Theorem bnj1501 31135

Description: Technical lemma for bnj1500 31136. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)

Hypotheses

Ref	Expression
bnj1501.1	|- B = { d | ( d C_ A /\ A. x e. d pred ( x , A , R ) C_ d ) }
bnj1501.2	|- Y = <. x , ( f |` pred ( x , A , R ) ) >.
bnj1501.3	|- C = { f | E. d e. B ( f Fn d /\ A. x e. d ( f ` x ) = ( G ` Y ) ) }
bnj1501.4	|- F = U. C
bnj1501.5	|- ( ph <-> ( R FrSe A /\ x e. A ) )
bnj1501.6	|- ( ps <-> ( ph /\ f e. C /\ x e. dom f ) )
bnj1501.7	|- ( ch <-> ( ps /\ d e. B /\ dom f = d ) )

Assertion

Ref	Expression
bnj1501	|- ( R FrSe A -> A. x e. A ( F ` x ) = ( G ` <. x , ( F |` pred ( x , A , R ) ) >. ) )

Distinct variable groups:   A, d, f, x   B, f   G, d, f, x   R, d, f, x   Y, d   ph, d, f

Allowed substitution hints:   ph( x)   ps( x, f, d)   ch( x, f, d)   B( x, d)   C( x, f, d)   F( x, f, d)   Y( x, f)

Proof of Theorem bnj1501

Dummy variable w is distinct from all other variables.

Step	Hyp	Ref	Expression
1		bnj1501.5	. 2 |- ( ph <-> ( R FrSe A /\ x e. A ) )
2	1	simprbi 480	. . . . . . . 8 |- ( ph -> x e. A )
3		bnj1501.1	. . . . . . . . . . 11 |- B = { d | ( d C_ A /\ A. x e. d pred ( x , A , R ) C_ d ) }
4		bnj1501.2	. . . . . . . . . . 11 |- Y = <. x , ( f |` pred ( x , A , R ) ) >.
5		bnj1501.3	. . . . . . . . . . 11 |- C = { f | E. d e. B ( f Fn d /\ A. x e. d ( f ` x ) = ( G ` Y ) ) }
6		bnj1501.4	. . . . . . . . . . 11 |- F = U. C
7	3, 4, 5, 6	bnj60 31130	. . . . . . . . . 10 |- ( R FrSe A -> F Fn A )
8		fndm 5990	. . . . . . . . . 10 |- ( F Fn A -> dom F = A )
9	7, 8	syl 17	. . . . . . . . 9 |- ( R FrSe A -> dom F = A )
10	1, 9	bnj832 30828	. . . . . . . 8 |- ( ph -> dom F = A )
11	2, 10	eleqtrrd 2704	. . . . . . 7 |- ( ph -> x e. dom F )
12	6	dmeqi 5325	. . . . . . . 8 |- dom F = dom U. C
13	5	bnj1317 30892	. . . . . . . . 9 |- ( w e. C -> A. f w e. C )
14	13	bnj1400 30906	. . . . . . . 8 |- dom U. C = U_ f e. C dom f
15	12, 14	eqtri 2644	. . . . . . 7 |- dom F = U_ f e. C dom f
16	11, 15	syl6eleq 2711	. . . . . 6 |- ( ph -> x e. U_ f e. C dom f )
17	16	bnj1405 30907	. . . . 5 |- ( ph -> E. f e. C x e. dom f )
18		bnj1501.6	. . . . 5 |- ( ps <-> ( ph /\ f e. C /\ x e. dom f ) )
19	17, 18	bnj1209 30867	. . . 4 |- ( ph -> E. f ps )
20	5	bnj1436 30910	. . . . . . . . . 10 |- ( f e. C -> E. d e. B ( f Fn d /\ A. x e. d ( f ` x ) = ( G ` Y ) ) )
21	20	bnj1299 30889	. . . . . . . . 9 |- ( f e. C -> E. d e. B f Fn d )
22		fndm 5990	. . . . . . . . 9 |- ( f Fn d -> dom f = d )
23	21, 22	bnj31 30785	. . . . . . . 8 |- ( f e. C -> E. d e. B dom f = d )
24	18, 23	bnj836 30830	. . . . . . 7 |- ( ps -> E. d e. B dom f = d )
25		bnj1501.7	. . . . . . 7 |- ( ch <-> ( ps /\ d e. B /\ dom f = d ) )
26	3, 4, 5, 6, 1, 18	bnj1518 31132	. . . . . . 7 |- ( ps -> A. d ps )
27	24, 25, 26	bnj1521 30921	. . . . . 6 |- ( ps -> E. d ch )
28	7	bnj930 30840	. . . . . . . . . . . 12 |- ( R FrSe A -> Fun F )
29	1, 28	bnj832 30828	. . . . . . . . . . 11 |- ( ph -> Fun F )
30	18, 29	bnj835 30829	. . . . . . . . . 10 |- ( ps -> Fun F )
31		elssuni 4467	. . . . . . . . . . . 12 |- ( f e. C -> f C_ U. C )
32	31, 6	syl6sseqr 3652	. . . . . . . . . . 11 |- ( f e. C -> f C_ F )
33	18, 32	bnj836 30830	. . . . . . . . . 10 |- ( ps -> f C_ F )
34	18	simp3bi 1078	. . . . . . . . . 10 |- ( ps -> x e. dom f )
35	30, 33, 34	bnj1502 30918	. . . . . . . . 9 |- ( ps -> ( F ` x ) = ( f ` x ) )
36	3, 4, 5	bnj1514 31131	. . . . . . . . . . 11 |- ( f e. C -> A. x e. dom f ( f ` x ) = ( G ` Y ) )
37	18, 36	bnj836 30830	. . . . . . . . . 10 |- ( ps -> A. x e. dom f ( f ` x ) = ( G ` Y ) )
38	37, 34	bnj1294 30888	. . . . . . . . 9 |- ( ps -> ( f ` x ) = ( G ` Y ) )
39	35, 38	eqtrd 2656	. . . . . . . 8 |- ( ps -> ( F ` x ) = ( G ` Y ) )
40	25, 39	bnj835 30829	. . . . . . 7 |- ( ch -> ( F ` x ) = ( G ` Y ) )
41	25, 30	bnj835 30829	. . . . . . . . . . 11 |- ( ch -> Fun F )
42	25, 33	bnj835 30829	. . . . . . . . . . 11 |- ( ch -> f C_ F )
43	3	bnj1517 30920	. . . . . . . . . . . . . 14 |- ( d e. B -> A. x e. d pred ( x , A , R ) C_ d )
44	25, 43	bnj836 30830	. . . . . . . . . . . . 13 |- ( ch -> A. x e. d pred ( x , A , R ) C_ d )
45	25, 34	bnj835 30829	. . . . . . . . . . . . . 14 |- ( ch -> x e. dom f )
46	25	simp3bi 1078	. . . . . . . . . . . . . 14 |- ( ch -> dom f = d )
47	45, 46	eleqtrd 2703	. . . . . . . . . . . . 13 |- ( ch -> x e. d )
48	44, 47	bnj1294 30888	. . . . . . . . . . . 12 |- ( ch -> pred ( x , A , R ) C_ d )
49	48, 46	sseqtr4d 3642	. . . . . . . . . . 11 |- ( ch -> pred ( x , A , R ) C_ dom f )
50	41, 42, 49	bnj1503 30919	. . . . . . . . . 10 |- ( ch -> ( F |` pred ( x , A , R ) ) = ( f |` pred ( x , A , R ) ) )
51	50	opeq2d 4409	. . . . . . . . 9 |- ( ch -> <. x , ( F |` pred ( x , A , R ) ) >. = <. x , ( f |` pred ( x , A , R ) ) >. )
52	51, 4	syl6eqr 2674	. . . . . . . 8 |- ( ch -> <. x , ( F |` pred ( x , A , R ) ) >. = Y )
53	52	fveq2d 6195	. . . . . . 7 |- ( ch -> ( G ` <. x , ( F |` pred ( x , A , R ) ) >. ) = ( G ` Y ) )
54	40, 53	eqtr4d 2659	. . . . . 6 |- ( ch -> ( F ` x ) = ( G ` <. x , ( F |` pred ( x , A , R ) ) >. ) )
55	27, 54	bnj593 30815	. . . . 5 |- ( ps -> E. d ( F ` x ) = ( G ` <. x , ( F |` pred ( x , A , R ) ) >. ) )
56	3, 4, 5, 6	bnj1519 31133	. . . . 5 |- ( ( F ` x ) = ( G ` <. x , ( F |` pred ( x , A , R ) ) >. ) -> A. d ( F ` x ) = ( G ` <. x , ( F |` pred ( x , A , R ) ) >. ) )
57	55, 56	bnj1397 30905	. . . 4 |- ( ps -> ( F ` x ) = ( G ` <. x , ( F |` pred ( x , A , R ) ) >. ) )
58	19, 57	bnj593 30815	. . 3 |- ( ph -> E. f ( F ` x ) = ( G ` <. x , ( F |` pred ( x , A , R ) ) >. ) )
59	3, 4, 5, 6	bnj1520 31134	. . 3 |- ( ( F ` x ) = ( G ` <. x , ( F |` pred ( x , A , R ) ) >. ) -> A. f ( F ` x ) = ( G ` <. x , ( F |` pred ( x , A , R ) ) >. ) )
60	58, 59	bnj1397 30905	. 2 |- ( ph -> ( F ` x ) = ( G ` <. x , ( F |` pred ( x , A , R ) ) >. ) )
61	1, 60	bnj1459 30913	1 |- ( R FrSe A -> A. x e. A ( F ` x ) = ( G ` <. x , ( F |` pred ( x , A , R ) ) >. ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   {cab 2608   A.wral 2912   E.wrex 2913   C_ wss 3574   <.cop 4183   U.cuni 4436   U_ciun 4520   dom cdm 5114   |` cres 5116   Fun wfun 5882   Fn wfn 5883   ` cfv 5888   predc-bnj14 30754   FrSe w-bnj15 30758

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-reg 8497  ax-inf2 8538

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-fal 1489  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-om 7066  df-1o 7560  df-bnj17 30753  df-bnj14 30755  df-bnj13 30757  df-bnj15 30759  df-bnj18 30761  df-bnj19 30763

This theorem is referenced by:  bnj1500  31136