Theorem marypha2lem2 8342

Description: Lemma for marypha2 8345. Properties of the used relation. (Contributed by Stefan O'Rear, 20-Feb-2015.)

Hypothesis

Ref	Expression
marypha2lem.t	|- T = U_ x e. A ( { x } X. ( F ` x ) )

Assertion

Ref	Expression
marypha2lem2	|- T = { <. x , y >. | ( x e. A /\ y e. ( F ` x ) ) }

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

Allowed substitution hints:   T( x, y)

Proof of Theorem marypha2lem2

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		marypha2lem.t	. 2 |- T = U_ x e. A ( { x } X. ( F ` x ) )
2		sneq 4187	. . . 4 |- ( x = z -> { x } = { z } )
3		fveq2 6191	. . . 4 |- ( x = z -> ( F ` x ) = ( F ` z ) )
4	2, 3	xpeq12d 5140	. . 3 |- ( x = z -> ( { x } X. ( F ` x ) ) = ( { z } X. ( F ` z ) ) )
5	4	cbviunv 4559	. 2 |- U_ x e. A ( { x } X. ( F ` x ) ) = U_ z e. A ( { z } X. ( F ` z ) )
6		df-xp 5120	. . . . 5 |- ( { z } X. ( F ` z ) ) = { <. x , y >. | ( x e. { z } /\ y e. ( F ` z ) ) }
7	6	a1i 11	. . . 4 |- ( z e. A -> ( { z } X. ( F ` z ) ) = { <. x , y >. | ( x e. { z } /\ y e. ( F ` z ) ) } )
8	7	iuneq2i 4539	. . 3 |- U_ z e. A ( { z } X. ( F ` z ) ) = U_ z e. A { <. x , y >. | ( x e. { z } /\ y e. ( F ` z ) ) }
9		iunopab 5012	. . 3 |- U_ z e. A { <. x , y >. | ( x e. { z } /\ y e. ( F ` z ) ) } = { <. x , y >. | E. z e. A ( x e. { z } /\ y e. ( F ` z ) ) }
10		velsn 4193	. . . . . . . 8 |- ( x e. { z } <-> x = z )
11		equcom 1945	. . . . . . . 8 |- ( x = z <-> z = x )
12	10, 11	bitri 264	. . . . . . 7 |- ( x e. { z } <-> z = x )
13	12	anbi1i 731	. . . . . 6 |- ( ( x e. { z } /\ y e. ( F ` z ) ) <-> ( z = x /\ y e. ( F ` z ) ) )
14	13	rexbii 3041	. . . . 5 |- ( E. z e. A ( x e. { z } /\ y e. ( F ` z ) ) <-> E. z e. A ( z = x /\ y e. ( F ` z ) ) )
15		fveq2 6191	. . . . . . 7 |- ( z = x -> ( F ` z ) = ( F ` x ) )
16	15	eleq2d 2687	. . . . . 6 |- ( z = x -> ( y e. ( F ` z ) <-> y e. ( F ` x ) ) )
17	16	ceqsrexbv 3337	. . . . 5 |- ( E. z e. A ( z = x /\ y e. ( F ` z ) ) <-> ( x e. A /\ y e. ( F ` x ) ) )
18	14, 17	bitri 264	. . . 4 |- ( E. z e. A ( x e. { z } /\ y e. ( F ` z ) ) <-> ( x e. A /\ y e. ( F ` x ) ) )
19	18	opabbii 4717	. . 3 |- { <. x , y >. | E. z e. A ( x e. { z } /\ y e. ( F ` z ) ) } = { <. x , y >. | ( x e. A /\ y e. ( F ` x ) ) }
20	8, 9, 19	3eqtri 2648	. 2 |- U_ z e. A ( { z } X. ( F ` z ) ) = { <. x , y >. | ( x e. A /\ y e. ( F ` x ) ) }
21	1, 5, 20	3eqtri 2648	1 |- T = { <. x , y >. | ( x e. A /\ y e. ( F ` x ) ) }

Syntax hints:   /\ wa 384   = wceq 1483   e. wcel 1990   E.wrex 2913   {csn 4177   U_ciun 4520   {copab 4712   X. cxp 5112   ` cfv 5888

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-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  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-iun 4522  df-br 4654  df-opab 4713  df-xp 5120  df-iota 5851  df-fv 5896

This theorem is referenced by:  marypha2lem3  8343  marypha2lem4  8344  eulerpartlemgu  30439