 3625  df-we 3644  df-ord 3660  df-on 3661  df-xp 4000  df-rel 4001  df-cnv 4002  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fv 4014  df-pred 13880

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