Theorem marypha2lem3 8343

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
marypha2lem3	|- ( ( F Fn A /\ G Fn A ) -> ( G C_ T <-> A. x e. A ( G ` x ) e. ( F ` x ) ) )

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

Allowed substitution hint:   T( x)

Proof of Theorem marypha2lem3

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		dffn5 6241	. . . . . . 7 |- ( G Fn A <-> G = ( x e. A |-> ( G ` x ) ) )
2	1	biimpi 206	. . . . . 6 |- ( G Fn A -> G = ( x e. A |-> ( G ` x ) ) )
3	2	adantl 482	. . . . 5 |- ( ( F Fn A /\ G Fn A ) -> G = ( x e. A |-> ( G ` x ) ) )
4		df-mpt 4730	. . . . 5 |- ( x e. A |-> ( G ` x ) ) = { <. x , y >. | ( x e. A /\ y = ( G ` x ) ) }
5	3, 4	syl6eq 2672	. . . 4 |- ( ( F Fn A /\ G Fn A ) -> G = { <. x , y >. | ( x e. A /\ y = ( G ` x ) ) } )
6		marypha2lem.t	. . . . . 6 |- T = U_ x e. A ( { x } X. ( F ` x ) )
7	6	marypha2lem2 8342	. . . . 5 |- T = { <. x , y >. | ( x e. A /\ y e. ( F ` x ) ) }
8	7	a1i 11	. . . 4 |- ( ( F Fn A /\ G Fn A ) -> T = { <. x , y >. | ( x e. A /\ y e. ( F ` x ) ) } )
9	5, 8	sseq12d 3634	. . 3 |- ( ( F Fn A /\ G Fn A ) -> ( G C_ T <-> { <. x , y >. | ( x e. A /\ y = ( G ` x ) ) } C_ { <. x , y >. | ( x e. A /\ y e. ( F ` x ) ) } ) )
10		ssopab2b 5002	. . 3 |- ( { <. x , y >. | ( x e. A /\ y = ( G ` x ) ) } C_ { <. x , y >. | ( x e. A /\ y e. ( F ` x ) ) } <-> A. x A. y ( ( x e. A /\ y = ( G ` x ) ) -> ( x e. A /\ y e. ( F ` x ) ) ) )
11	9, 10	syl6bb 276	. 2 |- ( ( F Fn A /\ G Fn A ) -> ( G C_ T <-> A. x A. y ( ( x e. A /\ y = ( G ` x ) ) -> ( x e. A /\ y e. ( F ` x ) ) ) ) )
12		19.21v 1868	. . . . 5 |- ( A. y ( x e. A -> ( y = ( G ` x ) -> y e. ( F ` x ) ) ) <-> ( x e. A -> A. y ( y = ( G ` x ) -> y e. ( F ` x ) ) ) )
13		imdistan 725	. . . . . 6 |- ( ( x e. A -> ( y = ( G ` x ) -> y e. ( F ` x ) ) ) <-> ( ( x e. A /\ y = ( G ` x ) ) -> ( x e. A /\ y e. ( F ` x ) ) ) )
14	13	albii 1747	. . . . 5 |- ( A. y ( x e. A -> ( y = ( G ` x ) -> y e. ( F ` x ) ) ) <-> A. y ( ( x e. A /\ y = ( G ` x ) ) -> ( x e. A /\ y e. ( F ` x ) ) ) )
15		fvex 6201	. . . . . . 7 |- ( G ` x ) e. _V
16		eleq1 2689	. . . . . . 7 |- ( y = ( G ` x ) -> ( y e. ( F ` x ) <-> ( G ` x ) e. ( F ` x ) ) )
17	15, 16	ceqsalv 3233	. . . . . 6 |- ( A. y ( y = ( G ` x ) -> y e. ( F ` x ) ) <-> ( G ` x ) e. ( F ` x ) )
18	17	imbi2i 326	. . . . 5 |- ( ( x e. A -> A. y ( y = ( G ` x ) -> y e. ( F ` x ) ) ) <-> ( x e. A -> ( G ` x ) e. ( F ` x ) ) )
19	12, 14, 18	3bitr3i 290	. . . 4 |- ( A. y ( ( x e. A /\ y = ( G ` x ) ) -> ( x e. A /\ y e. ( F ` x ) ) ) <-> ( x e. A -> ( G ` x ) e. ( F ` x ) ) )
20	19	albii 1747	. . 3 |- ( A. x A. y ( ( x e. A /\ y = ( G ` x ) ) -> ( x e. A /\ y e. ( F ` x ) ) ) <-> A. x ( x e. A -> ( G ` x ) e. ( F ` x ) ) )
21		df-ral 2917	. . 3 |- ( A. x e. A ( G ` x ) e. ( F ` x ) <-> A. x ( x e. A -> ( G ` x ) e. ( F ` x ) ) )
22	20, 21	bitr4i 267	. 2 |- ( A. x A. y ( ( x e. A /\ y = ( G ` x ) ) -> ( x e. A /\ y e. ( F ` x ) ) ) <-> A. x e. A ( G ` x ) e. ( F ` x ) )
23	11, 22	syl6bb 276	1 |- ( ( F Fn A /\ G Fn A ) -> ( G C_ T <-> A. x e. A ( G ` x ) e. ( F ` x ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   A.wal 1481   = wceq 1483   e. wcel 1990   A.wral 2912   C_ wss 3574   {csn 4177   U_ciun 4520   {copab 4712   |-> cmpt 4729   X. cxp 5112   Fn wfn 5883   ` 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-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-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-iota 5851  df-fun 5890  df-fn 5891  df-fv 5896

This theorem is referenced by:  marypha2  8345