Theorem bnj1498 31129

Description: Technical lemma for bnj60 31130. 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
bnj1498.1	|- B = { d | ( d C_ A /\ A. x e. d pred ( x , A , R ) C_ d ) }
bnj1498.2	|- Y = <. x , ( f |` pred ( x , A , R ) ) >.
bnj1498.3	|- C = { f | E. d e. B ( f Fn d /\ A. x e. d ( f ` x ) = ( G ` Y ) ) }
bnj1498.4	|- F = U. C

Assertion

Ref	Expression
bnj1498	|- ( R FrSe A -> dom F = A )

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

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

Proof of Theorem bnj1498

Dummy variables t z w are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eliun 4524	. . . . . . 7 |- ( z e. U_ f e. C dom f <-> E. f e. C z e. dom f )
2		bnj1498.3	. . . . . . . . . . . . . . . 16 |- C = { f | E. d e. B ( f Fn d /\ A. x e. d ( f ` x ) = ( G ` Y ) ) }
3	2	bnj1436 30910	. . . . . . . . . . . . . . 15 |- ( f e. C -> E. d e. B ( f Fn d /\ A. x e. d ( f ` x ) = ( G ` Y ) ) )
4	3	bnj1299 30889	. . . . . . . . . . . . . 14 |- ( f e. C -> E. d e. B f Fn d )
5		fndm 5990	. . . . . . . . . . . . . 14 |- ( f Fn d -> dom f = d )
6	4, 5	bnj31 30785	. . . . . . . . . . . . 13 |- ( f e. C -> E. d e. B dom f = d )
7	6	bnj1196 30865	. . . . . . . . . . . 12 |- ( f e. C -> E. d ( d e. B /\ dom f = d ) )
8		bnj1498.1	. . . . . . . . . . . . . . 15 |- B = { d | ( d C_ A /\ A. x e. d pred ( x , A , R ) C_ d ) }
9	8	bnj1436 30910	. . . . . . . . . . . . . 14 |- ( d e. B -> ( d C_ A /\ A. x e. d pred ( x , A , R ) C_ d ) )
10	9	simpld 475	. . . . . . . . . . . . 13 |- ( d e. B -> d C_ A )
11	10	anim1i 592	. . . . . . . . . . . 12 |- ( ( d e. B /\ dom f = d ) -> ( d C_ A /\ dom f = d ) )
12	7, 11	bnj593 30815	. . . . . . . . . . 11 |- ( f e. C -> E. d ( d C_ A /\ dom f = d ) )
13		sseq1 3626	. . . . . . . . . . . 12 |- ( dom f = d -> ( dom f C_ A <-> d C_ A ) )
14	13	biimparc 504	. . . . . . . . . . 11 |- ( ( d C_ A /\ dom f = d ) -> dom f C_ A )
15	12, 14	bnj593 30815	. . . . . . . . . 10 |- ( f e. C -> E. d dom f C_ A )
16	15	bnj937 30842	. . . . . . . . 9 |- ( f e. C -> dom f C_ A )
17	16	sselda 3603	. . . . . . . 8 |- ( ( f e. C /\ z e. dom f ) -> z e. A )
18	17	rexlimiva 3028	. . . . . . 7 |- ( E. f e. C z e. dom f -> z e. A )
19	1, 18	sylbi 207	. . . . . 6 |- ( z e. U_ f e. C dom f -> z e. A )
20	2	bnj1317 30892	. . . . . . 7 |- ( w e. C -> A. f w e. C )
21	20	bnj1400 30906	. . . . . 6 |- dom U. C = U_ f e. C dom f
22	19, 21	eleq2s 2719	. . . . 5 |- ( z e. dom U. C -> z e. A )
23		bnj1498.4	. . . . . 6 |- F = U. C
24	23	dmeqi 5325	. . . . 5 |- dom F = dom U. C
25	22, 24	eleq2s 2719	. . . 4 |- ( z e. dom F -> z e. A )
26	25	ssriv 3607	. . 3 |- dom F C_ A
27	26	a1i 11	. 2 |- ( R FrSe A -> dom F C_ A )
28		bnj1498.2	. . . . . . . 8 |- Y = <. x , ( f |` pred ( x , A , R ) ) >.
29	8, 28, 2	bnj1493 31127	. . . . . . 7 |- ( R FrSe A -> A. x e. A E. f e. C dom f = ( { x } u. trCl ( x , A , R ) ) )
30		vsnid 4209	. . . . . . . . . . 11 |- x e. { x }
31		elun1 3780	. . . . . . . . . . 11 |- ( x e. { x } -> x e. ( { x } u. trCl ( x , A , R ) ) )
32	30, 31	ax-mp 5	. . . . . . . . . 10 |- x e. ( { x } u. trCl ( x , A , R ) )
33		eleq2 2690	. . . . . . . . . 10 |- ( dom f = ( { x } u. trCl ( x , A , R ) ) -> ( x e. dom f <-> x e. ( { x } u. trCl ( x , A , R ) ) ) )
34	32, 33	mpbiri 248	. . . . . . . . 9 |- ( dom f = ( { x } u. trCl ( x , A , R ) ) -> x e. dom f )
35	34	reximi 3011	. . . . . . . 8 |- ( E. f e. C dom f = ( { x } u. trCl ( x , A , R ) ) -> E. f e. C x e. dom f )
36	35	ralimi 2952	. . . . . . 7 |- ( A. x e. A E. f e. C dom f = ( { x } u. trCl ( x , A , R ) ) -> A. x e. A E. f e. C x e. dom f )
37	29, 36	syl 17	. . . . . 6 |- ( R FrSe A -> A. x e. A E. f e. C x e. dom f )
38		eliun 4524	. . . . . . 7 |- ( x e. U_ f e. C dom f <-> E. f e. C x e. dom f )
39	38	ralbii 2980	. . . . . 6 |- ( A. x e. A x e. U_ f e. C dom f <-> A. x e. A E. f e. C x e. dom f )
40	37, 39	sylibr 224	. . . . 5 |- ( R FrSe A -> A. x e. A x e. U_ f e. C dom f )
41		nfcv 2764	. . . . . 6 |- F/_ x A
42	8	bnj1309 31090	. . . . . . . . 9 |- ( t e. B -> A. x t e. B )
43	2, 42	bnj1307 31091	. . . . . . . 8 |- ( t e. C -> A. x t e. C )
44	43	nfcii 2755	. . . . . . 7 |- F/_ x C
45		nfcv 2764	. . . . . . 7 |- F/_ x dom f
46	44, 45	nfiun 4548	. . . . . 6 |- F/_ x U_ f e. C dom f
47	41, 46	dfss3f 3595	. . . . 5 |- ( A C_ U_ f e. C dom f <-> A. x e. A x e. U_ f e. C dom f )
48	40, 47	sylibr 224	. . . 4 |- ( R FrSe A -> A C_ U_ f e. C dom f )
49	48, 21	syl6sseqr 3652	. . 3 |- ( R FrSe A -> A C_ dom U. C )
50	49, 24	syl6sseqr 3652	. 2 |- ( R FrSe A -> A C_ dom F )
51	27, 50	eqssd 3620	1 |- ( R FrSe A -> dom F = A )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   {cab 2608   A.wral 2912   E.wrex 2913   u. cun 3572   C_ wss 3574   {csn 4177   <.cop 4183   U.cuni 4436   U_ciun 4520   dom cdm 5114   |` cres 5116   Fn wfn 5883   ` cfv 5888   predc-bnj14 30754   FrSe w-bnj15 30758   trClc-bnj18 30760

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:  bnj60  31130