Theorem pofun 4771

Description: A function preserves a partial order relation. (Contributed by Jeff Madsen, 18-Jun-2011.)

Hypotheses

Ref	Expression
pofun.1	|- S = { <. x , y >. | X R Y }
pofun.2	|- ( x = y -> X = Y )

Assertion

Ref	Expression
pofun	|- ( ( R Po B /\ A. x e. A X e. B ) -> S Po A )

Distinct variable groups:   x, R, y   y, X   x, Y   x, A   x, B

Allowed substitution hints:   A( y)   B( y)   S( x, y)   X( x)   Y( y)

Proof of Theorem pofun

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

Step	Hyp	Ref	Expression
1		nfcsb1v 3379	. . . . . . 7 |- F/_ x [_ v / x ]_ X
2	1	nfel1 2606	. . . . . 6 |- F/ x [_ v / x ]_ X e. B
3		csbeq1a 3372	. . . . . . 7 |- ( x = v -> X = [_ v / x ]_ X )
4	3	eleq1d 2513	. . . . . 6 |- ( x = v -> ( X e. B <-> [_ v / x ]_ X e. B ) )
5	2, 4	rspc 3144	. . . . 5 |- ( v e. A -> ( A. x e. A X e. B -> [_ v / x ]_ X e. B ) )
6	5	impcom 432	. . . 4 |- ( ( A. x e. A X e. B /\ v e. A ) -> [_ v / x ]_ X e. B )
7		poirr 4766	. . . . 5 |- ( ( R Po B /\ [_ v / x ]_ X e. B ) -> -. [_ v / x ]_ X R [_ v / x ]_ X )
8		df-br 4403	. . . . . 6 |- ( v S v <-> <. v , v >. e. S )
9		pofun.1	. . . . . . 7 |- S = { <. x , y >. | X R Y }
10	9	eleq2i 2521	. . . . . 6 |- ( <. v , v >. e. S <-> <. v , v >. e. { <. x , y >. | X R Y } )
11		nfcv 2592	. . . . . . . 8 |- F/_ x R
12		nfcv 2592	. . . . . . . 8 |- F/_ x Y
13	1, 11, 12	nfbr 4447	. . . . . . 7 |- F/ x [_ v / x ]_ X R Y
14		nfv 1761	. . . . . . 7 |- F/ y [_ v / x ]_ X R [_ v / x ]_ X
15		vex 3048	. . . . . . 7 |- v e. _V
16	3	breq1d 4412	. . . . . . 7 |- ( x = v -> ( X R Y <-> [_ v / x ]_ X R Y ) )
17		vex 3048	. . . . . . . . . 10 |- y e. _V
18		pofun.2	. . . . . . . . . 10 |- ( x = y -> X = Y )
19	17, 18	csbie 3389	. . . . . . . . 9 |- [_ y / x ]_ X = Y
20		csbeq1 3366	. . . . . . . . 9 |- ( y = v -> [_ y / x ]_ X = [_ v / x ]_ X )
21	19, 20	syl5eqr 2499	. . . . . . . 8 |- ( y = v -> Y = [_ v / x ]_ X )
22	21	breq2d 4414	. . . . . . 7 |- ( y = v -> ( [_ v / x ]_ X R Y <-> [_ v / x ]_ X R [_ v / x ]_ X ) )
23	13, 14, 15, 15, 16, 22	opelopabf 4726	. . . . . 6 |- ( <. v , v >. e. { <. x , y >. | X R Y } <-> [_ v / x ]_ X R [_ v / x ]_ X )
24	8, 10, 23	3bitri 275	. . . . 5 |- ( v S v <-> [_ v / x ]_ X R [_ v / x ]_ X )
25	7, 24	sylnibr 307	. . . 4 |- ( ( R Po B /\ [_ v / x ]_ X e. B ) -> -. v S v )
26	6, 25	sylan2 477	. . 3 |- ( ( R Po B /\ ( A. x e. A X e. B /\ v e. A ) ) -> -. v S v )
27	26	anassrs 654	. 2 |- ( ( ( R Po B /\ A. x e. A X e. B ) /\ v e. A ) -> -. v S v )
28	5	com12 32	. . . . . 6 |- ( A. x e. A X e. B -> ( v e. A -> [_ v / x ]_ X e. B ) )
29		nfcsb1v 3379	. . . . . . . . 9 |- F/_ x [_ w / x ]_ X
30	29	nfel1 2606	. . . . . . . 8 |- F/ x [_ w / x ]_ X e. B
31		csbeq1a 3372	. . . . . . . . 9 |- ( x = w -> X = [_ w / x ]_ X )
32	31	eleq1d 2513	. . . . . . . 8 |- ( x = w -> ( X e. B <-> [_ w / x ]_ X e. B ) )
33	30, 32	rspc 3144	. . . . . . 7 |- ( w e. A -> ( A. x e. A X e. B -> [_ w / x ]_ X e. B ) )
34	33	com12 32	. . . . . 6 |- ( A. x e. A X e. B -> ( w e. A -> [_ w / x ]_ X e. B ) )
35		nfcsb1v 3379	. . . . . . . . 9 |- F/_ x [_ z / x ]_ X
36	35	nfel1 2606	. . . . . . . 8 |- F/ x [_ z / x ]_ X e. B
37		csbeq1a 3372	. . . . . . . . 9 |- ( x = z -> X = [_ z / x ]_ X )
38	37	eleq1d 2513	. . . . . . . 8 |- ( x = z -> ( X e. B <-> [_ z / x ]_ X e. B ) )
39	36, 38	rspc 3144	. . . . . . 7 |- ( z e. A -> ( A. x e. A X e. B -> [_ z / x ]_ X e. B ) )
40	39	com12 32	. . . . . 6 |- ( A. x e. A X e. B -> ( z e. A -> [_ z / x ]_ X e. B ) )
41	28, 34, 40	3anim123d 1346	. . . . 5 |- ( A. x e. A X e. B -> ( ( v e. A /\ w e. A /\ z e. A ) -> ( [_ v / x ]_ X e. B /\ [_ w / x ]_ X e. B /\ [_ z / x ]_ X e. B ) ) )
42	41	imp 431	. . . 4 |- ( ( A. x e. A X e. B /\ ( v e. A /\ w e. A /\ z e. A ) ) -> ( [_ v / x ]_ X e. B /\ [_ w / x ]_ X e. B /\ [_ z / x ]_ X e. B ) )
43	42	adantll 720	. . 3 |- ( ( ( R Po B /\ A. x e. A X e. B ) /\ ( v e. A /\ w e. A /\ z e. A ) ) -> ( [_ v / x ]_ X e. B /\ [_ w / x ]_ X e. B /\ [_ z / x ]_ X e. B ) )
44		potr 4767	. . . . 5 |- ( ( R Po B /\ ( [_ v / x ]_ X e. B /\ [_ w / x ]_ X e. B /\ [_ z / x ]_ X e. B ) ) -> ( ( [_ v / x ]_ X R [_ w / x ]_ X /\ [_ w / x ]_ X R [_ z / x ]_ X ) -> [_ v / x ]_ X R [_ z / x ]_ X ) )
45		df-br 4403	. . . . . . 7 |- ( v S w <-> <. v , w >. e. S )
46	9	eleq2i 2521	. . . . . . 7 |- ( <. v , w >. e. S <-> <. v , w >. e. { <. x , y >. | X R Y } )
47		nfv 1761	. . . . . . . 8 |- F/ y [_ v / x ]_ X R [_ w / x ]_ X
48		vex 3048	. . . . . . . 8 |- w e. _V
49		csbeq1 3366	. . . . . . . . . 10 |- ( y = w -> [_ y / x ]_ X = [_ w / x ]_ X )
50	19, 49	syl5eqr 2499	. . . . . . . . 9 |- ( y = w -> Y = [_ w / x ]_ X )
51	50	breq2d 4414	. . . . . . . 8 |- ( y = w -> ( [_ v / x ]_ X R Y <-> [_ v / x ]_ X R [_ w / x ]_ X ) )
52	13, 47, 15, 48, 16, 51	opelopabf 4726	. . . . . . 7 |- ( <. v , w >. e. { <. x , y >. | X R Y } <-> [_ v / x ]_ X R [_ w / x ]_ X )
53	45, 46, 52	3bitri 275	. . . . . 6 |- ( v S w <-> [_ v / x ]_ X R [_ w / x ]_ X )
54		df-br 4403	. . . . . . 7 |- ( w S z <-> <. w , z >. e. S )
55	9	eleq2i 2521	. . . . . . 7 |- ( <. w , z >. e. S <-> <. w , z >. e. { <. x , y >. | X R Y } )
56	29, 11, 12	nfbr 4447	. . . . . . . 8 |- F/ x [_ w / x ]_ X R Y
57		nfv 1761	. . . . . . . 8 |- F/ y [_ w / x ]_ X R [_ z / x ]_ X
58		vex 3048	. . . . . . . 8 |- z e. _V
59	31	breq1d 4412	. . . . . . . 8 |- ( x = w -> ( X R Y <-> [_ w / x ]_ X R Y ) )
60		csbeq1 3366	. . . . . . . . . 10 |- ( y = z -> [_ y / x ]_ X = [_ z / x ]_ X )
61	19, 60	syl5eqr 2499	. . . . . . . . 9 |- ( y = z -> Y = [_ z / x ]_ X )
62	61	breq2d 4414	. . . . . . . 8 |- ( y = z -> ( [_ w / x ]_ X R Y <-> [_ w / x ]_ X R [_ z / x ]_ X ) )
63	56, 57, 48, 58, 59, 62	opelopabf 4726	. . . . . . 7 |- ( <. w , z >. e. { <. x , y >. | X R Y } <-> [_ w / x ]_ X R [_ z / x ]_ X )
64	54, 55, 63	3bitri 275	. . . . . 6 |- ( w S z <-> [_ w / x ]_ X R [_ z / x ]_ X )
65	53, 64	anbi12i 703	. . . . 5 |- ( ( v S w /\ w S z ) <-> ( [_ v / x ]_ X R [_ w / x ]_ X /\ [_ w / x ]_ X R [_ z / x ]_ X ) )
66		df-br 4403	. . . . . 6 |- ( v S z <-> <. v , z >. e. S )
67	9	eleq2i 2521	. . . . . 6 |- ( <. v , z >. e. S <-> <. v , z >. e. { <. x , y >. | X R Y } )
68		nfv 1761	. . . . . . 7 |- F/ y [_ v / x ]_ X R [_ z / x ]_ X
69	61	breq2d 4414	. . . . . . 7 |- ( y = z -> ( [_ v / x ]_ X R Y <-> [_ v / x ]_ X R [_ z / x ]_ X ) )
70	13, 68, 15, 58, 16, 69	opelopabf 4726	. . . . . 6 |- ( <. v , z >. e. { <. x , y >. | X R Y } <-> [_ v / x ]_ X R [_ z / x ]_ X )
71	66, 67, 70	3bitri 275	. . . . 5 |- ( v S z <-> [_ v / x ]_ X R [_ z / x ]_ X )
72	44, 65, 71	3imtr4g 274	. . . 4 |- ( ( R Po B /\ ( [_ v / x ]_ X e. B /\ [_ w / x ]_ X e. B /\ [_ z / x ]_ X e. B ) ) -> ( ( v S w /\ w S z ) -> v S z ) )
73	72	adantlr 721	. . 3 |- ( ( ( R Po B /\ A. x e. A X e. B ) /\ ( [_ v / x ]_ X e. B /\ [_ w / x ]_ X e. B /\ [_ z / x ]_ X e. B ) ) -> ( ( v S w /\ w S z ) -> v S z ) )
74	43, 73	syldan 473	. 2 |- ( ( ( R Po B /\ A. x e. A X e. B ) /\ ( v e. A /\ w e. A /\ z e. A ) ) -> ( ( v S w /\ w S z ) -> v S z ) )
75	27, 74	ispod 4763	1 |- ( ( R Po B /\ A. x e. A X e. B ) -> S Po A )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 371   /\ w3a 985   = wceq 1444   e. wcel 1887   A.wral 2737   [_csb 3363   <.cop 3974   class class class wbr 4402   {copab 4460   Po wpo 4753

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-sep 4525  ax-nul 4534  ax-pr 4639

This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 987  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-ral 2742  df-rex 2743  df-rab 2746  df-v 3047  df-sbc 3268  df-csb 3364  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-nul 3732  df-if 3882  df-sn 3969  df-pr 3971  df-op 3975  df-br 4403  df-opab 4462  df-po 4755

This theorem is referenced by: (None)