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Theorem isosolem 6244
Description: Lemma for isoso 6245. (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 6242 . . 3  |-  ( H 
Isom  R ,  S  ( A ,  B )  ->  ( S  Po  B  ->  R  Po  A
) )
2 isof1o 6222 . . . . . . . 8  |-  ( H 
Isom  R ,  S  ( A ,  B )  ->  H : A -1-1-onto-> B
)
3 f1of 5822 . . . . . . . 8  |-  ( H : A -1-1-onto-> B  ->  H : A
--> B )
4 ffvelrn 6030 . . . . . . . . . 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 6030 . . . . . . . . . 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 4459 . . . . . . . 8  |-  ( a  =  ( H `  c )  ->  (
a S b  <->  ( H `  c ) S b ) )
12 eqeq1 2461 . . . . . . . 8  |-  ( a  =  ( H `  c )  ->  (
a  =  b  <->  ( H `  c )  =  b ) )
13 breq2 4460 . . . . . . . 8  |-  ( a  =  ( H `  c )  ->  (
b S a  <->  b S
( H `  c
) ) )
1411, 12, 133orbi123d 1298 . . . . . . 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 4460 . . . . . . . 8  |-  ( b  =  ( H `  d )  ->  (
( H `  c
) S b  <->  ( H `  c ) S ( H `  d ) ) )
16 eqeq2 2472 . . . . . . . 8  |-  ( b  =  ( H `  d )  ->  (
( H `  c
)  =  b  <->  ( H `  c )  =  ( H `  d ) ) )
17 breq1 4459 . . . . . . . 8  |-  ( b  =  ( H `  d )  ->  (
b S ( H `
 c )  <->  ( H `  d ) S ( H `  c ) ) )
1815, 16, 173orbi123d 1298 . . . . . . 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 3219 . . . . . 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 6223 . . . . . 6  |-  ( ( H  Isom  R ,  S  ( A ,  B )  /\  (
c  e.  A  /\  d  e.  A )
)  ->  ( c R d  <->  ( H `  c ) S ( H `  d ) ) )
22 f1of1 5821 . . . . . . . . 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 6171 . . . . . . . 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 6223 . . . . . . 7  |-  ( ( H  Isom  R ,  S  ( A ,  B )  /\  (
d  e.  A  /\  c  e.  A )
)  ->  ( d R c  <->  ( H `  d ) S ( H `  c ) ) )
2827ancom2s 802 . . . . . 6  |-  ( ( H  Isom  R ,  S  ( A ,  B )  /\  (
c  e.  A  /\  d  e.  A )
)  ->  ( d R c  <->  ( H `  d ) S ( H `  c ) ) )
2921, 26, 283orbi123d 1298 . . . . 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 2881 . . 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 4810 . 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 4810 . 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 972    = wceq 1395    e. wcel 1819   A.wral 2807   class class class wbr 4456    Po wpo 4807    Or wor 4808   -->wf 5590   -1-1->wf1 5591   -1-1-onto->wf1o 5593   ` cfv 5594    Isom wiso 5595
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pr 4695
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-sbc 3328  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-br 4457  df-opab 4516  df-id 4804  df-po 4809  df-so 4810  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-f1o 5601  df-fv 5602  df-isom 5603
This theorem is referenced by:  isoso  6245  isowe2  6247
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