Theorem isocnv2 6202

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 3015	. . . 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 3109	. . . . . . 7 |- x e. _V
3		vex 3109	. . . . . . 7 |- y e. _V
4	2, 3	brcnv 5174	. . . . . 6 |- ( x `' R y <-> y R x )
5		fvex 5858	. . . . . . 7 |- ( H ` x ) e. _V
6		fvex 5858	. . . . . . 7 |- ( H ` y ) e. _V
7	5, 6	brcnv 5174	. . . . . 6 |- ( ( H ` x ) `' S ( H ` y ) <-> ( H ` y ) S ( H ` x ) )
8	4, 7	bibi12i 313	. . . . 5 |- ( ( x `' R y <-> ( H ` x ) `' S ( H ` y ) ) <-> ( y R x <-> ( H ` y ) S ( H ` x ) ) )
9	8	2ralbii 2886	. . . 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 252	. . 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 692	. 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 5579	. 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 5579	. 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 277	1 |- ( H Isom R , S ( A , B ) <-> H Isom `' R , `' S ( A , B ) )

Syntax hints:   <-> wb 184   /\ wa 367   A.wral 2804   class class class wbr 4439   `'ccnv 4987   -1-1-onto->wf1o 5569   ` cfv 5570   Isom wiso 5571

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pr 4676

This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-rab 2813  df-v 3108  df-sbc 3325  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-br 4440  df-opab 4498  df-cnv 4996  df-iota 5534  df-fv 5578  df-isom 5579

This theorem is referenced by:  wofib  7962  leiso  12492  gtiso  27747