Theorem frrlem5d 31787

Description: Lemma for founded recursion. The domain of the union of a subset of B is a subset of A. (Contributed by Paul Chapman, 29-Apr-2012.)

Hypotheses

Ref	Expression
frrlem5.1	|- R Fr A
frrlem5.2	|- R Se A
frrlem5.3	|- B = { f | E. x ( f Fn x /\ ( x C_ A /\ A. y e. x Pred ( R , A , y ) C_ x /\ A. y e. x ( f ` y ) = ( y G ( f |` Pred ( R , A , y ) ) ) ) ) }

Assertion

Ref	Expression
frrlem5d	|- ( C C_ B -> dom U. C C_ A )

Distinct variable groups:   A, f, x, y   f, G, x, y   R, f, x, y   x, B

Allowed substitution hints:   B( y, f)   C( x, y, f)

Proof of Theorem frrlem5d

Dummy variable g is distinct from all other variables.

Step	Hyp	Ref	Expression
1		dmuni 5334	. 2 |- dom U. C = U_ g e. C dom g
2		ssel 3597	. . . . 5 |- ( C C_ B -> ( g e. C -> g e. B ) )
3		frrlem5.3	. . . . . 6 |- B = { f | E. x ( f Fn x /\ ( x C_ A /\ A. y e. x Pred ( R , A , y ) C_ x /\ A. y e. x ( f ` y ) = ( y G ( f |` Pred ( R , A , y ) ) ) ) ) }
4	3	frrlem3 31782	. . . . 5 |- ( g e. B -> dom g C_ A )
5	2, 4	syl6 35	. . . 4 |- ( C C_ B -> ( g e. C -> dom g C_ A ) )
6	5	ralrimiv 2965	. . 3 |- ( C C_ B -> A. g e. C dom g C_ A )
7		iunss 4561	. . 3 |- ( U_ g e. C dom g C_ A <-> A. g e. C dom g C_ A )
8	6, 7	sylibr 224	. 2 |- ( C C_ B -> U_ g e. C dom g C_ A )
9	1, 8	syl5eqss 3649	1 |- ( C C_ B -> dom U. C C_ A )

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   E.wex 1704   e. wcel 1990   {cab 2608   A.wral 2912   C_ wss 3574   U.cuni 4436   U_ciun 4520   Fr wfr 5070   Se wse 5071   dom cdm 5114   |` cres 5116   Predcpred 5679   Fn wfn 5883   ` cfv 5888  (class class class)co 6650

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

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-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-iota 5851  df-fun 5890  df-fn 5891  df-fv 5896  df-ov 6653

This theorem is referenced by: (None)