Theorem f1owe 6603

Description: Well-ordering of isomorphic relations. (Contributed by NM, 4-Mar-1997.)

Hypothesis

Ref	Expression
f1owe.1	|- R = { <. x , y >. | ( F ` x ) S ( F ` y ) }

Assertion

Ref	Expression
f1owe	|- ( F : A -1-1-onto-> B -> ( S We B -> R We A ) )

Distinct variable groups:   x, y, S   x, F, y

Allowed substitution hints:   A( x, y)   B( x, y)   R( x, y)

Proof of Theorem f1owe

Dummy variables z w are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		fveq2 6191	. . . . . 6 |- ( x = z -> ( F ` x ) = ( F ` z ) )
2	1	breq1d 4663	. . . . 5 |- ( x = z -> ( ( F ` x ) S ( F ` y ) <-> ( F ` z ) S ( F ` y ) ) )
3		fveq2 6191	. . . . . 6 |- ( y = w -> ( F ` y ) = ( F ` w ) )
4	3	breq2d 4665	. . . . 5 |- ( y = w -> ( ( F ` z ) S ( F ` y ) <-> ( F ` z ) S ( F ` w ) ) )
5		f1owe.1	. . . . 5 |- R = { <. x , y >. | ( F ` x ) S ( F ` y ) }
6	2, 4, 5	brabg 4994	. . . 4 |- ( ( z e. A /\ w e. A ) -> ( z R w <-> ( F ` z ) S ( F ` w ) ) )
7	6	rgen2a 2977	. . 3 |- A. z e. A A. w e. A ( z R w <-> ( F ` z ) S ( F ` w ) )
8		df-isom 5897	. . . 4 |- ( F Isom R , S ( A , B ) <-> ( F : A -1-1-onto-> B /\ A. z e. A A. w e. A ( z R w <-> ( F ` z ) S ( F ` w ) ) ) )
9		isowe 6599	. . . 4 |- ( F Isom R , S ( A , B ) -> ( R We A <-> S We B ) )
10	8, 9	sylbir 225	. . 3 |- ( ( F : A -1-1-onto-> B /\ A. z e. A A. w e. A ( z R w <-> ( F ` z ) S ( F ` w ) ) ) -> ( R We A <-> S We B ) )
11	7, 10	mpan2 707	. 2 |- ( F : A -1-1-onto-> B -> ( R We A <-> S We B ) )
12	11	biimprd 238	1 |- ( F : A -1-1-onto-> B -> ( S We B -> R We A ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   A.wral 2912   class class class wbr 4653   {copab 4712   We wwe 5072   -1-1-onto->wf1o 5887   ` cfv 5888   Isom wiso 5889

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-id 5024  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-isom 5897

This theorem is referenced by:  wemapwe  8594  dfac8b  8854  ac10ct  8857  dnwech  37618