Theorem fvmptss2 5571

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 5510	. 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 5237	. . . . 5 |- Fun F
4		funrel 5215	. . . . 5 |- ( Fun F -> Rel F )
5	3, 4	ax-mp 5	. . . 4 |- Rel F
6	5	brrelex1i 4654	. . 3 |- ( D F y -> D e. _V )
7		nfcv 2312	. . . 4 |- F/_ x D
8		nfmpt1 4082	. . . . . . 7 |- F/_ x ( x e. A |-> B )
9	2, 8	nfcxfr 2309	. . . . . 6 |- F/_ x F
10		nfcv 2312	. . . . . 6 |- F/_ x y
11	7, 9, 10	nfbr 4035	. . . . 5 |- F/ x D F y
12		nfv 1521	. . . . 5 |- F/ x y C_ C
13	11, 12	nfim 1565	. . . 4 |- F/ x ( D F y -> y C_ C )
14		breq1 3992	. . . . 5 |- ( x = D -> ( x F y <-> D F y ) )
15		fvmptss2.1	. . . . . 6 |- ( x = D -> B = C )
16	15	sseq2d 3177	. . . . 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 3990	. . . . 5 |- ( x F y <-> <. x , y >. e. F )
19		opabid 4242	. . . . . . 7 |- ( <. x , y >. e. { <. x , y >. | ( x e. A /\ y = B ) } <-> ( x e. A /\ y = B ) )
20		eqimss 3201	. . . . . . . 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 4052	. . . . . . 7 |- ( x e. A |-> B ) = { <. x , y >. | ( x e. A /\ y = B ) }
24	2, 23	eqtri 2191	. . . . . 6 |- F = { <. x , y >. | ( x e. A /\ y = B ) }
25	22, 24	eleq2s 2265	. . . . 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 2788	. . 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 1444	1 |- ( F ` D ) C_ C

Syntax hints:   -> wi 4   /\ wa 103   = wceq 1348   e. wcel 2141   _Vcvv 2730   C_ wss 3121   <.cop 3586   class class class wbr 3989   {copab 4049   |-> cmpt 4050   Rel wrel 4616   Fun wfun 5192   ` cfv 5198

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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-v 2732  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-br 3990  df-opab 4051  df-mpt 4052  df-id 4278  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-iota 5160  df-fun 5200  df-fv 5206

This theorem is referenced by:  mptfvex  5581