Theorem frrlem5b 31785

Description: Lemma for founded recursion. The union of a subclass of B is a relationship. (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
frrlem5b	|- ( C C_ B -> Rel U. C )

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 frrlem5b

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ssel 3597	. . . 4 |- ( C C_ B -> ( z e. C -> z e. B ) )
2		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 ) ) ) ) ) }
3	2	frrlem2 31781	. . . . 5 |- ( z e. B -> Fun z )
4		funrel 5905	. . . . 5 |- ( Fun z -> Rel z )
5	3, 4	syl 17	. . . 4 |- ( z e. B -> Rel z )
6	1, 5	syl6 35	. . 3 |- ( C C_ B -> ( z e. C -> Rel z ) )
7	6	ralrimiv 2965	. 2 |- ( C C_ B -> A. z e. C Rel z )
8		reluni 5241	. 2 |- ( Rel U. C <-> A. z e. C Rel z )
9	7, 8	sylibr 224	1 |- ( C C_ B -> Rel U. C )

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   Fr wfr 5070   Se wse 5071   |` cres 5116   Rel wrel 5119   Predcpred 5679   Fun wfun 5882   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:  frrlem5c  31786