Theorem isores1 5723

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 5720	. . . . 5 |- ( H Isom R , S ( A , B ) -> `' H Isom S , R ( B , A ) )
2		isores2 5722	. . . . 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 5720	. . . 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 5716	. . . 4 |- ( H Isom R , S ( A , B ) -> H : A -1-1-onto-> B )
7		f1orel 5378	. . . 4 |- ( H : A -1-1-onto-> B -> Rel H )
8		dfrel2 4997	. . . . 5 |- ( Rel H <-> `' `' H = H )
9		isoeq1 5710	. . . . 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 5720	. . . . 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 5720	. . . 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 5716	. . . 4 |- ( H Isom ( R i^i ( A X. A ) ) , S ( A , B ) -> H : A -1-1-onto-> B )
18		isoeq1 5710	. . . . 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 1332   i^i cin 3075   X. cxp 4545   `'ccnv 4546   Rel wrel 4552   -1-1-onto->wf1o 5130   Isom wiso 5132

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 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4054  ax-pow 4106  ax-pr 4139

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-rex 2423  df-v 2691  df-sbc 2914  df-un 3080  df-in 3082  df-ss 3089  df-pw 3517  df-sn 3538  df-pr 3539  df-op 3541  df-uni 3745  df-br 3938  df-opab 3998  df-id 4223  df-xp 4553  df-rel 4554  df-cnv 4555  df-co 4556  df-dm 4557  df-rn 4558  df-res 4559  df-ima 4560  df-iota 5096  df-fun 5133  df-fn 5134  df-f 5135  df-f1 5136  df-fo 5137  df-f1o 5138  df-fv 5139  df-isom 5140

This theorem is referenced by: (None)