Theorem fvmptss2 5268

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 5209	. 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 4959	. . . . 5 |- Fun F
4		funrel 4939	. . . . 5 |- ( Fun F -> Rel F )
5	3, 4	ax-mp 7	. . . 4 |- Rel F
6	5	brrelexi 4402	. . 3 |- ( D F y -> D e. _V )
7		nfcv 2219	. . . 4 |- F/_ x D
8		nfmpt1 3871	. . . . . . 7 |- F/_ x ( x e. A |-> B )
9	2, 8	nfcxfr 2216	. . . . . 6 |- F/_ x F
10		nfcv 2219	. . . . . 6 |- F/_ x y
11	7, 9, 10	nfbr 3829	. . . . 5 |- F/ x D F y
12		nfv 1461	. . . . 5 |- F/ x y C_ C
13	11, 12	nfim 1504	. . . 4 |- F/ x ( D F y -> y C_ C )
14		breq1 3788	. . . . 5 |- ( x = D -> ( x F y <-> D F y ) )
15		fvmptss2.1	. . . . . 6 |- ( x = D -> B = C )
16	15	sseq2d 3027	. . . . 5 |- ( x = D -> ( y C_ B <-> y C_ C ) )
17	14, 16	imbi12d 232	. . . 4 |- ( x = D -> ( ( x F y -> y C_ B ) <-> ( D F y -> y C_ C ) ) )
18		df-br 3786	. . . . 5 |- ( x F y <-> <. x , y >. e. F )
19		opabid 4012	. . . . . . 7 |- ( <. x , y >. e. { <. x , y >. | ( x e. A /\ y = B ) } <-> ( x e. A /\ y = B ) )
20		eqimss 3051	. . . . . . . 8 |- ( y = B -> y C_ B )
21	20	adantl 271	. . . . . . 7 |- ( ( x e. A /\ y = B ) -> y C_ B )
22	19, 21	sylbi 119	. . . . . 6 |- ( <. x , y >. e. { <. x , y >. | ( x e. A /\ y = B ) } -> y C_ B )
23		df-mpt 3841	. . . . . . 7 |- ( x e. A |-> B ) = { <. x , y >. | ( x e. A /\ y = B ) }
24	2, 23	eqtri 2101	. . . . . 6 |- F = { <. x , y >. | ( x e. A /\ y = B ) }
25	22, 24	eleq2s 2173	. . . . 5 |- ( <. x , y >. e. F -> y C_ B )
26	18, 25	sylbi 119	. . . 4 |- ( x F y -> y C_ B )
27	7, 13, 17, 26	vtoclgf 2657	. . 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 1380	1 |- ( F ` D ) C_ C

Syntax hints:   -> wi 4   /\ wa 102   = wceq 1284   e. wcel 1433   _Vcvv 2601   C_ wss 2973   <.cop 3401   class class class wbr 3785   {copab 3838   |-> cmpt 3839   Rel wrel 4368   Fun wfun 4916   ` cfv 4922

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-sep 3896  ax-pow 3948  ax-pr 3964

This theorem depends on definitions:  df-bi 115  df-3an 921  df-tru 1287  df-nf 1390  df-sb 1686  df-eu 1944  df-mo 1945  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ral 2353  df-rex 2354  df-v 2603  df-un 2977  df-in 2979  df-ss 2986  df-pw 3384  df-sn 3404  df-pr 3405  df-op 3407  df-uni 3602  df-br 3786  df-opab 3840  df-mpt 3841  df-id 4048  df-xp 4369  df-rel 4370  df-cnv 4371  df-co 4372  df-iota 4887  df-fun 4924  df-fv 4930

This theorem is referenced by:  mptfvex  5277