Theorem isores1 5965

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 5962	. . . . 5 |- ( H Isom R , S ( A , B ) -> `' H Isom S , R ( B , A ) )
2		isores2 5964	. . . . 5 |- ( `' H Isom S , R ( B , A ) <-> `' H Isom S , ( R i^i ( A X. A ) ) ( B , A ) )
3	1, 2	sylib 122	. . . 4 |- ( H Isom R , S ( A , B ) -> `' H Isom S , ( R i^i ( A X. A ) ) ( B , A ) )
4		isocnv 5962	. . . 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 5958	. . . 4 |- ( H Isom R , S ( A , B ) -> H : A -1-1-onto-> B )
7		f1orel 5595	. . . 4 |- ( H : A -1-1-onto-> B -> Rel H )
8		dfrel2 5194	. . . . 5 |- ( Rel H <-> `' `' H = H )
9		isoeq1 5952	. . . . 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 121	. . . 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 147	. 2 |- ( H Isom R , S ( A , B ) -> H Isom ( R i^i ( A X. A ) ) , S ( A , B ) )
13		isocnv 5962	. . . . 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 134	. . . 4 |- ( H Isom ( R i^i ( A X. A ) ) , S ( A , B ) -> `' H Isom S , R ( B , A ) )
15		isocnv 5962	. . . 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 5958	. . . 4 |- ( H Isom ( R i^i ( A X. A ) ) , S ( A , B ) -> H : A -1-1-onto-> B )
18		isoeq1 5952	. . . . 5 |- ( `' `' H = H -> ( `' `' H Isom R , S ( A , B ) <-> H Isom R , S ( A , B ) ) )
19	8, 18	sylbi 121	. . . 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 147	. 2 |- ( H Isom ( R i^i ( A X. A ) ) , S ( A , B ) -> H Isom R , S ( A , B ) )
22	12, 21	impbii 126	1 |- ( H Isom R , S ( A , B ) <-> H Isom ( R i^i ( A X. A ) ) , S ( A , B ) )

Syntax hints:   <-> wb 105   = wceq 1398   i^i cin 3200   X. cxp 4729   `'ccnv 4730   Rel wrel 4736   -1-1-onto->wf1o 5332   Isom wiso 5334

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2205  ax-ext 2213  ax-sep 4212  ax-pow 4270  ax-pr 4305

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ral 2516  df-rex 2517  df-v 2805  df-sbc 3033  df-un 3205  df-in 3207  df-ss 3214  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-br 4094  df-opab 4156  df-id 4396  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-res 4743  df-ima 4744  df-iota 5293  df-fun 5335  df-fn 5336  df-f 5337  df-f1 5338  df-fo 5339  df-f1o 5340  df-fv 5341  df-isom 5342

This theorem is referenced by: (None)