Theorem fvmptss2 5653

Description: A mapping always evaluates to a subset of the substituted expression in the mapping, even if this is a proper class, or we are out of the domain. (Contributed by Mario Carneiro, 13-Feb-2015.) (Revised by Mario Carneiro, 3-Jul-2019.)

Hypotheses

Ref	Expression
fvmptss2.1	|- ( x = D -> B = C )
fvmptss2.2	|- F = ( x e. A |-> B )

Assertion

Ref	Expression
fvmptss2	|- ( F ` D ) C_ C

Distinct variable groups:   x, A   x, C   x, D

Allowed substitution hints:   B( x)   F( x)

Proof of Theorem fvmptss2

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		fvss 5589	. 2 |- ( A. y ( D F y -> y C_ C ) -> ( F ` D ) C_ C )
2		fvmptss2.2	. . . . . 6 |- F = ( x e. A |-> B )
3	2	funmpt2 5309	. . . . 5 |- Fun F
4		funrel 5287	. . . . 5 |- ( Fun F -> Rel F )
5	3, 4	ax-mp 5	. . . 4 |- Rel F
6	5	brrelex1i 4717	. . 3 |- ( D F y -> D e. _V )
7		nfcv 2347	. . . 4 |- F/_ x D
8		nfmpt1 4136	. . . . . . 7 |- F/_ x ( x e. A |-> B )
9	2, 8	nfcxfr 2344	. . . . . 6 |- F/_ x F
10		nfcv 2347	. . . . . 6 |- F/_ x y
11	7, 9, 10	nfbr 4089	. . . . 5 |- F/ x D F y
12		nfv 1550	. . . . 5 |- F/ x y C_ C
13	11, 12	nfim 1594	. . . 4 |- F/ x ( D F y -> y C_ C )
14		breq1 4046	. . . . 5 |- ( x = D -> ( x F y <-> D F y ) )
15		fvmptss2.1	. . . . . 6 |- ( x = D -> B = C )
16	15	sseq2d 3222	. . . . 5 |- ( x = D -> ( y C_ B <-> y C_ C ) )
17	14, 16	imbi12d 234	. . . 4 |- ( x = D -> ( ( x F y -> y C_ B ) <-> ( D F y -> y C_ C ) ) )
18		df-br 4044	. . . . 5 |- ( x F y <-> <. x , y >. e. F )
19		opabid 4301	. . . . . . 7 |- ( <. x , y >. e. { <. x , y >. | ( x e. A /\ y = B ) } <-> ( x e. A /\ y = B ) )
20		eqimss 3246	. . . . . . . 8 |- ( y = B -> y C_ B )
21	20	adantl 277	. . . . . . 7 |- ( ( x e. A /\ y = B ) -> y C_ B )
22	19, 21	sylbi 121	. . . . . 6 |- ( <. x , y >. e. { <. x , y >. | ( x e. A /\ y = B ) } -> y C_ B )
23		df-mpt 4106	. . . . . . 7 |- ( x e. A |-> B ) = { <. x , y >. | ( x e. A /\ y = B ) }
24	2, 23	eqtri 2225	. . . . . 6 |- F = { <. x , y >. | ( x e. A /\ y = B ) }
25	22, 24	eleq2s 2299	. . . . 5 |- ( <. x , y >. e. F -> y C_ B )
26	18, 25	sylbi 121	. . . 4 |- ( x F y -> y C_ B )
27	7, 13, 17, 26	vtoclgf 2830	. . 3 |- ( D e. _V -> ( D F y -> y C_ C ) )
28	6, 27	mpcom 36	. 2 |- ( D F y -> y C_ C )
29	1, 28	mpg 1473	1 |- ( F ` D ) C_ C

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1372   e. wcel 2175   _Vcvv 2771   C_ wss 3165   <.cop 3635   class class class wbr 4043   {copab 4103   |-> cmpt 4104   Rel wrel 4679   Fun wfun 5264   ` cfv 5270

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 710  ax-5 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-14 2178  ax-ext 2186  ax-sep 4161  ax-pow 4217  ax-pr 4252

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1375  df-nf 1483  df-sb 1785  df-eu 2056  df-mo 2057  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-ral 2488  df-rex 2489  df-v 2773  df-un 3169  df-in 3171  df-ss 3178  df-pw 3617  df-sn 3638  df-pr 3639  df-op 3641  df-uni 3850  df-br 4044  df-opab 4105  df-mpt 4106  df-id 4339  df-xp 4680  df-rel 4681  df-cnv 4682  df-co 4683  df-iota 5231  df-fun 5272  df-fv 5278

This theorem is referenced by:  mptfvex  5664