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Theorem isosolem 6146
Description: Lemma for isoso 6147. (Contributed by Stefan O'Rear, 16-Nov-2014.)
Assertion
Ref Expression
isosolem  |-  ( H 
Isom  R ,  S  ( A ,  B )  ->  ( S  Or  B  ->  R  Or  A
) )

Proof of Theorem isosolem
Dummy variables  a 
b  c  d are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 isopolem 6144 . . 3  |-  ( H 
Isom  R ,  S  ( A ,  B )  ->  ( S  Po  B  ->  R  Po  A
) )
2 isof1o 6124 . . . . . . . 8  |-  ( H 
Isom  R ,  S  ( A ,  B )  ->  H : A -1-1-onto-> B
)
3 f1of 5748 . . . . . . . 8  |-  ( H : A -1-1-onto-> B  ->  H : A
--> B )
4 ffvelrn 5949 . . . . . . . . . 10  |-  ( ( H : A --> B  /\  c  e.  A )  ->  ( H `  c
)  e.  B )
54ex 434 . . . . . . . . 9  |-  ( H : A --> B  -> 
( c  e.  A  ->  ( H `  c
)  e.  B ) )
6 ffvelrn 5949 . . . . . . . . . 10  |-  ( ( H : A --> B  /\  d  e.  A )  ->  ( H `  d
)  e.  B )
76ex 434 . . . . . . . . 9  |-  ( H : A --> B  -> 
( d  e.  A  ->  ( H `  d
)  e.  B ) )
85, 7anim12d 563 . . . . . . . 8  |-  ( H : A --> B  -> 
( ( c  e.  A  /\  d  e.  A )  ->  (
( H `  c
)  e.  B  /\  ( H `  d )  e.  B ) ) )
92, 3, 83syl 20 . . . . . . 7  |-  ( H 
Isom  R ,  S  ( A ,  B )  ->  ( ( c  e.  A  /\  d  e.  A )  ->  (
( H `  c
)  e.  B  /\  ( H `  d )  e.  B ) ) )
109imp 429 . . . . . 6  |-  ( ( H  Isom  R ,  S  ( A ,  B )  /\  (
c  e.  A  /\  d  e.  A )
)  ->  ( ( H `  c )  e.  B  /\  ( H `  d )  e.  B ) )
11 breq1 4402 . . . . . . . 8  |-  ( a  =  ( H `  c )  ->  (
a S b  <->  ( H `  c ) S b ) )
12 eqeq1 2458 . . . . . . . 8  |-  ( a  =  ( H `  c )  ->  (
a  =  b  <->  ( H `  c )  =  b ) )
13 breq2 4403 . . . . . . . 8  |-  ( a  =  ( H `  c )  ->  (
b S a  <->  b S
( H `  c
) ) )
1411, 12, 133orbi123d 1289 . . . . . . 7  |-  ( a  =  ( H `  c )  ->  (
( a S b  \/  a  =  b  \/  b S a )  <->  ( ( H `
 c ) S b  \/  ( H `
 c )  =  b  \/  b S ( H `  c
) ) ) )
15 breq2 4403 . . . . . . . 8  |-  ( b  =  ( H `  d )  ->  (
( H `  c
) S b  <->  ( H `  c ) S ( H `  d ) ) )
16 eqeq2 2469 . . . . . . . 8  |-  ( b  =  ( H `  d )  ->  (
( H `  c
)  =  b  <->  ( H `  c )  =  ( H `  d ) ) )
17 breq1 4402 . . . . . . . 8  |-  ( b  =  ( H `  d )  ->  (
b S ( H `
 c )  <->  ( H `  d ) S ( H `  c ) ) )
1815, 16, 173orbi123d 1289 . . . . . . 7  |-  ( b  =  ( H `  d )  ->  (
( ( H `  c ) S b  \/  ( H `  c )  =  b  \/  b S ( H `  c ) )  <->  ( ( H `
 c ) S ( H `  d
)  \/  ( H `
 c )  =  ( H `  d
)  \/  ( H `
 d ) S ( H `  c
) ) ) )
1914, 18rspc2v 3184 . . . . . 6  |-  ( ( ( H `  c
)  e.  B  /\  ( H `  d )  e.  B )  -> 
( A. a  e.  B  A. b  e.  B  ( a S b  \/  a  =  b  \/  b S a )  ->  (
( H `  c
) S ( H `
 d )  \/  ( H `  c
)  =  ( H `
 d )  \/  ( H `  d
) S ( H `
 c ) ) ) )
2010, 19syl 16 . . . . 5  |-  ( ( H  Isom  R ,  S  ( A ,  B )  /\  (
c  e.  A  /\  d  e.  A )
)  ->  ( A. a  e.  B  A. b  e.  B  (
a S b  \/  a  =  b  \/  b S a )  ->  ( ( H `
 c ) S ( H `  d
)  \/  ( H `
 c )  =  ( H `  d
)  \/  ( H `
 d ) S ( H `  c
) ) ) )
21 isorel 6125 . . . . . 6  |-  ( ( H  Isom  R ,  S  ( A ,  B )  /\  (
c  e.  A  /\  d  e.  A )
)  ->  ( c R d  <->  ( H `  c ) S ( H `  d ) ) )
22 f1of1 5747 . . . . . . . . 9  |-  ( H : A -1-1-onto-> B  ->  H : A -1-1-> B )
232, 22syl 16 . . . . . . . 8  |-  ( H 
Isom  R ,  S  ( A ,  B )  ->  H : A -1-1-> B )
24 f1fveq 6083 . . . . . . . 8  |-  ( ( H : A -1-1-> B  /\  ( c  e.  A  /\  d  e.  A
) )  ->  (
( H `  c
)  =  ( H `
 d )  <->  c  =  d ) )
2523, 24sylan 471 . . . . . . 7  |-  ( ( H  Isom  R ,  S  ( A ,  B )  /\  (
c  e.  A  /\  d  e.  A )
)  ->  ( ( H `  c )  =  ( H `  d )  <->  c  =  d ) )
2625bicomd 201 . . . . . 6  |-  ( ( H  Isom  R ,  S  ( A ,  B )  /\  (
c  e.  A  /\  d  e.  A )
)  ->  ( c  =  d  <->  ( H `  c )  =  ( H `  d ) ) )
27 isorel 6125 . . . . . . 7  |-  ( ( H  Isom  R ,  S  ( A ,  B )  /\  (
d  e.  A  /\  c  e.  A )
)  ->  ( d R c  <->  ( H `  d ) S ( H `  c ) ) )
2827ancom2s 800 . . . . . 6  |-  ( ( H  Isom  R ,  S  ( A ,  B )  /\  (
c  e.  A  /\  d  e.  A )
)  ->  ( d R c  <->  ( H `  d ) S ( H `  c ) ) )
2921, 26, 283orbi123d 1289 . . . . 5  |-  ( ( H  Isom  R ,  S  ( A ,  B )  /\  (
c  e.  A  /\  d  e.  A )
)  ->  ( (
c R d  \/  c  =  d  \/  d R c )  <-> 
( ( H `  c ) S ( H `  d )  \/  ( H `  c )  =  ( H `  d )  \/  ( H `  d ) S ( H `  c ) ) ) )
3020, 29sylibrd 234 . . . 4  |-  ( ( H  Isom  R ,  S  ( A ,  B )  /\  (
c  e.  A  /\  d  e.  A )
)  ->  ( A. a  e.  B  A. b  e.  B  (
a S b  \/  a  =  b  \/  b S a )  ->  ( c R d  \/  c  =  d  \/  d R c ) ) )
3130ralrimdvva 2915 . . 3  |-  ( H 
Isom  R ,  S  ( A ,  B )  ->  ( A. a  e.  B  A. b  e.  B  ( a S b  \/  a  =  b  \/  b S a )  ->  A. c  e.  A  A. d  e.  A  ( c R d  \/  c  =  d  \/  d R c ) ) )
321, 31anim12d 563 . 2  |-  ( H 
Isom  R ,  S  ( A ,  B )  ->  ( ( S  Po  B  /\  A. a  e.  B  A. b  e.  B  (
a S b  \/  a  =  b  \/  b S a ) )  ->  ( R  Po  A  /\  A. c  e.  A  A. d  e.  A  ( c R d  \/  c  =  d  \/  d R c ) ) ) )
33 df-so 4749 . 2  |-  ( S  Or  B  <->  ( S  Po  B  /\  A. a  e.  B  A. b  e.  B  ( a S b  \/  a  =  b  \/  b S a ) ) )
34 df-so 4749 . 2  |-  ( R  Or  A  <->  ( R  Po  A  /\  A. c  e.  A  A. d  e.  A  ( c R d  \/  c  =  d  \/  d R c ) ) )
3532, 33, 343imtr4g 270 1  |-  ( H 
Isom  R ,  S  ( A ,  B )  ->  ( S  Or  B  ->  R  Or  A
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    \/ w3o 964    = wceq 1370    e. wcel 1758   A.wral 2798   class class class wbr 4399    Po wpo 4746    Or wor 4747   -->wf 5521   -1-1->wf1 5522   -1-1-onto->wf1o 5524   ` cfv 5525    Isom wiso 5526
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4520  ax-nul 4528  ax-pr 4638
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2649  df-ral 2803  df-rex 2804  df-rab 2807  df-v 3078  df-sbc 3293  df-dif 3438  df-un 3440  df-in 3442  df-ss 3449  df-nul 3745  df-if 3899  df-sn 3985  df-pr 3987  df-op 3991  df-uni 4199  df-br 4400  df-opab 4458  df-id 4743  df-po 4748  df-so 4749  df-xp 4953  df-rel 4954  df-cnv 4955  df-co 4956  df-dm 4957  df-rn 4958  df-iota 5488  df-fun 5527  df-fn 5528  df-f 5529  df-f1 5530  df-f1o 5532  df-fv 5533  df-isom 5534
This theorem is referenced by:  isoso  6147  isowe2  6149
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