Theorem relmpt2opab 7259

Description: Any function to sets of ordered pairs produces a relation on function value unconditionally. (Contributed by Mario Carneiro, 9-Feb-2015.)

Hypothesis

Ref	Expression
relmpt2opab.1	|- F = ( x e. A , y e. B |-> { <. z , w >. | ph } )

Assertion

Ref	Expression
relmpt2opab	|- Rel ( C F D )

Distinct variable groups:   x, w, y, z   y, B   x, A, y

Allowed substitution hints:   ph( x, y, z, w)   A( z, w)   B( x, z, w)   C( x, y, z, w)   D( x, y, z, w)   F( x, y, z, w)

Proof of Theorem relmpt2opab

Step	Hyp	Ref	Expression
1		relopab 5247	. . . . 5 |- Rel { <. z , w >. | ph }
2		df-rel 5121	. . . . 5 |- ( Rel { <. z , w >. | ph } <-> { <. z , w >. | ph } C_ ( _V X. _V ) )
3	1, 2	mpbi 220	. . . 4 |- { <. z , w >. | ph } C_ ( _V X. _V )
4	3	rgen2w 2925	. . 3 |- A. x e. A A. y e. B { <. z , w >. | ph } C_ ( _V X. _V )
5		relmpt2opab.1	. . . 4 |- F = ( x e. A , y e. B |-> { <. z , w >. | ph } )
6	5	ovmptss 7258	. . 3 |- ( A. x e. A A. y e. B { <. z , w >. | ph } C_ ( _V X. _V ) -> ( C F D ) C_ ( _V X. _V ) )
7	4, 6	ax-mp 5	. 2 |- ( C F D ) C_ ( _V X. _V )
8		df-rel 5121	. 2 |- ( Rel ( C F D ) <-> ( C F D ) C_ ( _V X. _V ) )
9	7, 8	mpbir 221	1 |- Rel ( C F D )

Syntax hints:   = wceq 1483   A.wral 2912   _Vcvv 3200   C_ wss 3574   {copab 4712   X. cxp 5112   Rel wrel 5119  (class class class)co 6650   |-> cmpt2 6652

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  ax-un 6949

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-csb 3534  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-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-1st 7168  df-2nd 7169

This theorem is referenced by:  brovmpt2ex  7349  relfunc  16522  releqg  17641