Theorem marypha2lem1 8341

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
marypha2lem1	|- T C_ ( A X. U. ran F )

Distinct variable groups:   x, A   x, F

Allowed substitution hint:   T( x)

Proof of Theorem marypha2lem1

Step	Hyp	Ref	Expression
1		marypha2lem.t	. 2 |- T = U_ x e. A ( { x } X. ( F ` x ) )
2		iunss 4561	. . 3 |- ( U_ x e. A ( { x } X. ( F ` x ) ) C_ ( A X. U. ran F ) <-> A. x e. A ( { x } X. ( F ` x ) ) C_ ( A X. U. ran F ) )
3		snssi 4339	. . . 4 |- ( x e. A -> { x } C_ A )
4		fvssunirn 6217	. . . 4 |- ( F ` x ) C_ U. ran F
5		xpss12 5225	. . . 4 |- ( ( { x } C_ A /\ ( F ` x ) C_ U. ran F ) -> ( { x } X. ( F ` x ) ) C_ ( A X. U. ran F ) )
6	3, 4, 5	sylancl 694	. . 3 |- ( x e. A -> ( { x } X. ( F ` x ) ) C_ ( A X. U. ran F ) )
7	2, 6	mprgbir 2927	. 2 |- U_ x e. A ( { x } X. ( F ` x ) ) C_ ( A X. U. ran F )
8	1, 7	eqsstri 3635	1 |- T C_ ( A X. U. ran F )

Syntax hints:   = wceq 1483   e. wcel 1990   C_ wss 3574   {csn 4177   U.cuni 4436   U_ciun 4520   X. cxp 5112   ran crn 5115   ` 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-ne 2795  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-xp 5120  df-cnv 5122  df-dm 5124  df-rn 5125  df-iota 5851  df-fv 5896

This theorem is referenced by:  marypha2  8345