Theorem bnj1321 31095

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
bnj1321.1	|- B = { d | ( d C_ A /\ A. x e. d pred ( x , A , R ) C_ d ) }
bnj1321.2	|- Y = <. x , ( f |` pred ( x , A , R ) ) >.
bnj1321.3	|- C = { f | E. d e. B ( f Fn d /\ A. x e. d ( f ` x ) = ( G ` Y ) ) }
bnj1321.4	|- ( ta <-> ( f e. C /\ dom f = ( { x } u. trCl ( x , A , R ) ) ) )

Assertion

Ref	Expression
bnj1321	|- ( ( R FrSe A /\ E. f ta ) -> E! f ta )

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

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

Proof of Theorem bnj1321

Dummy variables g z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpr 477	. 2 |- ( ( R FrSe A /\ E. f ta ) -> E. f ta )
2		simp1 1061	. . . . . . 7 |- ( ( R FrSe A /\ ta /\ [ g / f ] ta ) -> R FrSe A )
3		bnj1321.4	. . . . . . . . 9 |- ( ta <-> ( f e. C /\ dom f = ( { x } u. trCl ( x , A , R ) ) ) )
4	3	simplbi 476	. . . . . . . 8 |- ( ta -> f e. C )
5	4	3ad2ant2 1083	. . . . . . 7 |- ( ( R FrSe A /\ ta /\ [ g / f ] ta ) -> f e. C )
6		bnj1321.3	. . . . . . . . . . . . 13 |- C = { f | E. d e. B ( f Fn d /\ A. x e. d ( f ` x ) = ( G ` Y ) ) }
7		nfab1 2766	. . . . . . . . . . . . 13 |- F/_ f { f | E. d e. B ( f Fn d /\ A. x e. d ( f ` x ) = ( G ` Y ) ) }
8	6, 7	nfcxfr 2762	. . . . . . . . . . . 12 |- F/_ f C
9	8	nfcri 2758	. . . . . . . . . . 11 |- F/ f g e. C
10		nfv 1843	. . . . . . . . . . 11 |- F/ f dom g = ( { x } u. trCl ( x , A , R ) )
11	9, 10	nfan 1828	. . . . . . . . . 10 |- F/ f ( g e. C /\ dom g = ( { x } u. trCl ( x , A , R ) ) )
12		eleq1 2689	. . . . . . . . . . . 12 |- ( f = g -> ( f e. C <-> g e. C ) )
13		dmeq 5324	. . . . . . . . . . . . 13 |- ( f = g -> dom f = dom g )
14	13	eqeq1d 2624	. . . . . . . . . . . 12 |- ( f = g -> ( dom f = ( { x } u. trCl ( x , A , R ) ) <-> dom g = ( { x } u. trCl ( x , A , R ) ) ) )
15	12, 14	anbi12d 747	. . . . . . . . . . 11 |- ( f = g -> ( ( f e. C /\ dom f = ( { x } u. trCl ( x , A , R ) ) ) <-> ( g e. C /\ dom g = ( { x } u. trCl ( x , A , R ) ) ) ) )
16	3, 15	syl5bb 272	. . . . . . . . . 10 |- ( f = g -> ( ta <-> ( g e. C /\ dom g = ( { x } u. trCl ( x , A , R ) ) ) ) )
17	11, 16	sbie 2408	. . . . . . . . 9 |- ( [ g / f ] ta <-> ( g e. C /\ dom g = ( { x } u. trCl ( x , A , R ) ) ) )
18	17	simplbi 476	. . . . . . . 8 |- ( [ g / f ] ta -> g e. C )
19	18	3ad2ant3 1084	. . . . . . 7 |- ( ( R FrSe A /\ ta /\ [ g / f ] ta ) -> g e. C )
20		bnj1321.1	. . . . . . . 8 |- B = { d | ( d C_ A /\ A. x e. d pred ( x , A , R ) C_ d ) }
21		bnj1321.2	. . . . . . . 8 |- Y = <. x , ( f |` pred ( x , A , R ) ) >.
22		eqid 2622	. . . . . . . 8 |- ( dom f i^i dom g ) = ( dom f i^i dom g )
23	20, 21, 6, 22	bnj1326 31094	. . . . . . 7 |- ( ( R FrSe A /\ f e. C /\ g e. C ) -> ( f |` ( dom f i^i dom g ) ) = ( g |` ( dom f i^i dom g ) ) )
24	2, 5, 19, 23	syl3anc 1326	. . . . . 6 |- ( ( R FrSe A /\ ta /\ [ g / f ] ta ) -> ( f |` ( dom f i^i dom g ) ) = ( g |` ( dom f i^i dom g ) ) )
25	3	simprbi 480	. . . . . . . . . 10 |- ( ta -> dom f = ( { x } u. trCl ( x , A , R ) ) )
26	25	3ad2ant2 1083	. . . . . . . . 9 |- ( ( R FrSe A /\ ta /\ [ g / f ] ta ) -> dom f = ( { x } u. trCl ( x , A , R ) ) )
27	17	simprbi 480	. . . . . . . . . 10 |- ( [ g / f ] ta -> dom g = ( { x } u. trCl ( x , A , R ) ) )
28	27	3ad2ant3 1084	. . . . . . . . 9 |- ( ( R FrSe A /\ ta /\ [ g / f ] ta ) -> dom g = ( { x } u. trCl ( x , A , R ) ) )
29	26, 28	eqtr4d 2659	. . . . . . . 8 |- ( ( R FrSe A /\ ta /\ [ g / f ] ta ) -> dom f = dom g )
30		bnj1322 30893	. . . . . . . . 9 |- ( dom f = dom g -> ( dom f i^i dom g ) = dom f )
31	30	reseq2d 5396	. . . . . . . 8 |- ( dom f = dom g -> ( f |` ( dom f i^i dom g ) ) = ( f |` dom f ) )
32	29, 31	syl 17	. . . . . . 7 |- ( ( R FrSe A /\ ta /\ [ g / f ] ta ) -> ( f |` ( dom f i^i dom g ) ) = ( f |` dom f ) )
33		releq 5201	. . . . . . . . 9 |- ( z = f -> ( Rel z <-> Rel f ) )
34	20, 21, 6	bnj66 30930	. . . . . . . . 9 |- ( z e. C -> Rel z )
35	33, 34	vtoclga 3272	. . . . . . . 8 |- ( f e. C -> Rel f )
36		resdm 5441	. . . . . . . 8 |- ( Rel f -> ( f |` dom f ) = f )
37	5, 35, 36	3syl 18	. . . . . . 7 |- ( ( R FrSe A /\ ta /\ [ g / f ] ta ) -> ( f |` dom f ) = f )
38	32, 37	eqtrd 2656	. . . . . 6 |- ( ( R FrSe A /\ ta /\ [ g / f ] ta ) -> ( f |` ( dom f i^i dom g ) ) = f )
39		eqeq2 2633	. . . . . . . . . 10 |- ( dom f = dom g -> ( ( dom f i^i dom g ) = dom f <-> ( dom f i^i dom g ) = dom g ) )
40	30, 39	mpbid 222	. . . . . . . . 9 |- ( dom f = dom g -> ( dom f i^i dom g ) = dom g )
41	40	reseq2d 5396	. . . . . . . 8 |- ( dom f = dom g -> ( g |` ( dom f i^i dom g ) ) = ( g |` dom g ) )
42	29, 41	syl 17	. . . . . . 7 |- ( ( R FrSe A /\ ta /\ [ g / f ] ta ) -> ( g |` ( dom f i^i dom g ) ) = ( g |` dom g ) )
43	20, 21, 6	bnj66 30930	. . . . . . . 8 |- ( g e. C -> Rel g )
44		resdm 5441	. . . . . . . 8 |- ( Rel g -> ( g |` dom g ) = g )
45	19, 43, 44	3syl 18	. . . . . . 7 |- ( ( R FrSe A /\ ta /\ [ g / f ] ta ) -> ( g |` dom g ) = g )
46	42, 45	eqtrd 2656	. . . . . 6 |- ( ( R FrSe A /\ ta /\ [ g / f ] ta ) -> ( g |` ( dom f i^i dom g ) ) = g )
47	24, 38, 46	3eqtr3d 2664	. . . . 5 |- ( ( R FrSe A /\ ta /\ [ g / f ] ta ) -> f = g )
48	47	3expib 1268	. . . 4 |- ( R FrSe A -> ( ( ta /\ [ g / f ] ta ) -> f = g ) )
49	48	alrimivv 1856	. . 3 |- ( R FrSe A -> A. f A. g ( ( ta /\ [ g / f ] ta ) -> f = g ) )
50	49	adantr 481	. 2 |- ( ( R FrSe A /\ E. f ta ) -> A. f A. g ( ( ta /\ [ g / f ] ta ) -> f = g ) )
51		nfv 1843	. . 3 |- F/ g ta
52	51	eu2 2509	. 2 |- ( E! f ta <-> ( E. f ta /\ A. f A. g ( ( ta /\ [ g / f ] ta ) -> f = g ) ) )
53	1, 50, 52	sylanbrc 698	1 |- ( ( R FrSe A /\ E. f ta ) -> E! f ta )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   A.wal 1481   = wceq 1483   E.wex 1704   [wsb 1880   e. wcel 1990   E!weu 2470   {cab 2608   A.wral 2912   E.wrex 2913   u. cun 3572   i^i cin 3573   C_ wss 3574   {csn 4177   <.cop 4183   dom cdm 5114   |` cres 5116   Rel wrel 5119   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:  bnj1489  31124