Theorem fvmptss2 5561

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 5500	. 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 5227	. . . . 5 |- Fun F
4		funrel 5205	. . . . 5 |- ( Fun F -> Rel F )
5	3, 4	ax-mp 5	. . . 4 |- Rel F
6	5	brrelex1i 4647	. . 3 |- ( D F y -> D e. _V )
7		nfcv 2308	. . . 4 |- F/_ x D
8		nfmpt1 4075	. . . . . . 7 |- F/_ x ( x e. A |-> B )
9	2, 8	nfcxfr 2305	. . . . . 6 |- F/_ x F
10		nfcv 2308	. . . . . 6 |- F/_ x y
11	7, 9, 10	nfbr 4028	. . . . 5 |- F/ x D F y
12		nfv 1516	. . . . 5 |- F/ x y C_ C
13	11, 12	nfim 1560	. . . 4 |- F/ x ( D F y -> y C_ C )
14		breq1 3985	. . . . 5 |- ( x = D -> ( x F y <-> D F y ) )
15		fvmptss2.1	. . . . . 6 |- ( x = D -> B = C )
16	15	sseq2d 3172	. . . . 5 |- ( x = D -> ( y C_ B <-> y C_ C ) )
17	14, 16	imbi12d 233	. . . 4 |- ( x = D -> ( ( x F y -> y C_ B ) <-> ( D F y -> y C_ C ) ) )
18		df-br 3983	. . . . 5 |- ( x F y <-> <. x , y >. e. F )
19		opabid 4235	. . . . . . 7 |- ( <. x , y >. e. { <. x , y >. | ( x e. A /\ y = B ) } <-> ( x e. A /\ y = B ) )
20		eqimss 3196	. . . . . . . 8 |- ( y = B -> y C_ B )
21	20	adantl 275	. . . . . . 7 |- ( ( x e. A /\ y = B ) -> y C_ B )
22	19, 21	sylbi 120	. . . . . 6 |- ( <. x , y >. e. { <. x , y >. | ( x e. A /\ y = B ) } -> y C_ B )
23		df-mpt 4045	. . . . . . 7 |- ( x e. A |-> B ) = { <. x , y >. | ( x e. A /\ y = B ) }
24	2, 23	eqtri 2186	. . . . . 6 |- F = { <. x , y >. | ( x e. A /\ y = B ) }
25	22, 24	eleq2s 2261	. . . . 5 |- ( <. x , y >. e. F -> y C_ B )
26	18, 25	sylbi 120	. . . 4 |- ( x F y -> y C_ B )
27	7, 13, 17, 26	vtoclgf 2784	. . 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 1439	1 |- ( F ` D ) C_ C

Syntax hints:   -> wi 4   /\ wa 103   = wceq 1343   e. wcel 2136   _Vcvv 2726   C_ wss 3116   <.cop 3579   class class class wbr 3982   {copab 4042   |-> cmpt 4043   Rel wrel 4609   Fun wfun 5182   ` cfv 5188

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-pow 4153  ax-pr 4187

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-rex 2450  df-v 2728  df-un 3120  df-in 3122  df-ss 3129  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-br 3983  df-opab 4044  df-mpt 4045  df-id 4271  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-iota 5153  df-fun 5190  df-fv 5196

This theorem is referenced by:  mptfvex  5571