Theorem fvmptss2 5496

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 5435	. 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 5162	. . . . 5 |- Fun F
4		funrel 5140	. . . . 5 |- ( Fun F -> Rel F )
5	3, 4	ax-mp 5	. . . 4 |- Rel F
6	5	brrelex1i 4582	. . 3 |- ( D F y -> D e. _V )
7		nfcv 2281	. . . 4 |- F/_ x D
8		nfmpt1 4021	. . . . . . 7 |- F/_ x ( x e. A |-> B )
9	2, 8	nfcxfr 2278	. . . . . 6 |- F/_ x F
10		nfcv 2281	. . . . . 6 |- F/_ x y
11	7, 9, 10	nfbr 3974	. . . . 5 |- F/ x D F y
12		nfv 1508	. . . . 5 |- F/ x y C_ C
13	11, 12	nfim 1551	. . . 4 |- F/ x ( D F y -> y C_ C )
14		breq1 3932	. . . . 5 |- ( x = D -> ( x F y <-> D F y ) )
15		fvmptss2.1	. . . . . 6 |- ( x = D -> B = C )
16	15	sseq2d 3127	. . . . 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 3930	. . . . 5 |- ( x F y <-> <. x , y >. e. F )
19		opabid 4179	. . . . . . 7 |- ( <. x , y >. e. { <. x , y >. | ( x e. A /\ y = B ) } <-> ( x e. A /\ y = B ) )
20		eqimss 3151	. . . . . . . 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 3991	. . . . . . 7 |- ( x e. A |-> B ) = { <. x , y >. | ( x e. A /\ y = B ) }
24	2, 23	eqtri 2160	. . . . . 6 |- F = { <. x , y >. | ( x e. A /\ y = B ) }
25	22, 24	eleq2s 2234	. . . . 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 2744	. . 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 1427	1 |- ( F ` D ) C_ C

Syntax hints:   -> wi 4   /\ wa 103   = wceq 1331   e. wcel 1480   _Vcvv 2686   C_ wss 3071   <.cop 3530   class class class wbr 3929   {copab 3988   |-> cmpt 3989   Rel wrel 4544   Fun wfun 5117   ` cfv 5123

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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-pow 4098  ax-pr 4131

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422  df-v 2688  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-br 3930  df-opab 3990  df-mpt 3991  df-id 4215  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-iota 5088  df-fun 5125  df-fv 5131

This theorem is referenced by:  mptfvex  5506