Theorem isores1 5782

Description: An isomorphism from one well-order to another can be restricted on either well-order. (Contributed by Mario Carneiro, 15-Jan-2013.)

Assertion

Ref	Expression
isores1	|- ( H Isom R , S ( A , B ) <-> H Isom ( R i^i ( A X. A ) ) , S ( A , B ) )

Proof of Theorem isores1

Step	Hyp	Ref	Expression
1		isocnv 5779	. . . . 5 |- ( H Isom R , S ( A , B ) -> `' H Isom S , R ( B , A ) )
2		isores2 5781	. . . . 5 |- ( `' H Isom S , R ( B , A ) <-> `' H Isom S , ( R i^i ( A X. A ) ) ( B , A ) )
3	1, 2	sylib 121	. . . 4 |- ( H Isom R , S ( A , B ) -> `' H Isom S , ( R i^i ( A X. A ) ) ( B , A ) )
4		isocnv 5779	. . . 4 |- ( `' H Isom S , ( R i^i ( A X. A ) ) ( B , A ) -> `' `' H Isom ( R i^i ( A X. A ) ) , S ( A , B ) )
5	3, 4	syl 14	. . 3 |- ( H Isom R , S ( A , B ) -> `' `' H Isom ( R i^i ( A X. A ) ) , S ( A , B ) )
6		isof1o 5775	. . . 4 |- ( H Isom R , S ( A , B ) -> H : A -1-1-onto-> B )
7		f1orel 5435	. . . 4 |- ( H : A -1-1-onto-> B -> Rel H )
8		dfrel2 5054	. . . . 5 |- ( Rel H <-> `' `' H = H )
9		isoeq1 5769	. . . . 5 |- ( `' `' H = H -> ( `' `' H Isom ( R i^i ( A X. A ) ) , S ( A , B ) <-> H Isom ( R i^i ( A X. A ) ) , S ( A , B ) ) )
10	8, 9	sylbi 120	. . . 4 |- ( Rel H -> ( `' `' H Isom ( R i^i ( A X. A ) ) , S ( A , B ) <-> H Isom ( R i^i ( A X. A ) ) , S ( A , B ) ) )
11	6, 7, 10	3syl 17	. . 3 |- ( H Isom R , S ( A , B ) -> ( `' `' H Isom ( R i^i ( A X. A ) ) , S ( A , B ) <-> H Isom ( R i^i ( A X. A ) ) , S ( A , B ) ) )
12	5, 11	mpbid 146	. 2 |- ( H Isom R , S ( A , B ) -> H Isom ( R i^i ( A X. A ) ) , S ( A , B ) )
13		isocnv 5779	. . . . 5 |- ( H Isom ( R i^i ( A X. A ) ) , S ( A , B ) -> `' H Isom S , ( R i^i ( A X. A ) ) ( B , A ) )
14	13, 2	sylibr 133	. . . 4 |- ( H Isom ( R i^i ( A X. A ) ) , S ( A , B ) -> `' H Isom S , R ( B , A ) )
15		isocnv 5779	. . . 4 |- ( `' H Isom S , R ( B , A ) -> `' `' H Isom R , S ( A , B ) )
16	14, 15	syl 14	. . 3 |- ( H Isom ( R i^i ( A X. A ) ) , S ( A , B ) -> `' `' H Isom R , S ( A , B ) )
17		isof1o 5775	. . . 4 |- ( H Isom ( R i^i ( A X. A ) ) , S ( A , B ) -> H : A -1-1-onto-> B )
18		isoeq1 5769	. . . . 5 |- ( `' `' H = H -> ( `' `' H Isom R , S ( A , B ) <-> H Isom R , S ( A , B ) ) )
19	8, 18	sylbi 120	. . . 4 |- ( Rel H -> ( `' `' H Isom R , S ( A , B ) <-> H Isom R , S ( A , B ) ) )
20	17, 7, 19	3syl 17	. . 3 |- ( H Isom ( R i^i ( A X. A ) ) , S ( A , B ) -> ( `' `' H Isom R , S ( A , B ) <-> H Isom R , S ( A , B ) ) )
21	16, 20	mpbid 146	. 2 |- ( H Isom ( R i^i ( A X. A ) ) , S ( A , B ) -> H Isom R , S ( A , B ) )
22	12, 21	impbii 125	1 |- ( H Isom R , S ( A , B ) <-> H Isom ( R i^i ( A X. A ) ) , S ( A , B ) )

Syntax hints:   <-> wb 104   = wceq 1343   i^i cin 3115   X. cxp 4602   `'ccnv 4603   Rel wrel 4609   -1-1-onto->wf1o 5187   Isom wiso 5189

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-pow 4153  ax-pr 4187

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-rex 2450  df-v 2728  df-sbc 2952  df-un 3120  df-in 3122  df-ss 3129  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-br 3983  df-opab 4044  df-id 4271  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-rn 4615  df-res 4616  df-ima 4617  df-iota 5153  df-fun 5190  df-fn 5191  df-f 5192  df-f1 5193  df-fo 5194  df-f1o 5195  df-fv 5196  df-isom 5197

This theorem is referenced by: (None)