Theorem pofun 4409

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 3160	. . . . . . 7 |- F/_ x [_ v / x ]_ X
2	1	nfel1 2385	. . . . . 6 |- F/ x [_ v / x ]_ X e. B
3		csbeq1a 3136	. . . . . . 7 |- ( x = v -> X = [_ v / x ]_ X )
4	3	eleq1d 2300	. . . . . 6 |- ( x = v -> ( X e. B <-> [_ v / x ]_ X e. B ) )
5	2, 4	rspc 2904	. . . . 5 |- ( v e. A -> ( A. x e. A X e. B -> [_ v / x ]_ X e. B ) )
6	5	impcom 125	. . . 4 |- ( ( A. x e. A X e. B /\ v e. A ) -> [_ v / x ]_ X e. B )
7		poirr 4404	. . . . 5 |- ( ( R Po B /\ [_ v / x ]_ X e. B ) -> -. [_ v / x ]_ X R [_ v / x ]_ X )
8		df-br 4089	. . . . . 6 |- ( v S v <-> <. v , v >. e. S )
9		pofun.1	. . . . . . 7 |- S = { <. x , y >. | X R Y }
10	9	eleq2i 2298	. . . . . 6 |- ( <. v , v >. e. S <-> <. v , v >. e. { <. x , y >. | X R Y } )
11		nfcv 2374	. . . . . . . 8 |- F/_ x R
12		nfcv 2374	. . . . . . . 8 |- F/_ x Y
13	1, 11, 12	nfbr 4135	. . . . . . 7 |- F/ x [_ v / x ]_ X R Y
14		nfv 1576	. . . . . . 7 |- F/ y [_ v / x ]_ X R [_ v / x ]_ X
15		vex 2805	. . . . . . 7 |- v e. _V
16	3	breq1d 4098	. . . . . . 7 |- ( x = v -> ( X R Y <-> [_ v / x ]_ X R Y ) )
17		vex 2805	. . . . . . . . . 10 |- y e. _V
18		pofun.2	. . . . . . . . . 10 |- ( x = y -> X = Y )
19	17, 12, 18	csbief 3172	. . . . . . . . 9 |- [_ y / x ]_ X = Y
20		csbeq1 3130	. . . . . . . . 9 |- ( y = v -> [_ y / x ]_ X = [_ v / x ]_ X )
21	19, 20	eqtr3id 2278	. . . . . . . 8 |- ( y = v -> Y = [_ v / x ]_ X )
22	21	breq2d 4100	. . . . . . 7 |- ( y = v -> ( [_ v / x ]_ X R Y <-> [_ v / x ]_ X R [_ v / x ]_ X ) )
23	13, 14, 15, 15, 16, 22	opelopabf 4369	. . . . . 6 |- ( <. v , v >. e. { <. x , y >. | X R Y } <-> [_ v / x ]_ X R [_ v / x ]_ X )
24	8, 10, 23	3bitri 206	. . . . 5 |- ( v S v <-> [_ v / x ]_ X R [_ v / x ]_ X )
25	7, 24	sylnibr 683	. . . 4 |- ( ( R Po B /\ [_ v / x ]_ X e. B ) -> -. v S v )
26	6, 25	sylan2 286	. . 3 |- ( ( R Po B /\ ( A. x e. A X e. B /\ v e. A ) ) -> -. v S v )
27	26	anassrs 400	. 2 |- ( ( ( R Po B /\ A. x e. A X e. B ) /\ v e. A ) -> -. v S v )
28	5	com12 30	. . . . . 6 |- ( A. x e. A X e. B -> ( v e. A -> [_ v / x ]_ X e. B ) )
29		nfcsb1v 3160	. . . . . . . . 9 |- F/_ x [_ w / x ]_ X
30	29	nfel1 2385	. . . . . . . 8 |- F/ x [_ w / x ]_ X e. B
31		csbeq1a 3136	. . . . . . . . 9 |- ( x = w -> X = [_ w / x ]_ X )
32	31	eleq1d 2300	. . . . . . . 8 |- ( x = w -> ( X e. B <-> [_ w / x ]_ X e. B ) )
33	30, 32	rspc 2904	. . . . . . 7 |- ( w e. A -> ( A. x e. A X e. B -> [_ w / x ]_ X e. B ) )
34	33	com12 30	. . . . . 6 |- ( A. x e. A X e. B -> ( w e. A -> [_ w / x ]_ X e. B ) )
35		nfcsb1v 3160	. . . . . . . . 9 |- F/_ x [_ z / x ]_ X
36	35	nfel1 2385	. . . . . . . 8 |- F/ x [_ z / x ]_ X e. B
37		csbeq1a 3136	. . . . . . . . 9 |- ( x = z -> X = [_ z / x ]_ X )
38	37	eleq1d 2300	. . . . . . . 8 |- ( x = z -> ( X e. B <-> [_ z / x ]_ X e. B ) )
39	36, 38	rspc 2904	. . . . . . 7 |- ( z e. A -> ( A. x e. A X e. B -> [_ z / x ]_ X e. B ) )
40	39	com12 30	. . . . . 6 |- ( A. x e. A X e. B -> ( z e. A -> [_ z / x ]_ X e. B ) )
41	28, 34, 40	3anim123d 1355	. . . . 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 124	. . . 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 476	. . 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 4405	. . . . 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 4089	. . . . . . 7 |- ( v S w <-> <. v , w >. e. S )
46	9	eleq2i 2298	. . . . . . 7 |- ( <. v , w >. e. S <-> <. v , w >. e. { <. x , y >. | X R Y } )
47		nfv 1576	. . . . . . . 8 |- F/ y [_ v / x ]_ X R [_ w / x ]_ X
48		vex 2805	. . . . . . . 8 |- w e. _V
49		csbeq1 3130	. . . . . . . . . 10 |- ( y = w -> [_ y / x ]_ X = [_ w / x ]_ X )
50	19, 49	eqtr3id 2278	. . . . . . . . 9 |- ( y = w -> Y = [_ w / x ]_ X )
51	50	breq2d 4100	. . . . . . . 8 |- ( y = w -> ( [_ v / x ]_ X R Y <-> [_ v / x ]_ X R [_ w / x ]_ X ) )
52	13, 47, 15, 48, 16, 51	opelopabf 4369	. . . . . . 7 |- ( <. v , w >. e. { <. x , y >. | X R Y } <-> [_ v / x ]_ X R [_ w / x ]_ X )
53	45, 46, 52	3bitri 206	. . . . . 6 |- ( v S w <-> [_ v / x ]_ X R [_ w / x ]_ X )
54		df-br 4089	. . . . . . 7 |- ( w S z <-> <. w , z >. e. S )
55	9	eleq2i 2298	. . . . . . 7 |- ( <. w , z >. e. S <-> <. w , z >. e. { <. x , y >. | X R Y } )
56	29, 11, 12	nfbr 4135	. . . . . . . 8 |- F/ x [_ w / x ]_ X R Y
57		nfv 1576	. . . . . . . 8 |- F/ y [_ w / x ]_ X R [_ z / x ]_ X
58		vex 2805	. . . . . . . 8 |- z e. _V
59	31	breq1d 4098	. . . . . . . 8 |- ( x = w -> ( X R Y <-> [_ w / x ]_ X R Y ) )
60		csbeq1 3130	. . . . . . . . . 10 |- ( y = z -> [_ y / x ]_ X = [_ z / x ]_ X )
61	19, 60	eqtr3id 2278	. . . . . . . . 9 |- ( y = z -> Y = [_ z / x ]_ X )
62	61	breq2d 4100	. . . . . . . 8 |- ( y = z -> ( [_ w / x ]_ X R Y <-> [_ w / x ]_ X R [_ z / x ]_ X ) )
63	56, 57, 48, 58, 59, 62	opelopabf 4369	. . . . . . 7 |- ( <. w , z >. e. { <. x , y >. | X R Y } <-> [_ w / x ]_ X R [_ z / x ]_ X )
64	54, 55, 63	3bitri 206	. . . . . 6 |- ( w S z <-> [_ w / x ]_ X R [_ z / x ]_ X )
65	53, 64	anbi12i 460	. . . . 5 |- ( ( v S w /\ w S z ) <-> ( [_ v / x ]_ X R [_ w / x ]_ X /\ [_ w / x ]_ X R [_ z / x ]_ X ) )
66		df-br 4089	. . . . . 6 |- ( v S z <-> <. v , z >. e. S )
67	9	eleq2i 2298	. . . . . 6 |- ( <. v , z >. e. S <-> <. v , z >. e. { <. x , y >. | X R Y } )
68		nfv 1576	. . . . . . 7 |- F/ y [_ v / x ]_ X R [_ z / x ]_ X
69	61	breq2d 4100	. . . . . . 7 |- ( y = z -> ( [_ v / x ]_ X R Y <-> [_ v / x ]_ X R [_ z / x ]_ X ) )
70	13, 68, 15, 58, 16, 69	opelopabf 4369	. . . . . 6 |- ( <. v , z >. e. { <. x , y >. | X R Y } <-> [_ v / x ]_ X R [_ z / x ]_ X )
71	66, 67, 70	3bitri 206	. . . . 5 |- ( v S z <-> [_ v / x ]_ X R [_ z / x ]_ X )
72	44, 65, 71	3imtr4g 205	. . . 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 477	. . 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 282	. 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 4401	1 |- ( ( R Po B /\ A. x e. A X e. B ) -> S Po A )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   /\ w3a 1004   = wceq 1397   e. wcel 2202   A.wral 2510   [_csb 3127   <.cop 3672   class class class wbr 4088   {copab 4149   Po wpo 4391

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ral 2515  df-rex 2516  df-v 2804  df-sbc 3032  df-csb 3128  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-br 4089  df-opab 4151  df-po 4393

This theorem is referenced by: (None)