Theorem fvmptss2 5392

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 5332	. 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 5066	. . . . 5 |- Fun F
4		funrel 5045	. . . . 5 |- ( Fun F -> Rel F )
5	3, 4	ax-mp 7	. . . 4 |- Rel F
6	5	brrelex1i 4494	. . 3 |- ( D F y -> D e. _V )
7		nfcv 2229	. . . 4 |- F/_ x D
8		nfmpt1 3937	. . . . . . 7 |- F/_ x ( x e. A |-> B )
9	2, 8	nfcxfr 2226	. . . . . 6 |- F/_ x F
10		nfcv 2229	. . . . . 6 |- F/_ x y
11	7, 9, 10	nfbr 3895	. . . . 5 |- F/ x D F y
12		nfv 1467	. . . . 5 |- F/ x y C_ C
13	11, 12	nfim 1510	. . . 4 |- F/ x ( D F y -> y C_ C )
14		breq1 3854	. . . . 5 |- ( x = D -> ( x F y <-> D F y ) )
15		fvmptss2.1	. . . . . 6 |- ( x = D -> B = C )
16	15	sseq2d 3055	. . . . 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 3852	. . . . 5 |- ( x F y <-> <. x , y >. e. F )
19		opabid 4093	. . . . . . 7 |- ( <. x , y >. e. { <. x , y >. | ( x e. A /\ y = B ) } <-> ( x e. A /\ y = B ) )
20		eqimss 3079	. . . . . . . 8 |- ( y = B -> y C_ B )
21	20	adantl 272	. . . . . . 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 3907	. . . . . . 7 |- ( x e. A |-> B ) = { <. x , y >. | ( x e. A /\ y = B ) }
24	2, 23	eqtri 2109	. . . . . 6 |- F = { <. x , y >. | ( x e. A /\ y = B ) }
25	22, 24	eleq2s 2183	. . . . 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 2678	. . 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 1386	1 |- ( F ` D ) C_ C

Syntax hints:   -> wi 4   /\ wa 103   = wceq 1290   e. wcel 1439   _Vcvv 2620   C_ wss 3000   <.cop 3453   class class class wbr 3851   {copab 3904   |-> cmpt 3905   Rel wrel 4457   Fun wfun 5022   ` cfv 5028

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-14 1451  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071  ax-sep 3963  ax-pow 4015  ax-pr 4045

This theorem depends on definitions:  df-bi 116  df-3an 927  df-tru 1293  df-nf 1396  df-sb 1694  df-eu 1952  df-mo 1953  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-ral 2365  df-rex 2366  df-v 2622  df-un 3004  df-in 3006  df-ss 3013  df-pw 3435  df-sn 3456  df-pr 3457  df-op 3459  df-uni 3660  df-br 3852  df-opab 3906  df-mpt 3907  df-id 4129  df-xp 4458  df-rel 4459  df-cnv 4460  df-co 4461  df-iota 4993  df-fun 5030  df-fv 5036

This theorem is referenced by:  mptfvex  5401