Theorem relmptopab 6281

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

Dummy variable r is distinct from all other variables.

Step	Hyp	Ref	Expression
1		relmptopab.1	. . . . . . . 8 |- F = ( x e. A |-> { <. y , z >. | ph } )
2	1	funmpt2 5411	. . . . . . 7 |- Fun F
3		funrel 5389	. . . . . . 7 |- ( Fun F -> Rel F )
4	2, 3	ax-mp 5	. . . . . 6 |- Rel F
5		relelfvdm 5722	. . . . . 6 |- ( ( Rel F /\ r e. ( F ` B ) ) -> B e. dom F )
6	4, 5	mpan 428	. . . . 5 |- ( r e. ( F ` B ) -> B e. dom F )
7		relopab 4901	. . . . . . 7 |- Rel { <. y , z >. | ph }
8		df-rel 4776	. . . . . . 7 |- ( Rel { <. y , z >. | ph } <-> { <. y , z >. | ph } C_ ( _V X. _V ) )
9	7, 8	mpbi 145	. . . . . 6 |- { <. y , z >. | ph } C_ ( _V X. _V )
10	9	rgenw 2605	. . . . 5 |- A. x e. A { <. y , z >. | ph } C_ ( _V X. _V )
11	1	fvmptssdm 5784	. . . . 5 |- ( ( B e. dom F /\ A. x e. A { <. y , z >. | ph } C_ ( _V X. _V ) ) -> ( F ` B ) C_ ( _V X. _V ) )
12	6, 10, 11	sylancl 417	. . . 4 |- ( r e. ( F ` B ) -> ( F ` B ) C_ ( _V X. _V ) )
13		ssel 3242	. . . 4 |- ( ( F ` B ) C_ ( _V X. _V ) -> ( r e. ( F ` B ) -> r e. ( _V X. _V ) ) )
14	12, 13	mpcom 36	. . 3 |- ( r e. ( F ` B ) -> r e. ( _V X. _V ) )
15	14	ssriv 3252	. 2 |- ( F ` B ) C_ ( _V X. _V )
16		df-rel 4776	. 2 |- ( Rel ( F ` B ) <-> ( F ` B ) C_ ( _V X. _V ) )
17	15, 16	mpbir 146	1 |- Rel ( F ` B )

Syntax hints:   = wceq 1402   e. wcel 2209   A.wral 2528   _Vcvv 2821   C_ wss 3220   {copab 4186   |-> cmpt 4187   X. cxp 4767   dom cdm 4769   Rel wrel 4774   Fun wfun 5366   ` cfv 5372

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4244  ax-pow 4306  ax-pr 4341

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-un 3224  df-in 3226  df-ss 3233  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-br 4126  df-opab 4188  df-mpt 4189  df-id 4433  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-res 4781  df-ima 4782  df-iota 5332  df-fun 5374  df-fv 5380

This theorem is referenced by:  reldvdsr  14371  lmrel  15215  relwlk  16502  reltrls  16537  releupth  16599