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Theorem isosolem 5860
Description: Lemma for isoso 5861. (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 5858 . . 3  |-  ( H 
Isom  R ,  S  ( A ,  B )  ->  ( S  Po  B  ->  R  Po  A
) )
2 isof1o 5838 . . . . . . . 8  |-  ( H 
Isom  R ,  S  ( A ,  B )  ->  H : A -1-1-onto-> B
)
3 f1of 5488 . . . . . . . 8  |-  ( H : A -1-1-onto-> B  ->  H : A
--> B )
4 ffvelrn 5679 . . . . . . . . . 10  |-  ( ( H : A --> B  /\  c  e.  A )  ->  ( H `  c
)  e.  B )
54ex 423 . . . . . . . . 9  |-  ( H : A --> B  -> 
( c  e.  A  ->  ( H `  c
)  e.  B ) )
6 ffvelrn 5679 . . . . . . . . . 10  |-  ( ( H : A --> B  /\  d  e.  A )  ->  ( H `  d
)  e.  B )
76ex 423 . . . . . . . . 9  |-  ( H : A --> B  -> 
( d  e.  A  ->  ( H `  d
)  e.  B ) )
85, 7anim12d 546 . . . . . . . 8  |-  ( H : A --> B  -> 
( ( c  e.  A  /\  d  e.  A )  ->  (
( H `  c
)  e.  B  /\  ( H `  d )  e.  B ) ) )
92, 3, 83syl 18 . . . . . . 7  |-  ( H 
Isom  R ,  S  ( A ,  B )  ->  ( ( c  e.  A  /\  d  e.  A )  ->  (
( H `  c
)  e.  B  /\  ( H `  d )  e.  B ) ) )
109imp 418 . . . . . 6  |-  ( ( H  Isom  R ,  S  ( A ,  B )  /\  (
c  e.  A  /\  d  e.  A )
)  ->  ( ( H `  c )  e.  B  /\  ( H `  d )  e.  B ) )
11 breq1 4042 . . . . . . . 8  |-  ( a  =  ( H `  c )  ->  (
a S b  <->  ( H `  c ) S b ) )
12 eqeq1 2302 . . . . . . . 8  |-  ( a  =  ( H `  c )  ->  (
a  =  b  <->  ( H `  c )  =  b ) )
13 breq2 4043 . . . . . . . 8  |-  ( a  =  ( H `  c )  ->  (
b S a  <->  b S
( H `  c
) ) )
1411, 12, 133orbi123d 1251 . . . . . . 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 4043 . . . . . . . 8  |-  ( b  =  ( H `  d )  ->  (
( H `  c
) S b  <->  ( H `  c ) S ( H `  d ) ) )
16 eqeq2 2305 . . . . . . . 8  |-  ( b  =  ( H `  d )  ->  (
( H `  c
)  =  b  <->  ( H `  c )  =  ( H `  d ) ) )
17 breq1 4042 . . . . . . . 8  |-  ( b  =  ( H `  d )  ->  (
b S ( H `
 c )  <->  ( H `  d ) S ( H `  c ) ) )
1815, 16, 173orbi123d 1251 . . . . . . 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 2903 . . . . . 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 15 . . . . 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 5839 . . . . . 6  |-  ( ( H  Isom  R ,  S  ( A ,  B )  /\  (
c  e.  A  /\  d  e.  A )
)  ->  ( c R d  <->  ( H `  c ) S ( H `  d ) ) )
22 f1of1 5487 . . . . . . . . 9  |-  ( H : A -1-1-onto-> B  ->  H : A -1-1-> B )
232, 22syl 15 . . . . . . . 8  |-  ( H 
Isom  R ,  S  ( A ,  B )  ->  H : A -1-1-> B )
24 f1fveq 5802 . . . . . . . 8  |-  ( ( H : A -1-1-> B  /\  ( c  e.  A  /\  d  e.  A
) )  ->  (
( H `  c
)  =  ( H `
 d )  <->  c  =  d ) )
2523, 24sylan 457 . . . . . . 7  |-  ( ( H  Isom  R ,  S  ( A ,  B )  /\  (
c  e.  A  /\  d  e.  A )
)  ->  ( ( H `  c )  =  ( H `  d )  <->  c  =  d ) )
2625bicomd 192 . . . . . 6  |-  ( ( H  Isom  R ,  S  ( A ,  B )  /\  (
c  e.  A  /\  d  e.  A )
)  ->  ( c  =  d  <->  ( H `  c )  =  ( H `  d ) ) )
27 isorel 5839 . . . . . . 7  |-  ( ( H  Isom  R ,  S  ( A ,  B )  /\  (
d  e.  A  /\  c  e.  A )
)  ->  ( d R c  <->  ( H `  d ) S ( H `  c ) ) )
2827ancom2s 777 . . . . . 6  |-  ( ( H  Isom  R ,  S  ( A ,  B )  /\  (
c  e.  A  /\  d  e.  A )
)  ->  ( d R c  <->  ( H `  d ) S ( H `  c ) ) )
2921, 26, 283orbi123d 1251 . . . . 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 225 . . . 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 2651 . . 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 546 . 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 4331 . 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 4331 . 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 261 1  |-  ( H 
Isom  R ,  S  ( A ,  B )  ->  ( S  Or  B  ->  R  Or  A
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    \/ w3o 933    = wceq 1632    e. wcel 1696   A.wral 2556   class class class wbr 4039    Po wpo 4328    Or wor 4329   -->wf 5267   -1-1->wf1 5268   -1-1-onto->wf1o 5270   ` cfv 5271    Isom wiso 5272
This theorem is referenced by:  isoso  5861  isowe2  5863
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-id 4325  df-po 4330  df-so 4331  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-f1o 5278  df-fv 5279  df-isom 5280
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