Theorem relmptopab 6883

Description: Any function to sets of ordered pairs produces a relation on function value unconditionally. (Contributed by Mario Carneiro, 7-Aug-2014.) (Proof shortened by Mario Carneiro, 24-Dec-2016.)

Hypothesis

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

Assertion

Ref	Expression
relmptopab	|- Rel ( F ` B )

Distinct variable group:   x, A

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

Proof of Theorem relmptopab

Step	Hyp	Ref	Expression
1		relmptopab.1	. . . 4 |- F = ( x e. A |-> { <. y , z >. | ph } )
2	1	fvmptss 6292	. . 3 |- ( A. x e. A { <. y , z >. | ph } C_ ( _V X. _V ) -> ( F ` B ) C_ ( _V X. _V ) )
3		relopab 5247	. . . . 5 |- Rel { <. y , z >. | ph }
4		df-rel 5121	. . . . 5 |- ( Rel { <. y , z >. | ph } <-> { <. y , z >. | ph } C_ ( _V X. _V ) )
5	3, 4	mpbi 220	. . . 4 |- { <. y , z >. | ph } C_ ( _V X. _V )
6	5	a1i 11	. . 3 |- ( x e. A -> { <. y , z >. | ph } C_ ( _V X. _V ) )
7	2, 6	mprg 2926	. 2 |- ( F ` B ) C_ ( _V X. _V )
8		df-rel 5121	. 2 |- ( Rel ( F ` B ) <-> ( F ` B ) C_ ( _V X. _V ) )
9	7, 8	mpbir 221	1 |- Rel ( F ` B )

Syntax hints:   = wceq 1483   e. wcel 1990   _Vcvv 3200   C_ wss 3574   {copab 4712   |-> cmpt 4729   X. cxp 5112   Rel wrel 5119   ` 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-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-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

This theorem is referenced by:  reldvdsr  18644  lmrel  21034  phtpcrel  22792  ulmrel  24132  ercgrg  25412  relwlk  26521  reltrls  26591  relpths  26616  releupth  27059