Theorem marypha2lem4 8344

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
marypha2lem4	|- ( ( F Fn A /\ X C_ A ) -> ( T " X ) = U. ( F " X ) )

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

Allowed substitution hint:   T( x)

Proof of Theorem marypha2lem4

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		marypha2lem.t	. . . . . 6 |- T = U_ x e. A ( { x } X. ( F ` x ) )
2	1	marypha2lem2 8342	. . . . 5 |- T = { <. x , y >. | ( x e. A /\ y e. ( F ` x ) ) }
3	2	imaeq1i 5463	. . . 4 |- ( T " X ) = ( { <. x , y >. | ( x e. A /\ y e. ( F ` x ) ) } " X )
4		df-ima 5127	. . . 4 |- ( { <. x , y >. | ( x e. A /\ y e. ( F ` x ) ) } " X ) = ran ( { <. x , y >. | ( x e. A /\ y e. ( F ` x ) ) } |` X )
5	3, 4	eqtri 2644	. . 3 |- ( T " X ) = ran ( { <. x , y >. | ( x e. A /\ y e. ( F ` x ) ) } |` X )
6		resopab2 5448	. . . . . 6 |- ( X C_ A -> ( { <. x , y >. | ( x e. A /\ y e. ( F ` x ) ) } |` X ) = { <. x , y >. | ( x e. X /\ y e. ( F ` x ) ) } )
7	6	adantl 482	. . . . 5 |- ( ( F Fn A /\ X C_ A ) -> ( { <. x , y >. | ( x e. A /\ y e. ( F ` x ) ) } |` X ) = { <. x , y >. | ( x e. X /\ y e. ( F ` x ) ) } )
8	7	rneqd 5353	. . . 4 |- ( ( F Fn A /\ X C_ A ) -> ran ( { <. x , y >. | ( x e. A /\ y e. ( F ` x ) ) } |` X ) = ran { <. x , y >. | ( x e. X /\ y e. ( F ` x ) ) } )
9		rnopab 5370	. . . . 5 |- ran { <. x , y >. | ( x e. X /\ y e. ( F ` x ) ) } = { y | E. x ( x e. X /\ y e. ( F ` x ) ) }
10		df-rex 2918	. . . . . . . . 9 |- ( E. x e. X y e. ( F ` x ) <-> E. x ( x e. X /\ y e. ( F ` x ) ) )
11	10	bicomi 214	. . . . . . . 8 |- ( E. x ( x e. X /\ y e. ( F ` x ) ) <-> E. x e. X y e. ( F ` x ) )
12	11	abbii 2739	. . . . . . 7 |- { y | E. x ( x e. X /\ y e. ( F ` x ) ) } = { y | E. x e. X y e. ( F ` x ) }
13		df-iun 4522	. . . . . . 7 |- U_ x e. X ( F ` x ) = { y | E. x e. X y e. ( F ` x ) }
14	12, 13	eqtr4i 2647	. . . . . 6 |- { y | E. x ( x e. X /\ y e. ( F ` x ) ) } = U_ x e. X ( F ` x )
15	14	a1i 11	. . . . 5 |- ( ( F Fn A /\ X C_ A ) -> { y | E. x ( x e. X /\ y e. ( F ` x ) ) } = U_ x e. X ( F ` x ) )
16	9, 15	syl5eq 2668	. . . 4 |- ( ( F Fn A /\ X C_ A ) -> ran { <. x , y >. | ( x e. X /\ y e. ( F ` x ) ) } = U_ x e. X ( F ` x ) )
17	8, 16	eqtrd 2656	. . 3 |- ( ( F Fn A /\ X C_ A ) -> ran ( { <. x , y >. | ( x e. A /\ y e. ( F ` x ) ) } |` X ) = U_ x e. X ( F ` x ) )
18	5, 17	syl5eq 2668	. 2 |- ( ( F Fn A /\ X C_ A ) -> ( T " X ) = U_ x e. X ( F ` x ) )
19		fnfun 5988	. . . 4 |- ( F Fn A -> Fun F )
20	19	adantr 481	. . 3 |- ( ( F Fn A /\ X C_ A ) -> Fun F )
21		funiunfv 6506	. . 3 |- ( Fun F -> U_ x e. X ( F ` x ) = U. ( F " X ) )
22	20, 21	syl 17	. 2 |- ( ( F Fn A /\ X C_ A ) -> U_ x e. X ( F ` x ) = U. ( F " X ) )
23	18, 22	eqtrd 2656	1 |- ( ( F Fn A /\ X C_ A ) -> ( T " X ) = U. ( F " X ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   E.wex 1704   e. wcel 1990   {cab 2608   E.wrex 2913   C_ wss 3574   {csn 4177   U.cuni 4436   U_ciun 4520   {copab 4712   X. cxp 5112   ran crn 5115   |` cres 5116   "cima 5117   Fun wfun 5882   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-8 1992  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-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-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-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-fv 5896

This theorem is referenced by:  marypha2  8345