Theorem bnj1326 31094

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
bnj1326.1	|- B = { d | ( d C_ A /\ A. x e. d pred ( x , A , R ) C_ d ) }
bnj1326.2	|- Y = <. x , ( f |` pred ( x , A , R ) ) >.
bnj1326.3	|- C = { f | E. d e. B ( f Fn d /\ A. x e. d ( f ` x ) = ( G ` Y ) ) }
bnj1326.4	|- D = ( dom g i^i dom h )

Assertion

Ref	Expression
bnj1326	|- ( ( R FrSe A /\ g e. C /\ h e. C ) -> ( g |` D ) = ( h |` D ) )

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

Allowed substitution hints:   A( g, h)   B( x, g, h, d)   C( x, f, g, h, d)   D( x, f, g, h, d)   R( g, h)   G( x, g, h)   Y( x, f, g, h, d)

Proof of Theorem bnj1326

Dummy variables p q are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eleq1 2689	. . . 4 |- ( q = h -> ( q e. C <-> h e. C ) )
2	1	3anbi3d 1405	. . 3 |- ( q = h -> ( ( R FrSe A /\ g e. C /\ q e. C ) <-> ( R FrSe A /\ g e. C /\ h e. C ) ) )
3		dmeq 5324	. . . . . . 7 |- ( q = h -> dom q = dom h )
4	3	ineq2d 3814	. . . . . 6 |- ( q = h -> ( dom g i^i dom q ) = ( dom g i^i dom h ) )
5	4	reseq2d 5396	. . . . 5 |- ( q = h -> ( g |` ( dom g i^i dom q ) ) = ( g |` ( dom g i^i dom h ) ) )
6		bnj1326.4	. . . . . 6 |- D = ( dom g i^i dom h )
7	6	reseq2i 5393	. . . . 5 |- ( g |` D ) = ( g |` ( dom g i^i dom h ) )
8	5, 7	syl6eqr 2674	. . . 4 |- ( q = h -> ( g |` ( dom g i^i dom q ) ) = ( g |` D ) )
9	4	reseq2d 5396	. . . . . 6 |- ( q = h -> ( q |` ( dom g i^i dom q ) ) = ( q |` ( dom g i^i dom h ) ) )
10		reseq1 5390	. . . . . 6 |- ( q = h -> ( q |` ( dom g i^i dom h ) ) = ( h |` ( dom g i^i dom h ) ) )
11	9, 10	eqtrd 2656	. . . . 5 |- ( q = h -> ( q |` ( dom g i^i dom q ) ) = ( h |` ( dom g i^i dom h ) ) )
12	6	reseq2i 5393	. . . . 5 |- ( h |` D ) = ( h |` ( dom g i^i dom h ) )
13	11, 12	syl6eqr 2674	. . . 4 |- ( q = h -> ( q |` ( dom g i^i dom q ) ) = ( h |` D ) )
14	8, 13	eqeq12d 2637	. . 3 |- ( q = h -> ( ( g |` ( dom g i^i dom q ) ) = ( q |` ( dom g i^i dom q ) ) <-> ( g |` D ) = ( h |` D ) ) )
15	2, 14	imbi12d 334	. 2 |- ( q = h -> ( ( ( R FrSe A /\ g e. C /\ q e. C ) -> ( g |` ( dom g i^i dom q ) ) = ( q |` ( dom g i^i dom q ) ) ) <-> ( ( R FrSe A /\ g e. C /\ h e. C ) -> ( g |` D ) = ( h |` D ) ) ) )
16		eleq1 2689	. . . . 5 |- ( p = g -> ( p e. C <-> g e. C ) )
17	16	3anbi2d 1404	. . . 4 |- ( p = g -> ( ( R FrSe A /\ p e. C /\ q e. C ) <-> ( R FrSe A /\ g e. C /\ q e. C ) ) )
18		dmeq 5324	. . . . . . . 8 |- ( p = g -> dom p = dom g )
19	18	ineq1d 3813	. . . . . . 7 |- ( p = g -> ( dom p i^i dom q ) = ( dom g i^i dom q ) )
20	19	reseq2d 5396	. . . . . 6 |- ( p = g -> ( p |` ( dom p i^i dom q ) ) = ( p |` ( dom g i^i dom q ) ) )
21		reseq1 5390	. . . . . 6 |- ( p = g -> ( p |` ( dom g i^i dom q ) ) = ( g |` ( dom g i^i dom q ) ) )
22	20, 21	eqtrd 2656	. . . . 5 |- ( p = g -> ( p |` ( dom p i^i dom q ) ) = ( g |` ( dom g i^i dom q ) ) )
23	19	reseq2d 5396	. . . . 5 |- ( p = g -> ( q |` ( dom p i^i dom q ) ) = ( q |` ( dom g i^i dom q ) ) )
24	22, 23	eqeq12d 2637	. . . 4 |- ( p = g -> ( ( p |` ( dom p i^i dom q ) ) = ( q |` ( dom p i^i dom q ) ) <-> ( g |` ( dom g i^i dom q ) ) = ( q |` ( dom g i^i dom q ) ) ) )
25	17, 24	imbi12d 334	. . 3 |- ( p = g -> ( ( ( R FrSe A /\ p e. C /\ q e. C ) -> ( p |` ( dom p i^i dom q ) ) = ( q |` ( dom p i^i dom q ) ) ) <-> ( ( R FrSe A /\ g e. C /\ q e. C ) -> ( g |` ( dom g i^i dom q ) ) = ( q |` ( dom g i^i dom q ) ) ) ) )
26		bnj1326.1	. . . 4 |- B = { d | ( d C_ A /\ A. x e. d pred ( x , A , R ) C_ d ) }
27		bnj1326.2	. . . 4 |- Y = <. x , ( f |` pred ( x , A , R ) ) >.
28		bnj1326.3	. . . 4 |- C = { f | E. d e. B ( f Fn d /\ A. x e. d ( f ` x ) = ( G ` Y ) ) }
29		eqid 2622	. . . 4 |- ( dom p i^i dom q ) = ( dom p i^i dom q )
30	26, 27, 28, 29	bnj1311 31092	. . 3 |- ( ( R FrSe A /\ p e. C /\ q e. C ) -> ( p |` ( dom p i^i dom q ) ) = ( q |` ( dom p i^i dom q ) ) )
31	25, 30	chvarv 2263	. 2 |- ( ( R FrSe A /\ g e. C /\ q e. C ) -> ( g |` ( dom g i^i dom q ) ) = ( q |` ( dom g i^i dom q ) ) )
32	15, 31	chvarv 2263	1 |- ( ( R FrSe A /\ g e. C /\ h e. C ) -> ( g |` D ) = ( h |` D ) )

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   {cab 2608   A.wral 2912   E.wrex 2913   i^i cin 3573   C_ wss 3574   <.cop 4183   dom cdm 5114   |` cres 5116   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:  bnj1321  31095  bnj1384  31100