Theorem isocnv2 6581

Description: Converse law for isomorphism. (Contributed by Mario Carneiro, 30-Jan-2014.)

Assertion

Ref	Expression
isocnv2	|- ( H Isom R , S ( A , B ) <-> H Isom `' R , `' S ( A , B ) )

Proof of Theorem isocnv2

Dummy variables x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ralcom 3098	. . . 4 |- ( A. y e. A A. x e. A ( y R x <-> ( H ` y ) S ( H ` x ) ) <-> A. x e. A A. y e. A ( y R x <-> ( H ` y ) S ( H ` x ) ) )
2		vex 3203	. . . . . . 7 |- x e. _V
3		vex 3203	. . . . . . 7 |- y e. _V
4	2, 3	brcnv 5305	. . . . . 6 |- ( x `' R y <-> y R x )
5		fvex 6201	. . . . . . 7 |- ( H ` x ) e. _V
6		fvex 6201	. . . . . . 7 |- ( H ` y ) e. _V
7	5, 6	brcnv 5305	. . . . . 6 |- ( ( H ` x ) `' S ( H ` y ) <-> ( H ` y ) S ( H ` x ) )
8	4, 7	bibi12i 329	. . . . 5 |- ( ( x `' R y <-> ( H ` x ) `' S ( H ` y ) ) <-> ( y R x <-> ( H ` y ) S ( H ` x ) ) )
9	8	2ralbii 2981	. . . 4 |- ( A. x e. A A. y e. A ( x `' R y <-> ( H ` x ) `' S ( H ` y ) ) <-> A. x e. A A. y e. A ( y R x <-> ( H ` y ) S ( H ` x ) ) )
10	1, 9	bitr4i 267	. . 3 |- ( A. y e. A A. x e. A ( y R x <-> ( H ` y ) S ( H ` x ) ) <-> A. x e. A A. y e. A ( x `' R y <-> ( H ` x ) `' S ( H ` y ) ) )
11	10	anbi2i 730	. 2 |- ( ( H : A -1-1-onto-> B /\ A. y e. A A. x e. A ( y R x <-> ( H ` y ) S ( H ` x ) ) ) <-> ( H : A -1-1-onto-> B /\ A. x e. A A. y e. A ( x `' R y <-> ( H ` x ) `' S ( H ` y ) ) ) )
12		df-isom 5897	. 2 |- ( H Isom R , S ( A , B ) <-> ( H : A -1-1-onto-> B /\ A. y e. A A. x e. A ( y R x <-> ( H ` y ) S ( H ` x ) ) ) )
13		df-isom 5897	. 2 |- ( H Isom `' R , `' S ( A , B ) <-> ( H : A -1-1-onto-> B /\ A. x e. A A. y e. A ( x `' R y <-> ( H ` x ) `' S ( H ` y ) ) ) )
14	11, 12, 13	3bitr4i 292	1 |- ( H Isom R , S ( A , B ) <-> H Isom `' R , `' S ( A , B ) )

Syntax hints:   <-> wb 196   /\ wa 384   A.wral 2912   class class class wbr 4653   `'ccnv 5113   -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-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  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-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-cnv 5122  df-iota 5851  df-fv 5896  df-isom 5897

This theorem is referenced by:  infiso  8413  wofib  8450  leiso  13243  gtiso  29478