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Theorem isosolem 6067
Description: Lemma for isoso 6068. (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 6065 . . 3  |-  ( H 
Isom  R ,  S  ( A ,  B )  ->  ( S  Po  B  ->  R  Po  A
) )
2 isof1o 6045 . . . . . . . 8  |-  ( H 
Isom  R ,  S  ( A ,  B )  ->  H : A -1-1-onto-> B
)
3 f1of 5674 . . . . . . . 8  |-  ( H : A -1-1-onto-> B  ->  H : A
--> B )
4 ffvelrn 5868 . . . . . . . . . 10  |-  ( ( H : A --> B  /\  c  e.  A )  ->  ( H `  c
)  e.  B )
54ex 424 . . . . . . . . 9  |-  ( H : A --> B  -> 
( c  e.  A  ->  ( H `  c
)  e.  B ) )
6 ffvelrn 5868 . . . . . . . . . 10  |-  ( ( H : A --> B  /\  d  e.  A )  ->  ( H `  d
)  e.  B )
76ex 424 . . . . . . . . 9  |-  ( H : A --> B  -> 
( d  e.  A  ->  ( H `  d
)  e.  B ) )
85, 7anim12d 547 . . . . . . . 8  |-  ( H : A --> B  -> 
( ( c  e.  A  /\  d  e.  A )  ->  (
( H `  c
)  e.  B  /\  ( H `  d )  e.  B ) ) )
92, 3, 83syl 19 . . . . . . 7  |-  ( H 
Isom  R ,  S  ( A ,  B )  ->  ( ( c  e.  A  /\  d  e.  A )  ->  (
( H `  c
)  e.  B  /\  ( H `  d )  e.  B ) ) )
109imp 419 . . . . . 6  |-  ( ( H  Isom  R ,  S  ( A ,  B )  /\  (
c  e.  A  /\  d  e.  A )
)  ->  ( ( H `  c )  e.  B  /\  ( H `  d )  e.  B ) )
11 breq1 4215 . . . . . . . 8  |-  ( a  =  ( H `  c )  ->  (
a S b  <->  ( H `  c ) S b ) )
12 eqeq1 2442 . . . . . . . 8  |-  ( a  =  ( H `  c )  ->  (
a  =  b  <->  ( H `  c )  =  b ) )
13 breq2 4216 . . . . . . . 8  |-  ( a  =  ( H `  c )  ->  (
b S a  <->  b S
( H `  c
) ) )
1411, 12, 133orbi123d 1253 . . . . . . 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 4216 . . . . . . . 8  |-  ( b  =  ( H `  d )  ->  (
( H `  c
) S b  <->  ( H `  c ) S ( H `  d ) ) )
16 eqeq2 2445 . . . . . . . 8  |-  ( b  =  ( H `  d )  ->  (
( H `  c
)  =  b  <->  ( H `  c )  =  ( H `  d ) ) )
17 breq1 4215 . . . . . . . 8  |-  ( b  =  ( H `  d )  ->  (
b S ( H `
 c )  <->  ( H `  d ) S ( H `  c ) ) )
1815, 16, 173orbi123d 1253 . . . . . . 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 3058 . . . . . 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 6046 . . . . . 6  |-  ( ( H  Isom  R ,  S  ( A ,  B )  /\  (
c  e.  A  /\  d  e.  A )
)  ->  ( c R d  <->  ( H `  c ) S ( H `  d ) ) )
22 f1of1 5673 . . . . . . . . 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 6008 . . . . . . . 8  |-  ( ( H : A -1-1-> B  /\  ( c  e.  A  /\  d  e.  A
) )  ->  (
( H `  c
)  =  ( H `
 d )  <->  c  =  d ) )
2523, 24sylan 458 . . . . . . 7  |-  ( ( H  Isom  R ,  S  ( A ,  B )  /\  (
c  e.  A  /\  d  e.  A )
)  ->  ( ( H `  c )  =  ( H `  d )  <->  c  =  d ) )
2625bicomd 193 . . . . . 6  |-  ( ( H  Isom  R ,  S  ( A ,  B )  /\  (
c  e.  A  /\  d  e.  A )
)  ->  ( c  =  d  <->  ( H `  c )  =  ( H `  d ) ) )
27 isorel 6046 . . . . . . 7  |-  ( ( H  Isom  R ,  S  ( A ,  B )  /\  (
d  e.  A  /\  c  e.  A )
)  ->  ( d R c  <->  ( H `  d ) S ( H `  c ) ) )
2827ancom2s 778 . . . . . 6  |-  ( ( H  Isom  R ,  S  ( A ,  B )  /\  (
c  e.  A  /\  d  e.  A )
)  ->  ( d R c  <->  ( H `  d ) S ( H `  c ) ) )
2921, 26, 283orbi123d 1253 . . . . 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 226 . . . 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 2801 . . 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 547 . 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 4504 . 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 4504 . 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 262 1  |-  ( H 
Isom  R ,  S  ( A ,  B )  ->  ( S  Or  B  ->  R  Or  A
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    \/ w3o 935    = wceq 1652    e. wcel 1725   A.wral 2705   class class class wbr 4212    Po wpo 4501    Or wor 4502   -->wf 5450   -1-1->wf1 5451   -1-1-onto->wf1o 5453   ` cfv 5454    Isom wiso 5455
This theorem is referenced by:  isoso  6068  isowe2  6070
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pr 4403
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-sbc 3162  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-br 4213  df-opab 4267  df-id 4498  df-po 4503  df-so 4504  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-f1o 5461  df-fv 5462  df-isom 5463
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