Theorem isores1 5454

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 5451	. . . . 5 |- ( H Isom R , S ( A , B ) -> `' H Isom S , R ( B , A ) )
2		isores2 5453	. . . . 5 |- ( `' H Isom S , R ( B , A ) <-> `' H Isom S , ( R i^i ( A X. A ) ) ( B , A ) )
3	1, 2	sylib 127	. . . 4 |- ( H Isom R , S ( A , B ) -> `' H Isom S , ( R i^i ( A X. A ) ) ( B , A ) )
4		isocnv 5451	. . . 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 5447	. . . 4 |- ( H Isom R , S ( A , B ) -> H : A -1-1-onto-> B )
7		f1orel 5129	. . . 4 |- ( H : A -1-1-onto-> B -> Rel H )
8		dfrel2 4771	. . . . 5 |- ( Rel H <-> `' `' H = H )
9		isoeq1 5441	. . . . 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 114	. . . 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 135	. 2 |- ( H Isom R , S ( A , B ) -> H Isom ( R i^i ( A X. A ) ) , S ( A , B ) )
13		isocnv 5451	. . . . 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 137	. . . 4 |- ( H Isom ( R i^i ( A X. A ) ) , S ( A , B ) -> `' H Isom S , R ( B , A ) )
15		isocnv 5451	. . . 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 5447	. . . 4 |- ( H Isom ( R i^i ( A X. A ) ) , S ( A , B ) -> H : A -1-1-onto-> B )
18		isoeq1 5441	. . . . 5 |- ( `' `' H = H -> ( `' `' H Isom R , S ( A , B ) <-> H Isom R , S ( A , B ) ) )
19	8, 18	sylbi 114	. . . 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 135	. 2 |- ( H Isom ( R i^i ( A X. A ) ) , S ( A , B ) -> H Isom R , S ( A , B ) )
22	12, 21	impbii 117	1 |- ( H Isom R , S ( A , B ) <-> H Isom ( R i^i ( A X. A ) ) , S ( A , B ) )

Syntax hints:   <-> wb 98   = wceq 1243   i^i cin 2916   X. cxp 4343   `'ccnv 4344   Rel wrel 4350   -1-1-onto->wf1o 4901   Isom wiso 4903

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944

This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-v 2559  df-sbc 2765  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-br 3765  df-opab 3819  df-id 4030  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-rn 4356  df-res 4357  df-ima 4358  df-iota 4867  df-fun 4904  df-fn 4905  df-f 4906  df-f1 4907  df-fo 4908  df-f1o 4909  df-fv 4910  df-isom 4911

This theorem is referenced by: (None)