Theorem isores1 5861

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 5858	. . . . 5 |- ( H Isom R , S ( A , B ) -> `' H Isom S , R ( B , A ) )
2		isores2 5860	. . . . 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 5858	. . . 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 5854	. . . 4 |- ( H Isom R , S ( A , B ) -> H : A -1-1-onto-> B )
7		f1orel 5507	. . . 4 |- ( H : A -1-1-onto-> B -> Rel H )
8		dfrel2 5120	. . . . 5 |- ( Rel H <-> `' `' H = H )
9		isoeq1 5848	. . . . 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 5858	. . . . 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 5858	. . . 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 5854	. . . 4 |- ( H Isom ( R i^i ( A X. A ) ) , S ( A , B ) -> H : A -1-1-onto-> B )
18		isoeq1 5848	. . . . 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 1364   i^i cin 3156   X. cxp 4661   `'ccnv 4662   Rel wrel 4668   -1-1-onto->wf1o 5257   Isom wiso 5259

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 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-14 2170  ax-ext 2178  ax-sep 4151  ax-pow 4207  ax-pr 4242

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480  df-rex 2481  df-v 2765  df-sbc 2990  df-un 3161  df-in 3163  df-ss 3170  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-uni 3840  df-br 4034  df-opab 4095  df-id 4328  df-xp 4669  df-rel 4670  df-cnv 4671  df-co 4672  df-dm 4673  df-rn 4674  df-res 4675  df-ima 4676  df-iota 5219  df-fun 5260  df-fn 5261  df-f 5262  df-f1 5263  df-fo 5264  df-f1o 5265  df-fv 5266  df-isom 5267

This theorem is referenced by: (None)