Theorem tfrlemiubacc 5885

Description: The union of B satisfies the recursion rule (lemma for tfrlemi1 5887). (Contributed by Jim Kingdon, 22-Apr-2019.) (Proof shortened by Mario Carneiro, 24-May-2019.)

Hypotheses

Ref	Expression
tfrlemisucfn.1	|- A = {f | E.x e. On (f Fn x /\ A.y e. x (f ` y) = ( F ` (f |` y))) }
tfrlemisucfn.2	|- (ph -> A.x( Fun F /\ ( F ` x) e. _V))
tfrlemi1.3	|- B = { h | E.z e. x E.g(g Fn z /\ g e. A /\ h = (g u. { <.z , ( F ` g) >. })) }
tfrlemi1.4	|- (ph -> x e. On)
tfrlemi1.5	|- (ph -> A.z e. x E.g(g Fn z /\ A.w e. z (g ` w) = ( F ` (g |` w))))

Assertion

Ref	Expression
tfrlemiubacc	|- (ph -> A.u e. x ( U.B ` u) = ( F ` ( U.B |` u)))

Distinct variable groups:   f,g, h,u,w,x,y,z,A   f, F,g, h,u,w,x,y,z   ph,w,y   u,B,w,f,g, h,z   ph,g, h,z

Allowed substitution hints:   ph(x,u,f)   B(x,y)

Proof of Theorem tfrlemiubacc

Step	Hyp	Ref	Expression
1		tfrlemisucfn.1	. . . . . . . . 9 |- A = {f | E.x e. On (f Fn x /\ A.y e. x (f ` y) = ( F ` (f |` y))) }
2		tfrlemisucfn.2	. . . . . . . . 9 |- (ph -> A.x( Fun F /\ ( F ` x) e. _V))
3		tfrlemi1.3	. . . . . . . . 9 |- B = { h | E.z e. x E.g(g Fn z /\ g e. A /\ h = (g u. { <.z , ( F ` g) >. })) }
4		tfrlemi1.4	. . . . . . . . 9 |- (ph -> x e. On)
5		tfrlemi1.5	. . . . . . . . 9 |- (ph -> A.z e. x E.g(g Fn z /\ A.w e. z (g ` w) = ( F ` (g |` w))))
6	1, 2, 3, 4, 5	tfrlemibfn 5883	. . . . . . . 8 |- (ph -> U.B Fn x)
7		fndm 4941	. . . . . . . 8 |- ( U.B Fn x -> dom U.B = x)
8	6, 7	syl 14	. . . . . . 7 |- (ph -> dom U.B = x)
9	1, 2, 3, 4, 5	tfrlemibacc 5881	. . . . . . . . . 10 |- (ph -> B C_ A)
10	9	unissd 3595	. . . . . . . . 9 |- (ph -> U.B C_ U.A)
11	1	recsfval 5872	. . . . . . . . 9 |- recs( F) = U.A
12	10, 11	syl6sseqr 2986	. . . . . . . 8 |- (ph -> U.B C_ recs( F))
13		dmss 4477	. . . . . . . 8 |- ( U.B C_ recs( F) -> dom U.B C_ dom recs( F))
14	12, 13	syl 14	. . . . . . 7 |- (ph -> dom U.B C_ dom recs( F))
15	8, 14	eqsstr3d 2974	. . . . . 6 |- (ph -> x C_ dom recs( F))
16	15	sselda 2939	. . . . 5 |- ((ph /\ w e. x) -> w e. dom recs( F))
17	1	tfrlem9 5876	. . . . 5 |- (w e. dom recs( F) -> (recs( F) ` w) = ( F ` (recs( F) |` w)))
18	16, 17	syl 14	. . . 4 |- ((ph /\ w e. x) -> (recs( F) ` w) = ( F ` (recs( F) |` w)))
19	1	tfrlem7 5874	. . . . . 6 |- Fun recs( F)
20	19	a1i 9	. . . . 5 |- ((ph /\ w e. x) -> Fun recs( F))
21	12	adantr 261	. . . . 5 |- ((ph /\ w e. x) -> U.B C_ recs( F))
22	8	eleq2d 2104	. . . . . 6 |- (ph -> (w e. dom U.B <-> w e. x))
23	22	biimpar 281	. . . . 5 |- ((ph /\ w e. x) -> w e. dom U.B)
24		funssfv 5142	. . . . 5 |- (( Fun recs( F) /\ U.B C_ recs( F) /\ w e. dom U.B) -> (recs( F) ` w) = ( U.B ` w))
25	20, 21, 23, 24	syl3anc 1134	. . . 4 |- ((ph /\ w e. x) -> (recs( F) ` w) = ( U.B ` w))
26		eloni 4078	. . . . . . . . 9 |- (x e. On -> Ord x)
27	4, 26	syl 14	. . . . . . . 8 |- (ph -> Ord x)
28		ordelss 4082	. . . . . . . 8 |- (( Ord x /\ w e. x) -> w C_ x)
29	27, 28	sylan 267	. . . . . . 7 |- ((ph /\ w e. x) -> w C_ x)
30	8	adantr 261	. . . . . . 7 |- ((ph /\ w e. x) -> dom U.B = x)
31	29, 30	sseqtr4d 2976	. . . . . 6 |- ((ph /\ w e. x) -> w C_ dom U.B)
32		fun2ssres 4886	. . . . . 6 |- (( Fun recs( F) /\ U.B C_ recs( F) /\ w C_ dom U.B) -> (recs( F) |` w) = ( U.B |` w))
33	20, 21, 31, 32	syl3anc 1134	. . . . 5 |- ((ph /\ w e. x) -> (recs( F) |` w) = ( U.B |` w))
34	33	fveq2d 5125	. . . 4 |- ((ph /\ w e. x) -> ( F ` (recs( F) |` w)) = ( F ` ( U.B |` w)))
35	18, 25, 34	3eqtr3d 2077	. . 3 |- ((ph /\ w e. x) -> ( U.B ` w) = ( F ` ( U.B |` w)))
36	35	ralrimiva 2386	. 2 |- (ph -> A.w e. x ( U.B ` w) = ( F ` ( U.B |` w)))
37		fveq2 5121	. . . 4 |- (u = w -> ( U.B ` u) = ( U.B ` w))
38		reseq2 4550	. . . . 5 |- (u = w -> ( U.B |` u) = ( U.B |` w))
39	38	fveq2d 5125	. . . 4 |- (u = w -> ( F ` ( U.B |` u)) = ( F ` ( U.B |` w)))
40	37, 39	eqeq12d 2051	. . 3 |- (u = w -> (( U.B ` u) = ( F ` ( U.B |` u)) <-> ( U.B ` w) = ( F ` ( U.B |` w))))
41	40	cbvralv 2527	. 2 |- (A.u e. x ( U.B ` u) = ( F ` ( U.B |` u)) <-> A.w e. x ( U.B ` w) = ( F ` ( U.B |` w)))
42	36, 41	sylibr 137	1 |- (ph -> A.u e. x ( U.B ` u) = ( F ` ( U.B |` u)))

Syntax hints:   -> wi 4   /\ wa 97   /\ w3a 884  A.wal 1240   = wceq 1242  E.wex 1378   e. wcel 1390   {cab 2023  A.wral 2300  E.wrex 2301   _Vcvv 2551   u. cun 2909   C_ wss 2911   {csn 3367   <.cop 3370   U.cuni 3571   Ord word 4065   Oncon0 4066   dom cdm 4288   |` cres 4290   Fun wfun 4839   Fn wfn 4840   ` cfv 4845  recscrecs 5860

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-13 1401  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935  ax-un 4136  ax-setind 4220

This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-fal 1248  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ne 2203  df-ral 2305  df-rex 2306  df-rab 2309  df-v 2553  df-sbc 2759  df-csb 2847  df-dif 2914  df-un 2916  df-in 2918  df-ss 2925  df-nul 3219  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-iun 3650  df-br 3756  df-opab 3810  df-mpt 3811  df-tr 3846  df-id 4021  df-iord 4069  df-on 4071  df-suc 4074  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-rn 4299  df-res 4300  df-iota 4810  df-fun 4847  df-fn 4848  df-f 4849  df-fv 4853  df-recs 5861

This theorem is referenced by:  tfrlemiex  5886