Theorem isofr 6592

Description: An isomorphism preserves well-foundedness. Proposition 6.32(1) of [TakeutiZaring] p. 33. (Contributed by NM, 30-Apr-2004.) (Revised by Mario Carneiro, 18-Nov-2014.)

Assertion

Ref	Expression
isofr	|- ( H Isom R , S ( A , B ) -> ( R Fr A <-> S Fr B ) )

Proof of Theorem isofr

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		isocnv 6580	. . 3 |- ( H Isom R , S ( A , B ) -> `' H Isom S , R ( B , A ) )
2		id 22	. . . 4 |- ( `' H Isom S , R ( B , A ) -> `' H Isom S , R ( B , A ) )
3		isof1o 6573	. . . . 5 |- ( `' H Isom S , R ( B , A ) -> `' H : B -1-1-onto-> A )
4		f1ofun 6139	. . . . 5 |- ( `' H : B -1-1-onto-> A -> Fun `' H )
5		vex 3203	. . . . . 6 |- x e. _V
6	5	funimaex 5976	. . . . 5 |- ( Fun `' H -> ( `' H " x ) e. _V )
7	3, 4, 6	3syl 18	. . . 4 |- ( `' H Isom S , R ( B , A ) -> ( `' H " x ) e. _V )
8	2, 7	isofrlem 6590	. . 3 |- ( `' H Isom S , R ( B , A ) -> ( R Fr A -> S Fr B ) )
9	1, 8	syl 17	. 2 |- ( H Isom R , S ( A , B ) -> ( R Fr A -> S Fr B ) )
10		id 22	. . 3 |- ( H Isom R , S ( A , B ) -> H Isom R , S ( A , B ) )
11		isof1o 6573	. . . 4 |- ( H Isom R , S ( A , B ) -> H : A -1-1-onto-> B )
12		f1ofun 6139	. . . 4 |- ( H : A -1-1-onto-> B -> Fun H )
13	5	funimaex 5976	. . . 4 |- ( Fun H -> ( H " x ) e. _V )
14	11, 12, 13	3syl 18	. . 3 |- ( H Isom R , S ( A , B ) -> ( H " x ) e. _V )
15	10, 14	isofrlem 6590	. 2 |- ( H Isom R , S ( A , B ) -> ( S Fr B -> R Fr A ) )
16	9, 15	impbid 202	1 |- ( H Isom R , S ( A , B ) -> ( R Fr A <-> S Fr B ) )

Syntax hints:   -> wi 4   <-> wb 196   e. wcel 1990   _Vcvv 3200   Fr wfr 5070   `'ccnv 5113   "cima 5117   Fun wfun 5882   -1-1-onto->wf1o 5887   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-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-fr 5073  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:  isowe  6599  wofib  8450  isfin1-4  9209